11
Research Article Some Remarks on Nonlinear Electrodynamics Patricio Gaete Departamento de F´ ısica and Centro Cient´ ıfico-Tecnol´ ogico de Valpara´ ıso, Universidad T´ ecnica Federico Santa Mar´ ıa, Casilla 110-V, Valpara´ ıso, Chile Correspondence should be addressed to Patricio Gaete; [email protected] Received 13 October 2015; Revised 15 December 2015; Accepted 27 December 2015 Academic Editor: Elias C. Vagenas Copyright © 2016 Patricio Gaete. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. e publication of this article was funded by SCOAP 3 . By using the gauge-invariant, but path-dependent, variables formalism, we study both massive Euler-Heisenberg-like and Euler- Heisenberg-like electrodynamics in the approximation of the strong-field limit. It is shown that massive Euler-Heisenberg-type electrodynamics displays the vacuum birefringence phenomenon. Subsequently, we calculate the lowest-order modifications to the interaction energy for both classes of electrodynamics. As a result, for the case of massive Euler-Heisenbeg-like electrodynamics (Wichmann-Kroll), unexpected features are found. We obtain a new long-range (1/ 3 -type) correction, apart from a long-range (1/ 5 -type) correction to the Coulomb potential. Furthermore, Euler-Heisenberg-like electrodynamics in the approximation of the strong-field limit (to the leading logarithmic order) displays a long-range (1/ 5 -type) correction to the Coulomb potential. Besides, for their noncommutative versions, the interaction energy is ultraviolet finite. 1. Introduction e phenomenon of vacuum polarization in quantum elec- trodynamics (QED), arising from the polarization of virtual electron-positron pairs and leading to nonlinear interactions between electromagnetic fields, remains as exciting as in the early days of QED [1–5]. An example that illustrates this is the scattering of photons by photons, which despite remarkable progress has not yet been confirmed [6–10]. Along the same line, we also recall that alternative scenarios such as Born- Infeld theory [11], millicharged particles [12], or axion-like particles [13–15] may have more significant contributions to photon-photon scattering physics. Interestingly, it should be recalled here that the physical effect of vacuum polarization appears as a modification in the interaction energy between heavy charged particles. In fact, this physical effect changes both the strength and the structural form of the interaction energy. is clearly requires the addition of correction terms in the Maxwell Lagrangian to incorporate the contributions from vacuum polarization process. Two important examples of such a class of con- tributions are the Uehling and Serber correction and the Wichmann-Kroll correction, which can be derived from the Euler-Heisenberg Lagrangian. Incidentally, as explained in [5], it is of interest to notice that the Euler-Heisenberg result extends the Euler-Kockel calculation (in the constant back- ground field limit), which contains nonlinear corrections in powers of the field strengths, whereas the Uehling and Serber result contains corrections linear in the fields (but nonlinear in the space-time dependence of the background fields). We further mention that, as in the Euler-Heisenbeg case, Born- Infeld (BI) electrodynamics also contains similar nonlinear corrections to Maxwell theory from a classical point of view, as is well known. Nevertheless, BI electrodynamics is distinguished, since BI-type effective actions arise in many different contexts in superstring theory [16, 17]. In addition to Born-Infeld theory, other types of nonlinear electrodynamics have been discussed in the literature [18–23]. In this perspective, we also point out that extensions of the Standard Model (SM) such as Lorentz invariance violating scenarios and fundamental length have become the focus of intense research activity [24–31]. is has its origin in the fact that the SM does not include a quantum theory of gravitation, so as to circumvent difficulties theoretical in the quantum gravity program. Within this context quantum field theories allowing noncommuting position operators have Hindawi Publishing Corporation Advances in High Energy Physics Volume 2016, Article ID 2463203, 10 pages http://dx.doi.org/10.1155/2016/2463203

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Research ArticleSome Remarks on Nonlinear Electrodynamics

Patricio Gaete

Departamento de Fısica and Centro Cientıfico-Tecnologico de Valparaıso Universidad Tecnica Federico Santa MarıaCasilla 110-V Valparaıso Chile

Correspondence should be addressed to Patricio Gaete patriciogaeteusmcl

Received 13 October 2015 Revised 15 December 2015 Accepted 27 December 2015

Academic Editor Elias C Vagenas

Copyright copy 2016 Patricio Gaete This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited Thepublication of this article was funded by SCOAP3

By using the gauge-invariant but path-dependent variables formalism we study both massive Euler-Heisenberg-like and Euler-Heisenberg-like electrodynamics in the approximation of the strong-field limit It is shown that massive Euler-Heisenberg-typeelectrodynamics displays the vacuum birefringence phenomenon Subsequently we calculate the lowest-order modifications to theinteraction energy for both classes of electrodynamics As a result for the case of massive Euler-Heisenbeg-like electrodynamics(Wichmann-Kroll) unexpected features are found We obtain a new long-range (11199033-type) correction apart from a long-range(1119903

5-type) correction to the Coulomb potential Furthermore Euler-Heisenberg-like electrodynamics in the approximation of thestrong-field limit (to the leading logarithmic order) displays a long-range (11199035-type) correction to the Coulomb potential Besidesfor their noncommutative versions the interaction energy is ultraviolet finite

1 Introduction

The phenomenon of vacuum polarization in quantum elec-trodynamics (QED) arising from the polarization of virtualelectron-positron pairs and leading to nonlinear interactionsbetween electromagnetic fields remains as exciting as in theearly days of QED [1ndash5] An example that illustrates this is thescattering of photons by photons which despite remarkableprogress has not yet been confirmed [6ndash10] Along the sameline we also recall that alternative scenarios such as Born-Infeld theory [11] millicharged particles [12] or axion-likeparticles [13ndash15] may have more significant contributions tophoton-photon scattering physics

Interestingly it should be recalled here that the physicaleffect of vacuum polarization appears as a modification inthe interaction energy between heavy charged particles Infact this physical effect changes both the strength and thestructural form of the interaction energyThis clearly requiresthe addition of correction terms in the Maxwell Lagrangianto incorporate the contributions from vacuum polarizationprocess Two important examples of such a class of con-tributions are the Uehling and Serber correction and theWichmann-Kroll correction which can be derived from the

Euler-Heisenberg Lagrangian Incidentally as explained in[5] it is of interest to notice that the Euler-Heisenberg resultextends the Euler-Kockel calculation (in the constant back-ground field limit) which contains nonlinear corrections inpowers of the field strengths whereas the Uehling and Serberresult contains corrections linear in the fields (but nonlinearin the space-time dependence of the background fields) Wefurther mention that as in the Euler-Heisenbeg case Born-Infeld (BI) electrodynamics also contains similar nonlinearcorrections to Maxwell theory from a classical point ofview as is well known Nevertheless BI electrodynamics isdistinguished since BI-type effective actions arise in manydifferent contexts in superstring theory [16 17] In addition toBorn-Infeld theory other types of nonlinear electrodynamicshave been discussed in the literature [18ndash23]

In this perspective we also point out that extensions ofthe StandardModel (SM) such as Lorentz invariance violatingscenarios and fundamental length have become the focusof intense research activity [24ndash31] This has its origin inthe fact that the SM does not include a quantum theory ofgravitation so as to circumvent difficulties theoretical in thequantum gravity programWithin this context quantum fieldtheories allowing noncommuting position operators have

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2016 Article ID 2463203 10 pageshttpdxdoiorg10115520162463203

2 Advances in High Energy Physics

been studied by using a star-product (Moyal product) [32ndash37] In this connection it becomes of interest in particularto recall that a novel way to formulate noncommutativequantum field theory has been proposed in [38ndash40] Thekey ingredient of this development is to introduce coherentstates of the quantum position operators [41] where amodified form of heat kernel asymptotic expansion whichdoes not suffer from short distance divergences has beenobtained We also point out that an alternative derivation ofthe coherent state approach has been implemented througha new multiplication rule which is known as Voros star-product [42] Anyhow physics turns out to be independentof the choice of the type of product [43] It is worthy tonote here that this type of noncommutativity (coherent stateapproach) leads to a smearing effect which is equivalent tothat encountered in a class of nonlocal theory In other wordsnoncommutativity is just a subclass of possible nonlocaldeformation [44 45] More recently this new approach hasbeen successfully extended to black holes physics [46] alsoin connection to holographic superconductors via AdS-CFT[47]

Inspired by these observations the purpose of this paperis to extend our previous studies [18 19] on nonlinear electro-dynamics to the case when vacuum polarization correctionsare taken into accountThepreceding studieswere done usingthe gauge-invariant but path-dependent variables formalismwhere the interaction potential energy between two staticcharges is determined by the geometrical condition of gaugeinvariance One important advantage of this approach isthat it provides a physically based alternative to the usualWilson loop approach Accordingly we will work out thestatic potential for electrodynamicswhich include apart fromthe Maxwell Lagrangian additional terms correspondingto the Uehling massive Euler-Heisenberg-like and Euler-Heisenberg electrodynamics in the approximation of thestrong-field limit (to the leading logarithmic order) and fortheir noncommutative versions Our results show a long-range 11198715-type correction to the Coulomb potential for bothmassive Euler-Heisenberg-like and Euler-Heisenberg elec-trodynamics in the approximation of the strong-field limit(to the leading logarithmic order) Interestingly enough formassive Euler-Heisenbeg-like electrodynamics (Wichmann-Kroll) we obtain a new long-range 1119871

3 correction tothe interaction energy Nevertheless for their noncom-mutative versions the static potential becomes ultravioletfinite

The organization of the paper is as follows In Section 2we reexamine Uehling electrodynamics in order to establisha framework for the computation of the static potential InSection 3 we consider Euler-Heisenberg-like (with a massterm) electrodynamics and show that it yields birefringencecomputing the interaction energy for a fermion-antifermionpair and its version in the presence of a minimal lengthIn Section 4 we repeat our analysis for Euler-Heisenbergelectrodynamics in the approximation of the strong-fieldlimit Finally in Section 5 we cast our final remarks

In our conventions the signature of the metric is(+1 minus1 minus1 minus1)

2 Brief Review on the Uehling Potential

As already expressed we now reexamine the interactionenergy for Maxwell theory with an additional term corre-sponding to the Uehling correction (Uehling electrodynam-ics)This would not only provide the setup theoretical for oursubsequent work but also fix the notation To do that we willcalculate the expectation value of the energy operator 119867 inthe physical state |Φ⟩ which we will denote by ⟨119867⟩

Φ We

start off our analysis by considering the effective Lagrangiandensity [48]

L = minus1

4119865120583] (1 minus

120572

3120587ΔM)119865

120583] (1)

where

M (119898 119909) = int

infin

41198982

1198891205911

120591 (120591 + Δ)(1 +

21198982

120591)radic1 minus

41198982

120591 (2)

with Δ equiv 120597120583120597120583 It should be noted thatM contains the effect

of vacuum polarization to first order in the fine structureconstant 120572 = 119890

2ℏ119888 and 119898 is the electron mass We may

parenthetically note here that the presence of Δ in (2) doesnot offer problems On the one hand as we will explain belowwe restrict ourselves to the static case On the other hand inorder to compute the interaction energy wewill make a seriesexpansion at leading order in 120572

Before going on two remarks are pertinent at this pointFirst the modification of Coulombrsquos law in (1) follows fromthe weak-field limit of the one-loop effective action ofquantum electrodynamics (QED) Indeed as was explainedin [48] this modification can be written as

Lwf =1

2int119860

120583(119909)Π

120583] (119909 119910)119860](119910) 119889

4119909119889

4119910 (3)

where wf denotes weak field and Π120583] is the usual order-119890

2

polarization tensor of QED As is well known in momentumspace this tensor is given by

Π120583] (119896) = (119896

2119892120583] minus 119896

120583119896])Π (119896

2) (4)

while the momentum space spectral representation of thepolarization function Π(119896

2) reads

Π(1198962) = minus

120572

31205871198962int

infin

41198982

119889119905120588 (119905)

119905

1

1198962+ 119905

(5)

with 120588(119905) = (1 + 21198982119905)radic1 minus 4119898

2119905 Next due to the tensor

structure of Π120583](119896) Lwf can then be expressed in a gauge-

invariantway that is in terms of119865120583] As a consequence of this

(3) reduces to the modification of Coulombrsquos law appearingin (1) We mention in passing that in [48] the signature of themetric is different from that used in this paper

Second it should be noted that the theory described by(1) contains higher time derivatives hence to construct theHamiltonian one must use for example the Ostrogradskymethod [49] Accordingly in the theory under considerationthe velocities have to be taken as independent canonical

Advances in High Energy Physics 3

variables Let us also mention here that in previous studies[50 51] we have shown that although theories like (1) containhigher derivatives in the electrostatic case the canonicalmomentum conjugate to velocities disappears Hence thenew Legendre transformation to construct the Hamiltonianreduces to the standard Legendre transformation It shouldhowever be emphasized here that the present paper is aimedat studying the static potential of the above theory sothat Δ can be replaced by minusnabla

2 Notice that for notationalconvenience we have maintained Δ in (1) and (2) but itshould be borne inmind that this paper essentially deals withthe static case

Now we move on to compute the canonical Hamilto-nian For this end we perform a Hamiltonian constraintanalysis The canonical momenta are found to be Π

120583=

(1 minus (1205723120587)ΔM)1198651205830 It is easy to see that Π0 vanishes we

then have the usual constraint equation which accordingto Diracrsquos theory is written as a weak (asymp) equation that isΠ

0asymp 0 Itmay be noted that the remaining nonzeromomenta

must also be written as weak equations This leads to Π119894asymp

(1minus(1205723120587)ΔM)119864119894 (with119864

119894= 119865

1198940) Accordingly the canonical

Hamiltonian119867119862is

119867119862asymp int119889

3119909 [Π

119894120597119894119860

0minus1

119894(1 minus

120572

3120587ΔM)

minus1

Π119894

+1

4119865119894119895(1 minus

120572

3120587ΔM)119865

119894119895]

(6)

which must also be written as a weak equation Next theprimary constraint Π0

asymp 0 must be satisfied for all timesAn immediate consequence of this is that using the equationof motion asymp [119885119867

119862] we obtain the secondary constraint

Γ1

equiv 120597119894Π

119894asymp 0 which must also be true for all time In

passingwe recall that we are considering the static case hencethis new constraint does not contain time derivatives It isstraightforward to check that there are no further constraintsin the theory Therefore in the case under considerationthere are two constraints which are first class Accordingto the general theory we obtain the extended Hamiltonianas an ordinary (or strong) equation by adding all the first-class constraints with arbitrary constraints We thus write119867 = 119867

119862+ int119889

3119909(119906

0(119909)Π

0(119909) + 119906

1(119909)Γ

1(119909)) where 119906

0(119909) and

1199061(119909) are arbitrary Lagrange multipliers It is also important

to observe that when this new Hamiltonian is employed theequation of motion of a dynamic variable may be written as astrong equation With the aid of (6) we find that

0(119909) =

[1198600(119909)119867] = 119906

0(119909) which is an arbitrary function Since

Π0asymp 0 always neither1198600 norΠ0 are of interest in describing

the system andmay be discarded from the theory In fact theterm containing 119860

0is redundant because it can be absorbed

by redefining the function 119908(119909) Therefore the Hamiltonianis now given as

119867 = int1198893119909 [119908 (119909) 120597

119894Π

119894minus1

119894(1 minus

120572

3120587ΔM)

minus1

Π119894

+1

4119865119894119895(1 minus

120572

3120587ΔM)119865

119894119895]

(7)

where 119908(119909) = 1199061(119909) minus 119860

0(119909)

It must be clear from this discussion that the presenceof the new arbitrary function 119908(119909) is undesirable since wehave no way of giving it a meaning in a quantum theoryHence according to the usual procedure we impose a gaugecondition such that the full set of constraints becomes secondclass A convenient choice is

Γ2(119909) equiv int

119862120585119909

119889119911]119860] (119911) equiv int

1

0

119889120582119909119894119860

119894(120582119909) = 0 (8)

where 120582 (0 le 120582 le 1) is the parameter describing the space-like straight path 119911

119894= 120585

119894+ 120582(119909 minus 120585)

119894 and 120585 is a fixed point(reference point) There is no essential loss of generality if werestrict our considerations to 120585

119894= 0 The Dirac brackets can

now be determined and we simply note the only nontrivialDirac bracket involving the canonical variables that is

119860119894(119909) Π

119895(119910)

lowast

= 120575119895

119894120575(3)

(119909 minus 119910)

minus 120597119909

119894int

1

0

119889120582119909119894120575(3)

(120582119909 minus 119910)

(9)

In passing we also recall that the transition to a quantumtheory is made by the replacement of the Dirac brackets bythe operator commutation relations according to 119860 119861

lowastrarr

(119894ℏ)[119860 119861]With the foregoing information we can now proceed to

obtain the interaction energy As alreadymentioned in orderto accomplish this purpose we will calculate the expectationvalue of the energy operator119867 in the physical state |Φ⟩ wherethe physical states |Φ⟩ are gauge-invariant ones The physicalstate can be written as

|Φ⟩ equiv10038161003816100381610038161003816Ψ (y) Ψ (y1015840)⟩

= 120595 (y) exp(119894119902ℏint

y

y1015840119889119911

119894119860

119894(119911))120595 (y1015840) |0⟩

(10)

where |0⟩ is the physical vacuum state and the line integralappearing in the above expression is along a space-like pathstarting at y1015840 and ending at y on a fixed time slice The pointwe wish to emphasize however is that the physical fermion(Ψ(y)) is not the Lagrangian fermion (120595(y)) which is neithergauge-invariant nor associated with an electric field In factthe physical fermion is the Lagrangian fermion together witha cloud (or dressing) of gauge fields

Making use of the above Hamiltonian structure [18] wefind that

Π119894 (119909)

10038161003816100381610038161003816Ψ (y) Ψ (y1015840)⟩ = Ψ (y) Ψ (y1015840)Π

119894 (119909) |0⟩

+ 119902int

y1015840

y119889119911

119894120575(3)

(z minus x) |Φ⟩ (11)

With the aid of (11) and (7) the lowest-order modification in120572 of the interaction energy takes the form

⟨119867⟩Φ = ⟨119867⟩0 + 1198811+ 119881

2 (12)

4 Advances in High Energy Physics

where ⟨119867⟩0= ⟨0|119867|0⟩ The 119881

1 119881

2terms are given by

1198811=1199022

2int119889

3119909int

y

y1015840119889119911

119894120575(3)

(x minus z) inty

y1015840119889119911

1015840119894120575(3)

(x minus z1015840) (13)

1198812=1199022

2

120572

3120587int

infin

41198982

119889120591120588 (120591)

120591int 119889

3119909

sdot int

y1015840

y119889119911

1015840

119894120575(3)

(z1015840 minus x) nabla2

119909

1

120591 minus nabla2

119909

int

y1015840

y119889119911

119894120575(3)

(z minus x) (14)

where 120588(120591) = (1 + 21198982120591)radic1 minus 4119898

2120591

We note that term (13) may look peculiar but it is nothingbut the familiar Coulomb interaction plus a self-energy term[52] Now making use of the Green function 119866(z z1015840) =

(14120587)(119890minusradic120591|zminusz1015840|

|z minus z1015840|) the term (14) can be rewritten inthe form

1198812=1199022

2

120572

3120587int

infin

41198982

119889120591120588 (120591)

120591int

y1015840

y119889119911

1015840

119894nabla

2

1199111015840 int

y1015840

y119889119911

119894119866(z z1015840)

= minus120572

3120587

1199022

2int

infin

41198982

119889120591120588 (120591)

120591

119890minusradic120591|yminusy1015840|

1003816100381610038161003816y minus y10158401003816100381610038161003816

(15)

Since the second and third term on the right-hand sideof (12) are clearly dependent on the distance between theexternal static fields the potential for two opposite chargeslocated at y and y1015840 reads

119881 = minus1199022

4120587

1

119871(1 +

120572

3120587int

infin

41198982

119889120591120588 (120591)

120591119890minusradic120591119871

) (16)

where 119871 = |y minus y1015840| Accordingly one recovers the knownUehling potential which finds here an entirely differentderivation

Beforewe proceed further wewish to show that this resultcan be written alternatively in a more explicit form Makinguse of [53]

120594119899(119911) = int

infin

1

119889119905119890minus119905119911

119905119899

(1 +1

21199052)radic1 minus

1

1199052 (17)

we then get

119881 = minus1199022

4120587

1

119871(1 +

2120572

31205871205941 (2119898119871)) (18)

By the transformation 119905 = cosh 119906 [54] the functions 120594119899can

be reduced to the form [53]

120594119899(119911) = 119870119894

119899minus1(119911) minus

1

2119870119894

119899+1(119911) minus

1

2119870119894

119899+3(119911) (19)

where the functions 119870119894 denote Bessel function integralsHence we see that the interaction energy (with 119898 = 1)becomes

119881 = minus1199022

4120587

1

1198711 +

2120572

3120587[(1 +

1198712

3)119870

0 (2119871)

minus (5119871

3+2119871

3

3)119870

1(2119871)

+ (3119871

2+2119871

3

3)int

infin

2119871

1198891199051198700 (119905)]

(20)

where1198700(119911) and119870

1(119911) aremodified Bessel functions Finally

with the aid of asymptotic forms for Bessel functions it is asimple matter to find expressions for 119881 for large and small 119871

Before concluding this subsection we discuss an alterna-tive way of stating our previous result (16) which displayscertain distinctive features of our methodology We start byconsidering [52 55]

119881 equiv 119902 (A0(0) minusA

0(L)) (21)

where the physical scalar potential is given by

A0(119905 r) = int

1

0

119889120582119903119894119864

119894(119905 120582r) (22)

This follows from the vector gauge-invariant field expression

A120583 (119909) equiv 119860

120583 (119909) + 120597120583(minusint

119909

120585

119889119911120583119860

120583 (119911)) (23)

where the line integral is along a space-like path from thepoint 120585 to 119909 on a fixed slice time It is also important toobserve that the gauge-invariant variables (22) commute withthe sole first constraint (Gauss law) showing in this waythat these fields are physical variables In as much as we areinterested in estimating the lowest-order correction to theCoulomb energy we will retain only the leading term inexpression 119864

119894= (1 minus (1205723120587)ΔM)

minus1Π

119894 Making use of this lastexpression (22) gives

A0(119905 r)

= int

1

0

119889120582119903119894120597120582r119894(minus

1198690(120582r)nabla

2

120582r)

+120572

3120587int

infin

41198982

119889120591

120591120588 (120591) int

1

0

119889120582119903119894120597120582r119894(minus

1198690(120582r)

nabla2

120582r minus 120591)

(24)

to get the last line we used Gauss law for the present theorythat is 120597

119894Π

119894= 119869

0 (where we have included the externalcurrent 1198690 to represent the presence of two opposite charges)Accordingly for 1198690(119905 r) = 119902120575

(3)(r) the potential for a pair of

static point-like opposite charges located at 0 and L is givenby

119881 = minus1199022

4120587

1

119871(1 +

120572

3120587int

infin

41198982

119889120591120588 (120591)

120591119890minusradic120591119871

) (25)

after subtracting a self-energy term

Advances in High Energy Physics 5

3 Euler-Heisenberg-Like Model

Proceeding in the sameway aswe did in the foregoing sectionwe will now consider the interaction energy for Euler-Heisenberg-like electrodynamics Nevertheless in order toput our discussion into context it is useful to describe verybriefly the model under consideration In such a case theLagrangian density reads

L =1205732

21 minus [1 +

1

1205732F minus

1

12057321205742G

2]

119901

(26)

where we have included two parameters 120573 and 120574 As usualF = (14)119865

120583]119865120583] G = (14)119865

120583]119865120583] 119865

120583] = 120597120583119860] minus 120597]119860120583

and 119865

120583]= (12)120576

120583]120588120582119865120588120582 Let us also mention here that in

our previous paper [19] we have studied the domain 0 lt

119901 lt 1 Moreover it follows from (26) that when 119901 = 2

the model contains to orders O(11205732) and O(11205742

) a Euler-Heisenberg-likemodel with the appropriate identifications ofthe constants Interestingly we also observe that in the limit120574 rarr infin we obtain a Wichmann-Kroll model This remarkopens up the way to discuss the effect of these nonlinearcorrections on the interaction energy as we are going tostudy below In fact we will consider a massive Wichmann-Kroll systemThemotivation for this study comes from recentconsiderations in the context of dualities [56] where massiveBorn-Infeld systems play an important role

Having made these observations we can write immedi-ately the field equations for 119901 = 2

120597120583[Γ(119865

120583]minus

2

1205742G119865

120583])] = 0 (27)

while the Bianchi identities are given by

120597120583119865

120583]= 0 (28)

where

Γ = 1 +F

1205732minus

G2

12057321205742 (29)

Also it is straightforward to see that Gauss law becomes

nabla sdotD = 0 (30)

whereD is given by

D

= [1 minus

(E2minus B2

)

21205732

minus(E sdot B)2

12057321205742

](E +2

1205742(E sdot B)B)

(31)

Again from (30) for 1198690(119905 r) = 119890120575(3)(r) we findD = (119876119903

2)119903

where 119876 = 1198904120587 This then implies that for a point-likecharge 119890 at the origin the expression

119876

1199032= (1 minus

E2

21205732) |E| (32)

tells us that for 119903 rarr 0 the electrostatic field becomessingular at 119903 = 0 in contrast to the 0 lt 119901 lt 1 case wherethe electrostatic field is finite Even so in this theory thephenomenon of birefringence is present Before going intodetails we would like to recall that birefringence refers to theproperty that polarized light in a particular direction (opticalaxis) travels at a different velocity from that of light polarizedin a direction perpendicular to this axis Indeed due toquantum fluctuations the QED vacuum has this property aswe are going to show

To illustrate this important feature we introduce thevectorsD = 120597L120597E andH = minus120597L120597B

D = Γ(E + 2B (E sdot B)

1205742

)

H = Γ(B minus 2E (E sdot B)

1205742

)

(33)

where Γ = 1 + (121205732)(B2

minus E2) minus (1120573

21205742)(E sdot B)2 We thus

obtain the equations of motion

nabla sdotD = 0

120597D120597119905

minus nabla timesH = 0

nabla sdot B = 0

120597B120597119905

+ nabla times E = 0

(34)

With the aid from (33) we find the electric permittivity 120576119894119895

and the inverse magnetic permeability (120583minus1)119894119895 tensors of the

vacuum that is

120576119894119895= Γ(120575

119894119895+

2119861119894119861119895

1205742

)

(120583minus1)119894119895= Γ(120575

119894119895minus

2119864119894119864

119895

1205742

)

(35)

with119863119894= 120576

119894119895119864

119895and 119861

119894= 120583

119894119895119867

119895

In accordance with our previous procedure [18 19] wecan now linearize the above equations To do this it isadvantageous to introduce a weak electromagnetic wave(E

119901B

119901) propagating in the presence of a strong constant

external field (E0B

0) On these assumptions we readily find

that for the case of a purely magnetic field (E0= 0) the

vectorsD andH become

D = (1 +B2

0

21205732)[E

119901+

2

1205742(E

119901sdot B

0)B

0]

H = (1 +B2

0

21205732)

sdot [B119901+

1

1205732(1 + B2

02120573

2)(B

119901sdot B

0)B

0]

(36)

where we have to keep only linear terms in E119901B

119901 As

before we consider the 119911-axis as the direction of the external

6 Advances in High Energy Physics

magnetic field (B0= 119861

0e3) and assuming that the light wave

moves along the 119909-axis the decomposition into a plane wavefor the fields E

119901and B

119901can be written as

E119901 (x 119905) = E119890minus119894(119908119905minusksdotx)

B119901(x 119905) = B119890minus119894(119908119905minusksdotx)

(37)

In this case it clearly follows that

(1198962

1199082minus 120576

2212058333)119864

2= 0 (38)

(1198962

1199082minus 120576

3312058322)119864

3= 0 (39)

As a consequence we have two different situations Firstif E perp B

0(perpendicular polarization) from (39) 119864

3= 0 and

from (38) we get 1198962119908

2= 120576

2212058333 This then means that the

dispersion relation of the photon takes the form

119899perp= radic

1 + B2

02120573

2

1 + 3B2

02120573

2 (40)

Second if E B0(parallel polarization) from (38) 119864

2= 0

and from (39) we get 1198962119908

2= 120576

3312058322 This leads to

119899= radic1 +

2B2

0

1205742 (41)

Thus we verify that in the case of a generalized Euler-Heisenberg electrodynamics the phenomenon of birefrin-gence is present

We now pass to the calculation of the interaction energybetween static point-like sources for a massive Wichmann-Kroll-like model our analysis follows closely that of [18 19]The corresponding theory is governed by the Lagrangiandensity

L = minus1

4119865

2

120583] +1

321205732(119865

120583]119865120583])2

+119898

2

2119860

120583119860

120583 (42)

Next in order to handle the second term on the right hand in(42) we introduce an auxiliary field 120585 such that its equation ofmotion gives back the original theory This allows us to writethe Lagrangian density as

L = minus1

4119865120583]119865

120583]+

120585

321205732119865120583]119865

120583]minus

1

12812057321205852

+119898

2

2119860

120583119860

120583

(43)

With the redefinition 120578 = 1 minus 12058581205732 (43) becomes

L = minus1

4120578119865

120583]119865120583]minus1

2(1 minus 120578)

2+119898

2

2119860

120583119860

120583 (44)

Before we proceed to work out explicitly the interactionenergy we will first restore the gauge invariance in (44)

Following an earlier procedure we readily verify that thecanonical momenta read Π

120583= minus120578119865

0120583 which results in theusual primary constraint Π0

= 0 and Π119894= 120578119865

1198940 In this wayone obtains

119867119862= int119889

3119909Π

119894120597119894119860

0+

1

2120578Π

2+120578

2B2

minus119898

2

2119860

120583119860

120583

+1

2(1 minus 120578)

2

(45)

The consistency condition Π = 0 leads to the constraint Γ equiv

120597119894Π

119894+119898

2119860

0= 0 As a result both constraints are second-class

To convert the second-class system into first-class we willadopt the procedure described previously Thus we enlargethe original phase space by introducing a canonical pair offields 120579 and Π

120579 It follows therefore that a new set of first-

class constraints can be defined in this extended space Λ1equiv

Π0+ 119898

2120579 = 0 and Λ

2equiv Γ + Π

120579= 0 Notice that this new

120579-field is not to be confused with the usual noncommutativeparameter This then shows that the new constraints arefirst-class and therefore restore the gauge symmetry As iswell known this procedure reproduces the usual Stuckelbergformalism From this the new effective Lagrangian densityafter integrating out the 120579 fields becomes

L = minus1

4119865120583] (120578 +

1198982

Δ)119865

120583]minus1

2(1 minus 120578)

2 (46)

Now writing 120590 = 120578 + 1198982Δ expression (46) can be brought

to the form

L = minus1

4119865120583]120590119865

120583]minus

119896

128(1 minus 120590 +

1198982

Δ)

2

(47)

where 119896 = 641205732

We are now ready to compute the interaction energy Inthis case the canonical momenta are Π120583

= minus1205901198650120583 with the

usual primary constraint Π0= 0 and Π

119894= 120590119865

1198940 Hence thecanonical Hamiltonian is expressed as

119867119862= int119889

3119909Π

119894120597119894119860

0+

1

2120590Π

2+120590

2B2

+119896

128(1 minus 120590 +

1198982

Δ)

2

(48)

Time conservation of the primary constraint Π0 yields thesecondary constraint Γ

1equiv 120597

119894Π

119894= 0 Similarly the P

120590

constraint yields no further constraints and just determinesthe field 120590 In this case at leading order in 120573 the field 120590 isgiven by

120590 = (1 +119898

2

Δminus

B2

21205732)

sdot [1 minus3

21205732

1

(1 + 1198982Δ minus B2

21205732)3Π

2]

(49)

Advances in High Energy Physics 7

which will be used to eliminate 120590 As before the corre-sponding total (first-class) Hamiltonian that generates thetime evolution of the dynamical variables is 119867 = 119867

119862+

int1198893119909(119906

0(119909)Π

0(119909) + 119906

1(119909)Γ

1(119909)) where 119906

0(119909) and 119906

1(119909) are

the Lagrangemultiplier utilized to implement the constraintsIn the same way as was done in the previous subsection

the expectation value of the energy operator119867 in the physicalstate |Φ⟩ becomes

⟨119867⟩Φ = ⟨Φ|int1198893119909

1

119894(1 +

1198982

Δ)

minus1

Π119894+

15

81205732Π

4

minus15119898

2

21205732Π

2 1

ΔΠ

2 |Φ⟩

(50)

in this last line we have considered only quadratic terms in119898

2In such a case by employing (50) the lowest-order

modification in 1205732 and1198982 of the interaction energy takes theform

⟨119867⟩Φ = ⟨119867⟩0 + 1198811+ 119881

2+ 119881

3 (51)

where ⟨119867⟩0= ⟨0|119867|0⟩ The 119881

1 119881

2 and 119881

3terms are given by

1198811=1199022

2int119889

3119909int

y

y1015840119889119911

1015840119894120575(3)

(x minus z1015840)(1 minus 1198982

nabla2)

minus1

sdot int

y

y1015840119889119911

119894120575(3)

(x minus z)

1198812= minus

151199024

81205732int119889

3119909int

y

y1015840119889119911

119894120575(3)

(x minus z)

sdot int

y

y1015840119889119911

1015840119894120575(3)

(x minus z1015840)inty

y1015840119889119906

119896120575(3)

(x minus u)

sdot int

y

y1015840119889V

119896120575(3)

(x minus k)

1198813=15119898

21199024

21205732

int1198893119909int

y

y1015840119889119911

119894120575(3)

(x minus z)

sdot int

y

y1015840119889119911

1015840119894120575(3)

(x minus z1015840)inty

y1015840119889119906

119896120575(3)

(x minus u)

sdot int

y

y1015840119889V

119896120575(3)

(x minus k)

(52)

Finally with the aid of expressions (52) the potential fora pair of static point-like opposite charges located at 0 and Lis given by

119881 = minus1199022

4120587

119890minus119898119871

119871+

1199024

161205871205732(3

8120587

1

1198712minus 5119898

2)

1

1198713 (53)

observe that when 119898 = 0 profile (53) reduces to the knownWichmann-Kroll interaction energy On the other hand for119898 = 0 the key role played by the mass term in transformingthe Coulomb potential into the Yukawa one should be notedInterestingly enough an unexpected feature is found In factprofile (53) displays a new long-range 11198713 correction whereits strength is proportional to 119898

2 It is also important toobserve that an analogous correction has been found in Born-Infeld electrodynamics in the context of very special relativity[57] In this way we establish a new connection betweennonlinear effectives theories

Before we proceed further we should comment on ourresult In the case of QED (Euler-Heisenberg Lagrangiandensity) the parameter 11205732 is given by 11205732

= (1645)(1198904ℏ

1198984

1198901198887) where 119898

119890is the electron mass In this context we

also recall the currently accepted upper limit for the photonmass that is 119898

120574sim 2 times 10

minus16 eV Thus for the QED casefrom (53) it follows that the second term on the right handside would be detectable in long-range distances (sim109 m) Inother words we see that detectable corrections induced byvacuum polarization with a mass term would be present atlow energy scales

From (53) it clearly follows that the interaction energybetween heavy charged charges at leading order in 120573 is notfinite at the origin Motivated by this one may consider theabove calculation in a noncommutative geometry based onfindings of our previous studies [18 19] In such a case theelectric field at leading order in 120573

2 and1198982 takes the form

119864119894= [(1 +

1198982

Δ)

minus1

+3

21205732Π

2minus6119898

2

1205732Π

2 1

Δ]

sdot 120597119894(minus

119890120579nabla2

120575(3)

(x)nabla

2)

(54)

where it may be recalled that we are now replacing the source120575(3)(x minus y) by the smeared source 119890

120579nabla2

120575(3)(x minus y) with the

parameter 120579 being noncommutative Now making use of(22) we readily find that

A0(119905 r) = A

(1)

0(119905 r) +A

(2)

0(119905 r) +A

(3)

0(119905 r) (55)

The term A(1)

0(119905 r) was first calculated in [50] we can

therefore write only the result

A(1)

0(119905 r)

= 1199021198901198982120579

4120587

1

119903[119890

minus119898119903minus

1

radic120587int

infin

11990324120579

1198891199061

radic119906119890minus119906minus119898

211990324119906

]

minus 119902119898

41205871198901198982120579

(56)

8 Advances in High Energy Physics

Meanwhile the terms A(2)

0(119905 r) and A

(3)

0(119905 r) after some

manipulation can be brought to the form

A(2)

0(119905 r) =

121199023

120573212058732

119899119894int

119909

0

119889119906119894 1

11990661205743(3

21199062

4120579)

A(3)

0(119905 r) =

311989821199023

120573212058752

119899119894int

119909

0

119889119906119894 1

11990641205742(3

21199062

4120579)

sdot [4120579

1199062120574(

3

21199062

4120579) minus 120574(

1

21199062

4120579)]

(57)

where 120574(32 11990324120579) is the lower incomplete Gamma functiondefined by 120574(119886119887 119909) equiv int

119909

0(119889119906119906)119906

119886119887119890minus119906

Inserting these expressions in (21) we finally obtain thestatic potential for two opposite charges 119902 located at 0 and Las

119881 = minus119902

4120587

1198901198982120579

119871[119890

minus119898119871minus

1

radic120587int

infin

11987124120579

1198891199061

radic119906119890minus119906minus119898

211990624119906

]

minus12119902

4

120573212058732

119899119894int

119871

0

119889119906119894 1

11990661205743(3

21199062

4120579) minus

311989821199024

120573212058752

sdot 119899119894int

119909

0

119889119906119894 1

11990641205742(3

21199062

4120579)

sdot [4120579

1199062120574(

3

21199062

4120579) minus 120574(

1

21199062

4120579)]

(58)

which is finite for 119871 rarr 0 It is a simple matter to verify thatin the limit 120579 rarr 0 we recover our above result

4 Logarithmic Correction

We now want to extend what we have done to Euler-Heisenberg-like electrodynamics at strong fields As alreadymentioned such theories show a power behavior that istypical for critical phenomena [58] In such a case theLagrangian density reads

L = minus1

4119865120583]119865

120583]minus1198870

8119865120583]119865

120583] log(119865120583]119865

120583]

41205822

) (59)

where 1198870and 120582 are constants In fact by choosing 119887

0=

11989026120587

2 and 120582 = 1198982

1198901198883119890ℏ we recover the Euler-Heisenberg

electrodynamics at strong fields [58]In the same way as was done in the previous section one

can introduce an auxiliary field 120585 to handle the logarithm in(59) This leads to

L = minus1

41205721119865120583]119865

120583]minus 120572

2(119865

120583]119865120583])2

(60)

where 1205721= 1 minus (119887

02)(1 + log 120585) and 120572

2= 119887

012058532120582

2A similar procedure can be used to manipulate the

quadratic term in (60) Accordingly by introducing a secondauxiliary field 120578 one easily finds

L = minus1

4119865120583] (1205721

+ 41205722120578) 119865

120583]+1205782

41205722 (61)

By setting 120590 = 1205721+ 4120572

2120578 we then have

L = minus1

4120590119865

120583]119865120583]+

1

641205722

(120590 minus 1205721)2 (62)

It is once again straightforward to apply the gauge-invariant formalism discussed in the foregoing section Thecanonical momenta read Π

120583= minus120590119865

0120583 and at once werecognize the two primary constraints Π0

= 0 and P120590

equiv

120597119871120597 = 0The canonical Hamiltonian corresponding to (62)is

119867119862= int119889

3119909

sdot Π119894120597119894119860

0+

1

2120590Π

2+120590

2B2

minus1

641205722

(120590 minus 1205721)2

(63)

Requiring the primary constraint Π0 to be preserved intime one obtains the secondary constraint Γ

1= 120597

119894Π

119894= 0 In

the same way for the constraintP120590 we get the auxiliary field

120590 as

120590 = (1 minus1198870

2(1 + ln 120585) +

1198870B2

21205822120585)[1

+3119887

0B2

21205822

120585

(1 minus (11988702) (1 + ln 120585) + (119887

0B2

21205822) 120585)

3]

(64)

Hence we obtain

119867119862= int119889

3119909

sdot Π119894120597119894119860

0+1

2+1198870

4(1 + log 120585)Π2 minus

31198870120585

21205822Π

4

(65)

As before requiring the primary constraint P120585to be

preserved in time one obtains the auxiliary field 120585 In this case120585 = 1205826Π

2 Consequently we get

119867119862= int119889

3119909Π

119894120597119894119860

0+1

2(1 + 119887

0)Π

2minus6119887

0

1205822Π

4 (66)

Following the same steps that led to (50) we find that

⟨119867⟩(1)

Φ= ⟨Φ|int119889

3119909

1

2minus

3

81205732Π

4 |Φ⟩ (67)

It should be noted that this expression is similar to (50) in thelimit 119898 rarr 0 except by the changed sign in front of the Π4termHencewe see that the potential for two opposite chargesin 0 and L is given by

119881 = minus1199022

4120587

1

119871minus

1199024

604012057321205872

1

1198715 (68)

5 Final Remarks

Finally within the gauge-invariant but path-dependent vari-ables formalism we have considered the confinement versus

Advances in High Energy Physics 9

screening issue for both massive Euler-Heisenberg-like andEuler-Heisenberg electrodynamics in the approximation ofthe strong-field limit Once again a correct identificationof physical degrees of freedom has been fundamental forunderstanding the physics hidden in gauge theories Inter-estingly enough their noncommutative version displays anultraviolet finite static potential The analysis above revealsthe key role played by the new quantum of length in ouranalysis In a general perspective the benefit of consideringthe present approach is to provide a unification scenarioamong different models as well as exploiting the equivalencein explicit calculations as we have illustrated in the course ofthis work

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

It is a pleasure for the author to thank J A Helayel-Netofor helpful comments on the paper This work was partiallysupported by Fondecyt (Chile) Grant 1130426 and DGIP(UTFSM) internal project USM 111458

References

[1] G Breit and J A Wheeler ldquoCollision of two light quantardquoPhysical Review vol 46 no 12 pp 1087ndash1091 1934

[2] S L Adler ldquoPhoton splitting and photon dispersion in a strongmagnetic fieldrdquo Annals of Physics vol 67 no 2 pp 599ndash6471971

[3] V Costantini B De Tollis and G Pistoni ldquoNonlinear effects inquantum electrodynamicsrdquo Il Nuovo Cimento A vol 2 no 3pp 733ndash787 1971

[4] R Ruffini G Vereshchagin and S-S Xue ldquoElectron-positronpairs in physics and astrophysics from heavy nuclei to blackholesrdquo Physics Reports vol 487 no 1ndash4 pp 1ndash140 2010

[5] G V Dunne ldquoThe Heisenberg-Euler effective action 75 yearsonrdquo International Journal of Modern Physics Conference Seriesvol 14 pp 42ndash56 2012

[6] C Bamber S J Boege TKoffas et al ldquoStudies of nonlinearQEDin collisions of 466GeV electrons with intense laser pulsesrdquoPhysical Review D vol 60 no 9 Article ID 092004 1999

[7] D L Burke R C Field G Horton-Smith et al ldquoPositronproduction in multiphoton light-by-light scatteringrdquo PhysicalReview Letters vol 79 no 9 article 1626 1997

[8] O J Pike F MacKenroth E G Hill and S J Rose ldquoA photonndashphoton collider in a vacuum hohlraumrdquo Nature Photonics vol8 no 6 pp 434ndash436 2014

[9] D Tommasini A Ferrando H Michinel and M Seco ldquoPre-cision tests of QED and non-standard models by searchingphoton-photon scattering in vacuum with high power lasersrdquoJournal ofHigh Energy Physics vol 2009 no 11 article 043 2009

[10] D Tommasini A Ferrando HMichinel andM Seco ldquoDetect-ing photon-photon scattering in vacuum at exawatt lasersrdquoPhysical Review A vol 77 no 4 Article ID 042101 2008

[11] M Born and L Infeld ldquoFoundations of the new field theoryrdquoProceedings of the Royal Society of London Series A Containing

Papers of a Mathematical and Physical Character vol 144 no852 pp 425ndash451 1934

[12] H Gies J Jaeckel and A Ringwald ldquoPolarized light propagat-ing in a magnetic field as a probe for millicharged fermionsrdquoPhysical Review Letters vol 97 no 14 Article ID 140402 2006

[13] E Masso and R Toldra ldquoLight spinless particle coupled tophotonsrdquo Physical Review D vol 52 no 4 pp 1755ndash1763 1995

[14] P Gaete and E I Guendelman ldquoConfinement in the presenceof external fields and axionsrdquoModern Physics Letters A vol 20no 5 article 319 2005

[15] P Gaete and E Spallucci ldquoConfinement effects from interactingchromo-magnetic and axion fieldsrdquo Journal of Physics A Math-ematical and General vol 39 no 20 pp 6021ndash6029 2006

[16] E S Fradkin and A A Tseytlin ldquoNon-linear electrodynamicsfrom quantized stringsrdquo Physics Letters B vol 163 no 1ndash4 pp123ndash130 1985

[17] E Bergshoeff E Sezgin C N Pope and P K TownsendldquoThe Born-Infeld action from conformal invariance of the opensuperstringrdquo Physics Letters B vol 188 no 1 pp 70ndash74 1987

[18] P Gaete and J Helayel-Neto ldquoFinite field-energy and interpar-ticle potential in logarithmic electrodynamicsrdquo The EuropeanPhysical Journal C vol 74 no 3 article 2816 2014

[19] P Gaete and J Helayel-Neto ldquoRemarks on nonlinear electrody-namicsrdquo European Physical Journal C vol 74 no 11 article 31822014

[20] S H Hendi ldquoAsymptotic ReissnerndashNordstrom black holesrdquoAnnals of Physics vol 333 pp 282ndash289 2013

[21] Z Zhao Q Pan S Chen and J Jing ldquoNotes on holographicsuperconductor models with the nonlinear electrodynamicsrdquoNuclear Physics B vol 871 no 1 pp 98ndash110 2013

[22] O Miskovic and R Olea ldquoConserved charges for black holes inEinstein-Gauss-Bonnet gravity coupled to nonlinear electrody-namics in AdS spacerdquo Physical Review D vol 83 no 2 ArticleID 024011 2011

[23] S H Mazharimousavi and M Halilsoy ldquoBlack holes and theclassical model of a particle in Einstein non-linear electrody-namics theoryrdquo Physics Letters B vol 678 no 4 pp 407ndash4102009

[24] G Amelino-Camelia ldquoSpecial treatmentrdquo Nature vol 418 no6893 pp 34ndash35 2002

[25] T Jacobson S Liberati andDMattingly ldquoThreshold effects andPlanck scale Lorentz violation combined constraints from highenergy astrophysicsrdquo Physical Review D vol 67 no 12 ArticleID 124011 2003

[26] T J Konopka and S A Major ldquoObservational limits onquantum geometry effectsrdquo New Journal of Physics vol 4 pp571ndash5718 2002

[27] SHossenfelder ldquoInterpretation of quantumfield theories with aminimal length scalerdquo Physical Review D vol 73 no 10 ArticleID 105013 9 pages 2006

[28] P Nicolini ldquoNoncommutative black holes the final appeal toquantum gravity a reviewrdquo International Journal of ModernPhysics A vol 24 no 7 pp 1229ndash1308 2009

[29] S Pramanik S Ghosh and P Pal ldquoElectrodynamics of a gener-alized charged particle in doubly special relativity frameworkrdquoAnnals of Physics vol 346 pp 113ndash128 2014

[30] A F Ali S Das and E C Vagenas ldquoDiscreteness of space fromthe generalized uncertainty principlerdquo Physics Letters B vol678 no 5 pp 497ndash499 2009

[31] G Amelino-Camelia ldquoQuantum-spacetime phenomenologyrdquoLiving Reviews in Relativity vol 16 article 5 2013

10 Advances in High Energy Physics

[32] E Witten ldquoNoncommutative geometry and string field theoryrdquoNuclear Physics B vol 268 no 2 pp 253ndash294 1986

[33] N Seiberg and E Witten ldquoString theory and noncommutativegeometryrdquo Journal of High Energy Physics vol 1999 article 0321999

[34] M R Douglas and N A Nekrasov ldquoNoncommutative fieldtheoryrdquo Reviews of Modern Physics vol 73 no 4 pp 977ndash10292001

[35] R J Szabo ldquoQuantum field theory on noncommutative spacesrdquoPhysics Reports vol 378 no 4 pp 207ndash299 2003

[36] J Gomis K Kamimura and T Mateos ldquoGauge and BRST gen-erators for space-time non-commutative U(1) theoryrdquo Journalof High Energy Physics vol 2001 no 3 article 010 2001

[37] A A Bichl J M Grimstrup L Popp M Schweda and RWulkenhaar ldquoPerturbative analysis of the Seiberg-Witten maprdquoInternational Journal of Modern Physics A vol 17 no 16 pp2219ndash2231 2002

[38] A Smailagic and E Spallucci ldquoUV divergence-free QFT onnoncommutative planerdquo Journal of Physics AMathematical andGeneral vol 36 no 39 pp L517ndashL521 2003

[39] A Smailagic and E Spallucci ldquoFeynman path integral on thenon-commutative planerdquo Journal of Physics A Mathematicaland General vol 36 no 33 pp L467ndashL471 2003

[40] M Knezevic D Knezevic and D Spasojevic ldquoStatistics ofequally weighted random paths on a class of self-similarstructuresrdquo Journal of Physics A Mathematical and Generalvol 37 no 1 pp 1ndash8 2004 Erratum in Journal of Physics AMathematical and General vol 37 p 7169 2004

[41] E Spallucci A Smailagic and P Nicolini ldquoTrace anomaly ona quantum spacetime manifoldrdquo Physical Review D vol 73Article ID 084004 2006

[42] R Banerjee S Gangopadhyay and S K Modak ldquoVoros prod-uct noncommutative Schwarzschild black hole and correctedarea lawrdquo Physics Letters B vol 686 no 2-3 pp 181ndash187 2010

[43] A B Hammou M Lagraa and M M Sheikh-Jabbari ldquoCoher-ent state induced star product on 119877

3

120582and the fuzzy sphererdquo

Physical Review D vol 66 no 2 Article ID 025025 2002[44] L Modesto J W Moffat and P Nicolini ldquoBlack holes in an

ultraviolet complete quantum gravityrdquo Physics Letters B vol695 no 1ndash4 pp 397ndash400 2011

[45] P Nicolini ldquoNonlocal and generalizeduncertainty principleblack holesrdquo httparxivorgabs12022102

[46] P Nicolini A Smailagic and E Spallucci ldquoNoncommutativegeometry inspired Schwarzschild black holerdquo Physics Letters Bvol 632 no 4 pp 547ndash551 2006

[47] S Pramanik S Das and S Ghosh ldquoNoncommutative extensionof AdS-CFT and holographic superconductorsrdquo Physics LettersB vol 742 pp 266ndash273 2015

[48] W Dittrich and M Reuter Effective Lagrangians in QuantumElectrodynamics Springer Berlin Germany 1985

[49] M Ostrogradsky ldquoMemoires sur les equations differen-tielles relativrs au probleme des isoperimetersrdquo Memoires delrsquoAcademie Imperiale des Sciences de St Petersbourg vol 4 p385 1850

[50] P Gaete and E Spallucci ldquoFinite axionic electrodynamicsfrom a new non-commutative approachrdquo Journal of Physics AMathematical and Theoretical vol 45 no 6 Article ID 06540114 pages 2012

[51] P Gaete ldquoSome considerations about Podolsky-axionic electro-dynamicsrdquo International Journal of Modern Physics A vol 27no 11 Article ID 1250061 2012

[52] P Gaete ldquoRemarks on gauge-invariant variables and interactionenergy in QEDrdquo Physical Review D vol 59 no 12 Article ID127702 1999

[53] S Klarsfeld ldquoAnalytical expressions for the evaluation ofvacuum-polarization potentials in muonic atomsrdquo Physics Let-ters B vol 66 no 1 pp 86ndash88 1977

[54] A M Frolov and D M Wardlaw ldquoAnalytical formula for theUehling potentialrdquo The European Physical Journal B vol 85article 348 2012

[55] P Gaete ldquoOn gauge-invariant variables in QEDrdquo Zeitschrift furPhysik C Particles and Fields vol 76 no 2 pp 355ndash361 1997

[56] S Ferrara and A Sagnotti ldquoMassive Born-Infeld and other dualpairsrdquo Journal of High Energy Physics vol 2015 no 4 article 0322015

[57] R Bufalo ldquoBornndashinfeld electrodynamics in very special relativ-ityrdquo Physics Letters B vol 746 pp 251ndash256 2015

[58] HKleinert E Strobel and S-S Xue ldquoFractional effective actionat strong electromagnetic fieldsrdquo Physical Review D vol 88 no2 Article ID 025049 2013

Submit your manuscripts athttpwwwhindawicom

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ThermodynamicsJournal of

2 Advances in High Energy Physics

been studied by using a star-product (Moyal product) [32ndash37] In this connection it becomes of interest in particularto recall that a novel way to formulate noncommutativequantum field theory has been proposed in [38ndash40] Thekey ingredient of this development is to introduce coherentstates of the quantum position operators [41] where amodified form of heat kernel asymptotic expansion whichdoes not suffer from short distance divergences has beenobtained We also point out that an alternative derivation ofthe coherent state approach has been implemented througha new multiplication rule which is known as Voros star-product [42] Anyhow physics turns out to be independentof the choice of the type of product [43] It is worthy tonote here that this type of noncommutativity (coherent stateapproach) leads to a smearing effect which is equivalent tothat encountered in a class of nonlocal theory In other wordsnoncommutativity is just a subclass of possible nonlocaldeformation [44 45] More recently this new approach hasbeen successfully extended to black holes physics [46] alsoin connection to holographic superconductors via AdS-CFT[47]

Inspired by these observations the purpose of this paperis to extend our previous studies [18 19] on nonlinear electro-dynamics to the case when vacuum polarization correctionsare taken into accountThepreceding studieswere done usingthe gauge-invariant but path-dependent variables formalismwhere the interaction potential energy between two staticcharges is determined by the geometrical condition of gaugeinvariance One important advantage of this approach isthat it provides a physically based alternative to the usualWilson loop approach Accordingly we will work out thestatic potential for electrodynamicswhich include apart fromthe Maxwell Lagrangian additional terms correspondingto the Uehling massive Euler-Heisenberg-like and Euler-Heisenberg electrodynamics in the approximation of thestrong-field limit (to the leading logarithmic order) and fortheir noncommutative versions Our results show a long-range 11198715-type correction to the Coulomb potential for bothmassive Euler-Heisenberg-like and Euler-Heisenberg elec-trodynamics in the approximation of the strong-field limit(to the leading logarithmic order) Interestingly enough formassive Euler-Heisenbeg-like electrodynamics (Wichmann-Kroll) we obtain a new long-range 1119871

3 correction tothe interaction energy Nevertheless for their noncom-mutative versions the static potential becomes ultravioletfinite

The organization of the paper is as follows In Section 2we reexamine Uehling electrodynamics in order to establisha framework for the computation of the static potential InSection 3 we consider Euler-Heisenberg-like (with a massterm) electrodynamics and show that it yields birefringencecomputing the interaction energy for a fermion-antifermionpair and its version in the presence of a minimal lengthIn Section 4 we repeat our analysis for Euler-Heisenbergelectrodynamics in the approximation of the strong-fieldlimit Finally in Section 5 we cast our final remarks

In our conventions the signature of the metric is(+1 minus1 minus1 minus1)

2 Brief Review on the Uehling Potential

As already expressed we now reexamine the interactionenergy for Maxwell theory with an additional term corre-sponding to the Uehling correction (Uehling electrodynam-ics)This would not only provide the setup theoretical for oursubsequent work but also fix the notation To do that we willcalculate the expectation value of the energy operator 119867 inthe physical state |Φ⟩ which we will denote by ⟨119867⟩

Φ We

start off our analysis by considering the effective Lagrangiandensity [48]

L = minus1

4119865120583] (1 minus

120572

3120587ΔM)119865

120583] (1)

where

M (119898 119909) = int

infin

41198982

1198891205911

120591 (120591 + Δ)(1 +

21198982

120591)radic1 minus

41198982

120591 (2)

with Δ equiv 120597120583120597120583 It should be noted thatM contains the effect

of vacuum polarization to first order in the fine structureconstant 120572 = 119890

2ℏ119888 and 119898 is the electron mass We may

parenthetically note here that the presence of Δ in (2) doesnot offer problems On the one hand as we will explain belowwe restrict ourselves to the static case On the other hand inorder to compute the interaction energy wewill make a seriesexpansion at leading order in 120572

Before going on two remarks are pertinent at this pointFirst the modification of Coulombrsquos law in (1) follows fromthe weak-field limit of the one-loop effective action ofquantum electrodynamics (QED) Indeed as was explainedin [48] this modification can be written as

Lwf =1

2int119860

120583(119909)Π

120583] (119909 119910)119860](119910) 119889

4119909119889

4119910 (3)

where wf denotes weak field and Π120583] is the usual order-119890

2

polarization tensor of QED As is well known in momentumspace this tensor is given by

Π120583] (119896) = (119896

2119892120583] minus 119896

120583119896])Π (119896

2) (4)

while the momentum space spectral representation of thepolarization function Π(119896

2) reads

Π(1198962) = minus

120572

31205871198962int

infin

41198982

119889119905120588 (119905)

119905

1

1198962+ 119905

(5)

with 120588(119905) = (1 + 21198982119905)radic1 minus 4119898

2119905 Next due to the tensor

structure of Π120583](119896) Lwf can then be expressed in a gauge-

invariantway that is in terms of119865120583] As a consequence of this

(3) reduces to the modification of Coulombrsquos law appearingin (1) We mention in passing that in [48] the signature of themetric is different from that used in this paper

Second it should be noted that the theory described by(1) contains higher time derivatives hence to construct theHamiltonian one must use for example the Ostrogradskymethod [49] Accordingly in the theory under considerationthe velocities have to be taken as independent canonical

Advances in High Energy Physics 3

variables Let us also mention here that in previous studies[50 51] we have shown that although theories like (1) containhigher derivatives in the electrostatic case the canonicalmomentum conjugate to velocities disappears Hence thenew Legendre transformation to construct the Hamiltonianreduces to the standard Legendre transformation It shouldhowever be emphasized here that the present paper is aimedat studying the static potential of the above theory sothat Δ can be replaced by minusnabla

2 Notice that for notationalconvenience we have maintained Δ in (1) and (2) but itshould be borne inmind that this paper essentially deals withthe static case

Now we move on to compute the canonical Hamilto-nian For this end we perform a Hamiltonian constraintanalysis The canonical momenta are found to be Π

120583=

(1 minus (1205723120587)ΔM)1198651205830 It is easy to see that Π0 vanishes we

then have the usual constraint equation which accordingto Diracrsquos theory is written as a weak (asymp) equation that isΠ

0asymp 0 Itmay be noted that the remaining nonzeromomenta

must also be written as weak equations This leads to Π119894asymp

(1minus(1205723120587)ΔM)119864119894 (with119864

119894= 119865

1198940) Accordingly the canonical

Hamiltonian119867119862is

119867119862asymp int119889

3119909 [Π

119894120597119894119860

0minus1

119894(1 minus

120572

3120587ΔM)

minus1

Π119894

+1

4119865119894119895(1 minus

120572

3120587ΔM)119865

119894119895]

(6)

which must also be written as a weak equation Next theprimary constraint Π0

asymp 0 must be satisfied for all timesAn immediate consequence of this is that using the equationof motion asymp [119885119867

119862] we obtain the secondary constraint

Γ1

equiv 120597119894Π

119894asymp 0 which must also be true for all time In

passingwe recall that we are considering the static case hencethis new constraint does not contain time derivatives It isstraightforward to check that there are no further constraintsin the theory Therefore in the case under considerationthere are two constraints which are first class Accordingto the general theory we obtain the extended Hamiltonianas an ordinary (or strong) equation by adding all the first-class constraints with arbitrary constraints We thus write119867 = 119867

119862+ int119889

3119909(119906

0(119909)Π

0(119909) + 119906

1(119909)Γ

1(119909)) where 119906

0(119909) and

1199061(119909) are arbitrary Lagrange multipliers It is also important

to observe that when this new Hamiltonian is employed theequation of motion of a dynamic variable may be written as astrong equation With the aid of (6) we find that

0(119909) =

[1198600(119909)119867] = 119906

0(119909) which is an arbitrary function Since

Π0asymp 0 always neither1198600 norΠ0 are of interest in describing

the system andmay be discarded from the theory In fact theterm containing 119860

0is redundant because it can be absorbed

by redefining the function 119908(119909) Therefore the Hamiltonianis now given as

119867 = int1198893119909 [119908 (119909) 120597

119894Π

119894minus1

119894(1 minus

120572

3120587ΔM)

minus1

Π119894

+1

4119865119894119895(1 minus

120572

3120587ΔM)119865

119894119895]

(7)

where 119908(119909) = 1199061(119909) minus 119860

0(119909)

It must be clear from this discussion that the presenceof the new arbitrary function 119908(119909) is undesirable since wehave no way of giving it a meaning in a quantum theoryHence according to the usual procedure we impose a gaugecondition such that the full set of constraints becomes secondclass A convenient choice is

Γ2(119909) equiv int

119862120585119909

119889119911]119860] (119911) equiv int

1

0

119889120582119909119894119860

119894(120582119909) = 0 (8)

where 120582 (0 le 120582 le 1) is the parameter describing the space-like straight path 119911

119894= 120585

119894+ 120582(119909 minus 120585)

119894 and 120585 is a fixed point(reference point) There is no essential loss of generality if werestrict our considerations to 120585

119894= 0 The Dirac brackets can

now be determined and we simply note the only nontrivialDirac bracket involving the canonical variables that is

119860119894(119909) Π

119895(119910)

lowast

= 120575119895

119894120575(3)

(119909 minus 119910)

minus 120597119909

119894int

1

0

119889120582119909119894120575(3)

(120582119909 minus 119910)

(9)

In passing we also recall that the transition to a quantumtheory is made by the replacement of the Dirac brackets bythe operator commutation relations according to 119860 119861

lowastrarr

(119894ℏ)[119860 119861]With the foregoing information we can now proceed to

obtain the interaction energy As alreadymentioned in orderto accomplish this purpose we will calculate the expectationvalue of the energy operator119867 in the physical state |Φ⟩ wherethe physical states |Φ⟩ are gauge-invariant ones The physicalstate can be written as

|Φ⟩ equiv10038161003816100381610038161003816Ψ (y) Ψ (y1015840)⟩

= 120595 (y) exp(119894119902ℏint

y

y1015840119889119911

119894119860

119894(119911))120595 (y1015840) |0⟩

(10)

where |0⟩ is the physical vacuum state and the line integralappearing in the above expression is along a space-like pathstarting at y1015840 and ending at y on a fixed time slice The pointwe wish to emphasize however is that the physical fermion(Ψ(y)) is not the Lagrangian fermion (120595(y)) which is neithergauge-invariant nor associated with an electric field In factthe physical fermion is the Lagrangian fermion together witha cloud (or dressing) of gauge fields

Making use of the above Hamiltonian structure [18] wefind that

Π119894 (119909)

10038161003816100381610038161003816Ψ (y) Ψ (y1015840)⟩ = Ψ (y) Ψ (y1015840)Π

119894 (119909) |0⟩

+ 119902int

y1015840

y119889119911

119894120575(3)

(z minus x) |Φ⟩ (11)

With the aid of (11) and (7) the lowest-order modification in120572 of the interaction energy takes the form

⟨119867⟩Φ = ⟨119867⟩0 + 1198811+ 119881

2 (12)

4 Advances in High Energy Physics

where ⟨119867⟩0= ⟨0|119867|0⟩ The 119881

1 119881

2terms are given by

1198811=1199022

2int119889

3119909int

y

y1015840119889119911

119894120575(3)

(x minus z) inty

y1015840119889119911

1015840119894120575(3)

(x minus z1015840) (13)

1198812=1199022

2

120572

3120587int

infin

41198982

119889120591120588 (120591)

120591int 119889

3119909

sdot int

y1015840

y119889119911

1015840

119894120575(3)

(z1015840 minus x) nabla2

119909

1

120591 minus nabla2

119909

int

y1015840

y119889119911

119894120575(3)

(z minus x) (14)

where 120588(120591) = (1 + 21198982120591)radic1 minus 4119898

2120591

We note that term (13) may look peculiar but it is nothingbut the familiar Coulomb interaction plus a self-energy term[52] Now making use of the Green function 119866(z z1015840) =

(14120587)(119890minusradic120591|zminusz1015840|

|z minus z1015840|) the term (14) can be rewritten inthe form

1198812=1199022

2

120572

3120587int

infin

41198982

119889120591120588 (120591)

120591int

y1015840

y119889119911

1015840

119894nabla

2

1199111015840 int

y1015840

y119889119911

119894119866(z z1015840)

= minus120572

3120587

1199022

2int

infin

41198982

119889120591120588 (120591)

120591

119890minusradic120591|yminusy1015840|

1003816100381610038161003816y minus y10158401003816100381610038161003816

(15)

Since the second and third term on the right-hand sideof (12) are clearly dependent on the distance between theexternal static fields the potential for two opposite chargeslocated at y and y1015840 reads

119881 = minus1199022

4120587

1

119871(1 +

120572

3120587int

infin

41198982

119889120591120588 (120591)

120591119890minusradic120591119871

) (16)

where 119871 = |y minus y1015840| Accordingly one recovers the knownUehling potential which finds here an entirely differentderivation

Beforewe proceed further wewish to show that this resultcan be written alternatively in a more explicit form Makinguse of [53]

120594119899(119911) = int

infin

1

119889119905119890minus119905119911

119905119899

(1 +1

21199052)radic1 minus

1

1199052 (17)

we then get

119881 = minus1199022

4120587

1

119871(1 +

2120572

31205871205941 (2119898119871)) (18)

By the transformation 119905 = cosh 119906 [54] the functions 120594119899can

be reduced to the form [53]

120594119899(119911) = 119870119894

119899minus1(119911) minus

1

2119870119894

119899+1(119911) minus

1

2119870119894

119899+3(119911) (19)

where the functions 119870119894 denote Bessel function integralsHence we see that the interaction energy (with 119898 = 1)becomes

119881 = minus1199022

4120587

1

1198711 +

2120572

3120587[(1 +

1198712

3)119870

0 (2119871)

minus (5119871

3+2119871

3

3)119870

1(2119871)

+ (3119871

2+2119871

3

3)int

infin

2119871

1198891199051198700 (119905)]

(20)

where1198700(119911) and119870

1(119911) aremodified Bessel functions Finally

with the aid of asymptotic forms for Bessel functions it is asimple matter to find expressions for 119881 for large and small 119871

Before concluding this subsection we discuss an alterna-tive way of stating our previous result (16) which displayscertain distinctive features of our methodology We start byconsidering [52 55]

119881 equiv 119902 (A0(0) minusA

0(L)) (21)

where the physical scalar potential is given by

A0(119905 r) = int

1

0

119889120582119903119894119864

119894(119905 120582r) (22)

This follows from the vector gauge-invariant field expression

A120583 (119909) equiv 119860

120583 (119909) + 120597120583(minusint

119909

120585

119889119911120583119860

120583 (119911)) (23)

where the line integral is along a space-like path from thepoint 120585 to 119909 on a fixed slice time It is also important toobserve that the gauge-invariant variables (22) commute withthe sole first constraint (Gauss law) showing in this waythat these fields are physical variables In as much as we areinterested in estimating the lowest-order correction to theCoulomb energy we will retain only the leading term inexpression 119864

119894= (1 minus (1205723120587)ΔM)

minus1Π

119894 Making use of this lastexpression (22) gives

A0(119905 r)

= int

1

0

119889120582119903119894120597120582r119894(minus

1198690(120582r)nabla

2

120582r)

+120572

3120587int

infin

41198982

119889120591

120591120588 (120591) int

1

0

119889120582119903119894120597120582r119894(minus

1198690(120582r)

nabla2

120582r minus 120591)

(24)

to get the last line we used Gauss law for the present theorythat is 120597

119894Π

119894= 119869

0 (where we have included the externalcurrent 1198690 to represent the presence of two opposite charges)Accordingly for 1198690(119905 r) = 119902120575

(3)(r) the potential for a pair of

static point-like opposite charges located at 0 and L is givenby

119881 = minus1199022

4120587

1

119871(1 +

120572

3120587int

infin

41198982

119889120591120588 (120591)

120591119890minusradic120591119871

) (25)

after subtracting a self-energy term

Advances in High Energy Physics 5

3 Euler-Heisenberg-Like Model

Proceeding in the sameway aswe did in the foregoing sectionwe will now consider the interaction energy for Euler-Heisenberg-like electrodynamics Nevertheless in order toput our discussion into context it is useful to describe verybriefly the model under consideration In such a case theLagrangian density reads

L =1205732

21 minus [1 +

1

1205732F minus

1

12057321205742G

2]

119901

(26)

where we have included two parameters 120573 and 120574 As usualF = (14)119865

120583]119865120583] G = (14)119865

120583]119865120583] 119865

120583] = 120597120583119860] minus 120597]119860120583

and 119865

120583]= (12)120576

120583]120588120582119865120588120582 Let us also mention here that in

our previous paper [19] we have studied the domain 0 lt

119901 lt 1 Moreover it follows from (26) that when 119901 = 2

the model contains to orders O(11205732) and O(11205742

) a Euler-Heisenberg-likemodel with the appropriate identifications ofthe constants Interestingly we also observe that in the limit120574 rarr infin we obtain a Wichmann-Kroll model This remarkopens up the way to discuss the effect of these nonlinearcorrections on the interaction energy as we are going tostudy below In fact we will consider a massive Wichmann-Kroll systemThemotivation for this study comes from recentconsiderations in the context of dualities [56] where massiveBorn-Infeld systems play an important role

Having made these observations we can write immedi-ately the field equations for 119901 = 2

120597120583[Γ(119865

120583]minus

2

1205742G119865

120583])] = 0 (27)

while the Bianchi identities are given by

120597120583119865

120583]= 0 (28)

where

Γ = 1 +F

1205732minus

G2

12057321205742 (29)

Also it is straightforward to see that Gauss law becomes

nabla sdotD = 0 (30)

whereD is given by

D

= [1 minus

(E2minus B2

)

21205732

minus(E sdot B)2

12057321205742

](E +2

1205742(E sdot B)B)

(31)

Again from (30) for 1198690(119905 r) = 119890120575(3)(r) we findD = (119876119903

2)119903

where 119876 = 1198904120587 This then implies that for a point-likecharge 119890 at the origin the expression

119876

1199032= (1 minus

E2

21205732) |E| (32)

tells us that for 119903 rarr 0 the electrostatic field becomessingular at 119903 = 0 in contrast to the 0 lt 119901 lt 1 case wherethe electrostatic field is finite Even so in this theory thephenomenon of birefringence is present Before going intodetails we would like to recall that birefringence refers to theproperty that polarized light in a particular direction (opticalaxis) travels at a different velocity from that of light polarizedin a direction perpendicular to this axis Indeed due toquantum fluctuations the QED vacuum has this property aswe are going to show

To illustrate this important feature we introduce thevectorsD = 120597L120597E andH = minus120597L120597B

D = Γ(E + 2B (E sdot B)

1205742

)

H = Γ(B minus 2E (E sdot B)

1205742

)

(33)

where Γ = 1 + (121205732)(B2

minus E2) minus (1120573

21205742)(E sdot B)2 We thus

obtain the equations of motion

nabla sdotD = 0

120597D120597119905

minus nabla timesH = 0

nabla sdot B = 0

120597B120597119905

+ nabla times E = 0

(34)

With the aid from (33) we find the electric permittivity 120576119894119895

and the inverse magnetic permeability (120583minus1)119894119895 tensors of the

vacuum that is

120576119894119895= Γ(120575

119894119895+

2119861119894119861119895

1205742

)

(120583minus1)119894119895= Γ(120575

119894119895minus

2119864119894119864

119895

1205742

)

(35)

with119863119894= 120576

119894119895119864

119895and 119861

119894= 120583

119894119895119867

119895

In accordance with our previous procedure [18 19] wecan now linearize the above equations To do this it isadvantageous to introduce a weak electromagnetic wave(E

119901B

119901) propagating in the presence of a strong constant

external field (E0B

0) On these assumptions we readily find

that for the case of a purely magnetic field (E0= 0) the

vectorsD andH become

D = (1 +B2

0

21205732)[E

119901+

2

1205742(E

119901sdot B

0)B

0]

H = (1 +B2

0

21205732)

sdot [B119901+

1

1205732(1 + B2

02120573

2)(B

119901sdot B

0)B

0]

(36)

where we have to keep only linear terms in E119901B

119901 As

before we consider the 119911-axis as the direction of the external

6 Advances in High Energy Physics

magnetic field (B0= 119861

0e3) and assuming that the light wave

moves along the 119909-axis the decomposition into a plane wavefor the fields E

119901and B

119901can be written as

E119901 (x 119905) = E119890minus119894(119908119905minusksdotx)

B119901(x 119905) = B119890minus119894(119908119905minusksdotx)

(37)

In this case it clearly follows that

(1198962

1199082minus 120576

2212058333)119864

2= 0 (38)

(1198962

1199082minus 120576

3312058322)119864

3= 0 (39)

As a consequence we have two different situations Firstif E perp B

0(perpendicular polarization) from (39) 119864

3= 0 and

from (38) we get 1198962119908

2= 120576

2212058333 This then means that the

dispersion relation of the photon takes the form

119899perp= radic

1 + B2

02120573

2

1 + 3B2

02120573

2 (40)

Second if E B0(parallel polarization) from (38) 119864

2= 0

and from (39) we get 1198962119908

2= 120576

3312058322 This leads to

119899= radic1 +

2B2

0

1205742 (41)

Thus we verify that in the case of a generalized Euler-Heisenberg electrodynamics the phenomenon of birefrin-gence is present

We now pass to the calculation of the interaction energybetween static point-like sources for a massive Wichmann-Kroll-like model our analysis follows closely that of [18 19]The corresponding theory is governed by the Lagrangiandensity

L = minus1

4119865

2

120583] +1

321205732(119865

120583]119865120583])2

+119898

2

2119860

120583119860

120583 (42)

Next in order to handle the second term on the right hand in(42) we introduce an auxiliary field 120585 such that its equation ofmotion gives back the original theory This allows us to writethe Lagrangian density as

L = minus1

4119865120583]119865

120583]+

120585

321205732119865120583]119865

120583]minus

1

12812057321205852

+119898

2

2119860

120583119860

120583

(43)

With the redefinition 120578 = 1 minus 12058581205732 (43) becomes

L = minus1

4120578119865

120583]119865120583]minus1

2(1 minus 120578)

2+119898

2

2119860

120583119860

120583 (44)

Before we proceed to work out explicitly the interactionenergy we will first restore the gauge invariance in (44)

Following an earlier procedure we readily verify that thecanonical momenta read Π

120583= minus120578119865

0120583 which results in theusual primary constraint Π0

= 0 and Π119894= 120578119865

1198940 In this wayone obtains

119867119862= int119889

3119909Π

119894120597119894119860

0+

1

2120578Π

2+120578

2B2

minus119898

2

2119860

120583119860

120583

+1

2(1 minus 120578)

2

(45)

The consistency condition Π = 0 leads to the constraint Γ equiv

120597119894Π

119894+119898

2119860

0= 0 As a result both constraints are second-class

To convert the second-class system into first-class we willadopt the procedure described previously Thus we enlargethe original phase space by introducing a canonical pair offields 120579 and Π

120579 It follows therefore that a new set of first-

class constraints can be defined in this extended space Λ1equiv

Π0+ 119898

2120579 = 0 and Λ

2equiv Γ + Π

120579= 0 Notice that this new

120579-field is not to be confused with the usual noncommutativeparameter This then shows that the new constraints arefirst-class and therefore restore the gauge symmetry As iswell known this procedure reproduces the usual Stuckelbergformalism From this the new effective Lagrangian densityafter integrating out the 120579 fields becomes

L = minus1

4119865120583] (120578 +

1198982

Δ)119865

120583]minus1

2(1 minus 120578)

2 (46)

Now writing 120590 = 120578 + 1198982Δ expression (46) can be brought

to the form

L = minus1

4119865120583]120590119865

120583]minus

119896

128(1 minus 120590 +

1198982

Δ)

2

(47)

where 119896 = 641205732

We are now ready to compute the interaction energy Inthis case the canonical momenta are Π120583

= minus1205901198650120583 with the

usual primary constraint Π0= 0 and Π

119894= 120590119865

1198940 Hence thecanonical Hamiltonian is expressed as

119867119862= int119889

3119909Π

119894120597119894119860

0+

1

2120590Π

2+120590

2B2

+119896

128(1 minus 120590 +

1198982

Δ)

2

(48)

Time conservation of the primary constraint Π0 yields thesecondary constraint Γ

1equiv 120597

119894Π

119894= 0 Similarly the P

120590

constraint yields no further constraints and just determinesthe field 120590 In this case at leading order in 120573 the field 120590 isgiven by

120590 = (1 +119898

2

Δminus

B2

21205732)

sdot [1 minus3

21205732

1

(1 + 1198982Δ minus B2

21205732)3Π

2]

(49)

Advances in High Energy Physics 7

which will be used to eliminate 120590 As before the corre-sponding total (first-class) Hamiltonian that generates thetime evolution of the dynamical variables is 119867 = 119867

119862+

int1198893119909(119906

0(119909)Π

0(119909) + 119906

1(119909)Γ

1(119909)) where 119906

0(119909) and 119906

1(119909) are

the Lagrangemultiplier utilized to implement the constraintsIn the same way as was done in the previous subsection

the expectation value of the energy operator119867 in the physicalstate |Φ⟩ becomes

⟨119867⟩Φ = ⟨Φ|int1198893119909

1

119894(1 +

1198982

Δ)

minus1

Π119894+

15

81205732Π

4

minus15119898

2

21205732Π

2 1

ΔΠ

2 |Φ⟩

(50)

in this last line we have considered only quadratic terms in119898

2In such a case by employing (50) the lowest-order

modification in 1205732 and1198982 of the interaction energy takes theform

⟨119867⟩Φ = ⟨119867⟩0 + 1198811+ 119881

2+ 119881

3 (51)

where ⟨119867⟩0= ⟨0|119867|0⟩ The 119881

1 119881

2 and 119881

3terms are given by

1198811=1199022

2int119889

3119909int

y

y1015840119889119911

1015840119894120575(3)

(x minus z1015840)(1 minus 1198982

nabla2)

minus1

sdot int

y

y1015840119889119911

119894120575(3)

(x minus z)

1198812= minus

151199024

81205732int119889

3119909int

y

y1015840119889119911

119894120575(3)

(x minus z)

sdot int

y

y1015840119889119911

1015840119894120575(3)

(x minus z1015840)inty

y1015840119889119906

119896120575(3)

(x minus u)

sdot int

y

y1015840119889V

119896120575(3)

(x minus k)

1198813=15119898

21199024

21205732

int1198893119909int

y

y1015840119889119911

119894120575(3)

(x minus z)

sdot int

y

y1015840119889119911

1015840119894120575(3)

(x minus z1015840)inty

y1015840119889119906

119896120575(3)

(x minus u)

sdot int

y

y1015840119889V

119896120575(3)

(x minus k)

(52)

Finally with the aid of expressions (52) the potential fora pair of static point-like opposite charges located at 0 and Lis given by

119881 = minus1199022

4120587

119890minus119898119871

119871+

1199024

161205871205732(3

8120587

1

1198712minus 5119898

2)

1

1198713 (53)

observe that when 119898 = 0 profile (53) reduces to the knownWichmann-Kroll interaction energy On the other hand for119898 = 0 the key role played by the mass term in transformingthe Coulomb potential into the Yukawa one should be notedInterestingly enough an unexpected feature is found In factprofile (53) displays a new long-range 11198713 correction whereits strength is proportional to 119898

2 It is also important toobserve that an analogous correction has been found in Born-Infeld electrodynamics in the context of very special relativity[57] In this way we establish a new connection betweennonlinear effectives theories

Before we proceed further we should comment on ourresult In the case of QED (Euler-Heisenberg Lagrangiandensity) the parameter 11205732 is given by 11205732

= (1645)(1198904ℏ

1198984

1198901198887) where 119898

119890is the electron mass In this context we

also recall the currently accepted upper limit for the photonmass that is 119898

120574sim 2 times 10

minus16 eV Thus for the QED casefrom (53) it follows that the second term on the right handside would be detectable in long-range distances (sim109 m) Inother words we see that detectable corrections induced byvacuum polarization with a mass term would be present atlow energy scales

From (53) it clearly follows that the interaction energybetween heavy charged charges at leading order in 120573 is notfinite at the origin Motivated by this one may consider theabove calculation in a noncommutative geometry based onfindings of our previous studies [18 19] In such a case theelectric field at leading order in 120573

2 and1198982 takes the form

119864119894= [(1 +

1198982

Δ)

minus1

+3

21205732Π

2minus6119898

2

1205732Π

2 1

Δ]

sdot 120597119894(minus

119890120579nabla2

120575(3)

(x)nabla

2)

(54)

where it may be recalled that we are now replacing the source120575(3)(x minus y) by the smeared source 119890

120579nabla2

120575(3)(x minus y) with the

parameter 120579 being noncommutative Now making use of(22) we readily find that

A0(119905 r) = A

(1)

0(119905 r) +A

(2)

0(119905 r) +A

(3)

0(119905 r) (55)

The term A(1)

0(119905 r) was first calculated in [50] we can

therefore write only the result

A(1)

0(119905 r)

= 1199021198901198982120579

4120587

1

119903[119890

minus119898119903minus

1

radic120587int

infin

11990324120579

1198891199061

radic119906119890minus119906minus119898

211990324119906

]

minus 119902119898

41205871198901198982120579

(56)

8 Advances in High Energy Physics

Meanwhile the terms A(2)

0(119905 r) and A

(3)

0(119905 r) after some

manipulation can be brought to the form

A(2)

0(119905 r) =

121199023

120573212058732

119899119894int

119909

0

119889119906119894 1

11990661205743(3

21199062

4120579)

A(3)

0(119905 r) =

311989821199023

120573212058752

119899119894int

119909

0

119889119906119894 1

11990641205742(3

21199062

4120579)

sdot [4120579

1199062120574(

3

21199062

4120579) minus 120574(

1

21199062

4120579)]

(57)

where 120574(32 11990324120579) is the lower incomplete Gamma functiondefined by 120574(119886119887 119909) equiv int

119909

0(119889119906119906)119906

119886119887119890minus119906

Inserting these expressions in (21) we finally obtain thestatic potential for two opposite charges 119902 located at 0 and Las

119881 = minus119902

4120587

1198901198982120579

119871[119890

minus119898119871minus

1

radic120587int

infin

11987124120579

1198891199061

radic119906119890minus119906minus119898

211990624119906

]

minus12119902

4

120573212058732

119899119894int

119871

0

119889119906119894 1

11990661205743(3

21199062

4120579) minus

311989821199024

120573212058752

sdot 119899119894int

119909

0

119889119906119894 1

11990641205742(3

21199062

4120579)

sdot [4120579

1199062120574(

3

21199062

4120579) minus 120574(

1

21199062

4120579)]

(58)

which is finite for 119871 rarr 0 It is a simple matter to verify thatin the limit 120579 rarr 0 we recover our above result

4 Logarithmic Correction

We now want to extend what we have done to Euler-Heisenberg-like electrodynamics at strong fields As alreadymentioned such theories show a power behavior that istypical for critical phenomena [58] In such a case theLagrangian density reads

L = minus1

4119865120583]119865

120583]minus1198870

8119865120583]119865

120583] log(119865120583]119865

120583]

41205822

) (59)

where 1198870and 120582 are constants In fact by choosing 119887

0=

11989026120587

2 and 120582 = 1198982

1198901198883119890ℏ we recover the Euler-Heisenberg

electrodynamics at strong fields [58]In the same way as was done in the previous section one

can introduce an auxiliary field 120585 to handle the logarithm in(59) This leads to

L = minus1

41205721119865120583]119865

120583]minus 120572

2(119865

120583]119865120583])2

(60)

where 1205721= 1 minus (119887

02)(1 + log 120585) and 120572

2= 119887

012058532120582

2A similar procedure can be used to manipulate the

quadratic term in (60) Accordingly by introducing a secondauxiliary field 120578 one easily finds

L = minus1

4119865120583] (1205721

+ 41205722120578) 119865

120583]+1205782

41205722 (61)

By setting 120590 = 1205721+ 4120572

2120578 we then have

L = minus1

4120590119865

120583]119865120583]+

1

641205722

(120590 minus 1205721)2 (62)

It is once again straightforward to apply the gauge-invariant formalism discussed in the foregoing section Thecanonical momenta read Π

120583= minus120590119865

0120583 and at once werecognize the two primary constraints Π0

= 0 and P120590

equiv

120597119871120597 = 0The canonical Hamiltonian corresponding to (62)is

119867119862= int119889

3119909

sdot Π119894120597119894119860

0+

1

2120590Π

2+120590

2B2

minus1

641205722

(120590 minus 1205721)2

(63)

Requiring the primary constraint Π0 to be preserved intime one obtains the secondary constraint Γ

1= 120597

119894Π

119894= 0 In

the same way for the constraintP120590 we get the auxiliary field

120590 as

120590 = (1 minus1198870

2(1 + ln 120585) +

1198870B2

21205822120585)[1

+3119887

0B2

21205822

120585

(1 minus (11988702) (1 + ln 120585) + (119887

0B2

21205822) 120585)

3]

(64)

Hence we obtain

119867119862= int119889

3119909

sdot Π119894120597119894119860

0+1

2+1198870

4(1 + log 120585)Π2 minus

31198870120585

21205822Π

4

(65)

As before requiring the primary constraint P120585to be

preserved in time one obtains the auxiliary field 120585 In this case120585 = 1205826Π

2 Consequently we get

119867119862= int119889

3119909Π

119894120597119894119860

0+1

2(1 + 119887

0)Π

2minus6119887

0

1205822Π

4 (66)

Following the same steps that led to (50) we find that

⟨119867⟩(1)

Φ= ⟨Φ|int119889

3119909

1

2minus

3

81205732Π

4 |Φ⟩ (67)

It should be noted that this expression is similar to (50) in thelimit 119898 rarr 0 except by the changed sign in front of the Π4termHencewe see that the potential for two opposite chargesin 0 and L is given by

119881 = minus1199022

4120587

1

119871minus

1199024

604012057321205872

1

1198715 (68)

5 Final Remarks

Finally within the gauge-invariant but path-dependent vari-ables formalism we have considered the confinement versus

Advances in High Energy Physics 9

screening issue for both massive Euler-Heisenberg-like andEuler-Heisenberg electrodynamics in the approximation ofthe strong-field limit Once again a correct identificationof physical degrees of freedom has been fundamental forunderstanding the physics hidden in gauge theories Inter-estingly enough their noncommutative version displays anultraviolet finite static potential The analysis above revealsthe key role played by the new quantum of length in ouranalysis In a general perspective the benefit of consideringthe present approach is to provide a unification scenarioamong different models as well as exploiting the equivalencein explicit calculations as we have illustrated in the course ofthis work

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

It is a pleasure for the author to thank J A Helayel-Netofor helpful comments on the paper This work was partiallysupported by Fondecyt (Chile) Grant 1130426 and DGIP(UTFSM) internal project USM 111458

References

[1] G Breit and J A Wheeler ldquoCollision of two light quantardquoPhysical Review vol 46 no 12 pp 1087ndash1091 1934

[2] S L Adler ldquoPhoton splitting and photon dispersion in a strongmagnetic fieldrdquo Annals of Physics vol 67 no 2 pp 599ndash6471971

[3] V Costantini B De Tollis and G Pistoni ldquoNonlinear effects inquantum electrodynamicsrdquo Il Nuovo Cimento A vol 2 no 3pp 733ndash787 1971

[4] R Ruffini G Vereshchagin and S-S Xue ldquoElectron-positronpairs in physics and astrophysics from heavy nuclei to blackholesrdquo Physics Reports vol 487 no 1ndash4 pp 1ndash140 2010

[5] G V Dunne ldquoThe Heisenberg-Euler effective action 75 yearsonrdquo International Journal of Modern Physics Conference Seriesvol 14 pp 42ndash56 2012

[6] C Bamber S J Boege TKoffas et al ldquoStudies of nonlinearQEDin collisions of 466GeV electrons with intense laser pulsesrdquoPhysical Review D vol 60 no 9 Article ID 092004 1999

[7] D L Burke R C Field G Horton-Smith et al ldquoPositronproduction in multiphoton light-by-light scatteringrdquo PhysicalReview Letters vol 79 no 9 article 1626 1997

[8] O J Pike F MacKenroth E G Hill and S J Rose ldquoA photonndashphoton collider in a vacuum hohlraumrdquo Nature Photonics vol8 no 6 pp 434ndash436 2014

[9] D Tommasini A Ferrando H Michinel and M Seco ldquoPre-cision tests of QED and non-standard models by searchingphoton-photon scattering in vacuum with high power lasersrdquoJournal ofHigh Energy Physics vol 2009 no 11 article 043 2009

[10] D Tommasini A Ferrando HMichinel andM Seco ldquoDetect-ing photon-photon scattering in vacuum at exawatt lasersrdquoPhysical Review A vol 77 no 4 Article ID 042101 2008

[11] M Born and L Infeld ldquoFoundations of the new field theoryrdquoProceedings of the Royal Society of London Series A Containing

Papers of a Mathematical and Physical Character vol 144 no852 pp 425ndash451 1934

[12] H Gies J Jaeckel and A Ringwald ldquoPolarized light propagat-ing in a magnetic field as a probe for millicharged fermionsrdquoPhysical Review Letters vol 97 no 14 Article ID 140402 2006

[13] E Masso and R Toldra ldquoLight spinless particle coupled tophotonsrdquo Physical Review D vol 52 no 4 pp 1755ndash1763 1995

[14] P Gaete and E I Guendelman ldquoConfinement in the presenceof external fields and axionsrdquoModern Physics Letters A vol 20no 5 article 319 2005

[15] P Gaete and E Spallucci ldquoConfinement effects from interactingchromo-magnetic and axion fieldsrdquo Journal of Physics A Math-ematical and General vol 39 no 20 pp 6021ndash6029 2006

[16] E S Fradkin and A A Tseytlin ldquoNon-linear electrodynamicsfrom quantized stringsrdquo Physics Letters B vol 163 no 1ndash4 pp123ndash130 1985

[17] E Bergshoeff E Sezgin C N Pope and P K TownsendldquoThe Born-Infeld action from conformal invariance of the opensuperstringrdquo Physics Letters B vol 188 no 1 pp 70ndash74 1987

[18] P Gaete and J Helayel-Neto ldquoFinite field-energy and interpar-ticle potential in logarithmic electrodynamicsrdquo The EuropeanPhysical Journal C vol 74 no 3 article 2816 2014

[19] P Gaete and J Helayel-Neto ldquoRemarks on nonlinear electrody-namicsrdquo European Physical Journal C vol 74 no 11 article 31822014

[20] S H Hendi ldquoAsymptotic ReissnerndashNordstrom black holesrdquoAnnals of Physics vol 333 pp 282ndash289 2013

[21] Z Zhao Q Pan S Chen and J Jing ldquoNotes on holographicsuperconductor models with the nonlinear electrodynamicsrdquoNuclear Physics B vol 871 no 1 pp 98ndash110 2013

[22] O Miskovic and R Olea ldquoConserved charges for black holes inEinstein-Gauss-Bonnet gravity coupled to nonlinear electrody-namics in AdS spacerdquo Physical Review D vol 83 no 2 ArticleID 024011 2011

[23] S H Mazharimousavi and M Halilsoy ldquoBlack holes and theclassical model of a particle in Einstein non-linear electrody-namics theoryrdquo Physics Letters B vol 678 no 4 pp 407ndash4102009

[24] G Amelino-Camelia ldquoSpecial treatmentrdquo Nature vol 418 no6893 pp 34ndash35 2002

[25] T Jacobson S Liberati andDMattingly ldquoThreshold effects andPlanck scale Lorentz violation combined constraints from highenergy astrophysicsrdquo Physical Review D vol 67 no 12 ArticleID 124011 2003

[26] T J Konopka and S A Major ldquoObservational limits onquantum geometry effectsrdquo New Journal of Physics vol 4 pp571ndash5718 2002

[27] SHossenfelder ldquoInterpretation of quantumfield theories with aminimal length scalerdquo Physical Review D vol 73 no 10 ArticleID 105013 9 pages 2006

[28] P Nicolini ldquoNoncommutative black holes the final appeal toquantum gravity a reviewrdquo International Journal of ModernPhysics A vol 24 no 7 pp 1229ndash1308 2009

[29] S Pramanik S Ghosh and P Pal ldquoElectrodynamics of a gener-alized charged particle in doubly special relativity frameworkrdquoAnnals of Physics vol 346 pp 113ndash128 2014

[30] A F Ali S Das and E C Vagenas ldquoDiscreteness of space fromthe generalized uncertainty principlerdquo Physics Letters B vol678 no 5 pp 497ndash499 2009

[31] G Amelino-Camelia ldquoQuantum-spacetime phenomenologyrdquoLiving Reviews in Relativity vol 16 article 5 2013

10 Advances in High Energy Physics

[32] E Witten ldquoNoncommutative geometry and string field theoryrdquoNuclear Physics B vol 268 no 2 pp 253ndash294 1986

[33] N Seiberg and E Witten ldquoString theory and noncommutativegeometryrdquo Journal of High Energy Physics vol 1999 article 0321999

[34] M R Douglas and N A Nekrasov ldquoNoncommutative fieldtheoryrdquo Reviews of Modern Physics vol 73 no 4 pp 977ndash10292001

[35] R J Szabo ldquoQuantum field theory on noncommutative spacesrdquoPhysics Reports vol 378 no 4 pp 207ndash299 2003

[36] J Gomis K Kamimura and T Mateos ldquoGauge and BRST gen-erators for space-time non-commutative U(1) theoryrdquo Journalof High Energy Physics vol 2001 no 3 article 010 2001

[37] A A Bichl J M Grimstrup L Popp M Schweda and RWulkenhaar ldquoPerturbative analysis of the Seiberg-Witten maprdquoInternational Journal of Modern Physics A vol 17 no 16 pp2219ndash2231 2002

[38] A Smailagic and E Spallucci ldquoUV divergence-free QFT onnoncommutative planerdquo Journal of Physics AMathematical andGeneral vol 36 no 39 pp L517ndashL521 2003

[39] A Smailagic and E Spallucci ldquoFeynman path integral on thenon-commutative planerdquo Journal of Physics A Mathematicaland General vol 36 no 33 pp L467ndashL471 2003

[40] M Knezevic D Knezevic and D Spasojevic ldquoStatistics ofequally weighted random paths on a class of self-similarstructuresrdquo Journal of Physics A Mathematical and Generalvol 37 no 1 pp 1ndash8 2004 Erratum in Journal of Physics AMathematical and General vol 37 p 7169 2004

[41] E Spallucci A Smailagic and P Nicolini ldquoTrace anomaly ona quantum spacetime manifoldrdquo Physical Review D vol 73Article ID 084004 2006

[42] R Banerjee S Gangopadhyay and S K Modak ldquoVoros prod-uct noncommutative Schwarzschild black hole and correctedarea lawrdquo Physics Letters B vol 686 no 2-3 pp 181ndash187 2010

[43] A B Hammou M Lagraa and M M Sheikh-Jabbari ldquoCoher-ent state induced star product on 119877

3

120582and the fuzzy sphererdquo

Physical Review D vol 66 no 2 Article ID 025025 2002[44] L Modesto J W Moffat and P Nicolini ldquoBlack holes in an

ultraviolet complete quantum gravityrdquo Physics Letters B vol695 no 1ndash4 pp 397ndash400 2011

[45] P Nicolini ldquoNonlocal and generalizeduncertainty principleblack holesrdquo httparxivorgabs12022102

[46] P Nicolini A Smailagic and E Spallucci ldquoNoncommutativegeometry inspired Schwarzschild black holerdquo Physics Letters Bvol 632 no 4 pp 547ndash551 2006

[47] S Pramanik S Das and S Ghosh ldquoNoncommutative extensionof AdS-CFT and holographic superconductorsrdquo Physics LettersB vol 742 pp 266ndash273 2015

[48] W Dittrich and M Reuter Effective Lagrangians in QuantumElectrodynamics Springer Berlin Germany 1985

[49] M Ostrogradsky ldquoMemoires sur les equations differen-tielles relativrs au probleme des isoperimetersrdquo Memoires delrsquoAcademie Imperiale des Sciences de St Petersbourg vol 4 p385 1850

[50] P Gaete and E Spallucci ldquoFinite axionic electrodynamicsfrom a new non-commutative approachrdquo Journal of Physics AMathematical and Theoretical vol 45 no 6 Article ID 06540114 pages 2012

[51] P Gaete ldquoSome considerations about Podolsky-axionic electro-dynamicsrdquo International Journal of Modern Physics A vol 27no 11 Article ID 1250061 2012

[52] P Gaete ldquoRemarks on gauge-invariant variables and interactionenergy in QEDrdquo Physical Review D vol 59 no 12 Article ID127702 1999

[53] S Klarsfeld ldquoAnalytical expressions for the evaluation ofvacuum-polarization potentials in muonic atomsrdquo Physics Let-ters B vol 66 no 1 pp 86ndash88 1977

[54] A M Frolov and D M Wardlaw ldquoAnalytical formula for theUehling potentialrdquo The European Physical Journal B vol 85article 348 2012

[55] P Gaete ldquoOn gauge-invariant variables in QEDrdquo Zeitschrift furPhysik C Particles and Fields vol 76 no 2 pp 355ndash361 1997

[56] S Ferrara and A Sagnotti ldquoMassive Born-Infeld and other dualpairsrdquo Journal of High Energy Physics vol 2015 no 4 article 0322015

[57] R Bufalo ldquoBornndashinfeld electrodynamics in very special relativ-ityrdquo Physics Letters B vol 746 pp 251ndash256 2015

[58] HKleinert E Strobel and S-S Xue ldquoFractional effective actionat strong electromagnetic fieldsrdquo Physical Review D vol 88 no2 Article ID 025049 2013

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Advances in High Energy Physics 3

variables Let us also mention here that in previous studies[50 51] we have shown that although theories like (1) containhigher derivatives in the electrostatic case the canonicalmomentum conjugate to velocities disappears Hence thenew Legendre transformation to construct the Hamiltonianreduces to the standard Legendre transformation It shouldhowever be emphasized here that the present paper is aimedat studying the static potential of the above theory sothat Δ can be replaced by minusnabla

2 Notice that for notationalconvenience we have maintained Δ in (1) and (2) but itshould be borne inmind that this paper essentially deals withthe static case

Now we move on to compute the canonical Hamilto-nian For this end we perform a Hamiltonian constraintanalysis The canonical momenta are found to be Π

120583=

(1 minus (1205723120587)ΔM)1198651205830 It is easy to see that Π0 vanishes we

then have the usual constraint equation which accordingto Diracrsquos theory is written as a weak (asymp) equation that isΠ

0asymp 0 Itmay be noted that the remaining nonzeromomenta

must also be written as weak equations This leads to Π119894asymp

(1minus(1205723120587)ΔM)119864119894 (with119864

119894= 119865

1198940) Accordingly the canonical

Hamiltonian119867119862is

119867119862asymp int119889

3119909 [Π

119894120597119894119860

0minus1

119894(1 minus

120572

3120587ΔM)

minus1

Π119894

+1

4119865119894119895(1 minus

120572

3120587ΔM)119865

119894119895]

(6)

which must also be written as a weak equation Next theprimary constraint Π0

asymp 0 must be satisfied for all timesAn immediate consequence of this is that using the equationof motion asymp [119885119867

119862] we obtain the secondary constraint

Γ1

equiv 120597119894Π

119894asymp 0 which must also be true for all time In

passingwe recall that we are considering the static case hencethis new constraint does not contain time derivatives It isstraightforward to check that there are no further constraintsin the theory Therefore in the case under considerationthere are two constraints which are first class Accordingto the general theory we obtain the extended Hamiltonianas an ordinary (or strong) equation by adding all the first-class constraints with arbitrary constraints We thus write119867 = 119867

119862+ int119889

3119909(119906

0(119909)Π

0(119909) + 119906

1(119909)Γ

1(119909)) where 119906

0(119909) and

1199061(119909) are arbitrary Lagrange multipliers It is also important

to observe that when this new Hamiltonian is employed theequation of motion of a dynamic variable may be written as astrong equation With the aid of (6) we find that

0(119909) =

[1198600(119909)119867] = 119906

0(119909) which is an arbitrary function Since

Π0asymp 0 always neither1198600 norΠ0 are of interest in describing

the system andmay be discarded from the theory In fact theterm containing 119860

0is redundant because it can be absorbed

by redefining the function 119908(119909) Therefore the Hamiltonianis now given as

119867 = int1198893119909 [119908 (119909) 120597

119894Π

119894minus1

119894(1 minus

120572

3120587ΔM)

minus1

Π119894

+1

4119865119894119895(1 minus

120572

3120587ΔM)119865

119894119895]

(7)

where 119908(119909) = 1199061(119909) minus 119860

0(119909)

It must be clear from this discussion that the presenceof the new arbitrary function 119908(119909) is undesirable since wehave no way of giving it a meaning in a quantum theoryHence according to the usual procedure we impose a gaugecondition such that the full set of constraints becomes secondclass A convenient choice is

Γ2(119909) equiv int

119862120585119909

119889119911]119860] (119911) equiv int

1

0

119889120582119909119894119860

119894(120582119909) = 0 (8)

where 120582 (0 le 120582 le 1) is the parameter describing the space-like straight path 119911

119894= 120585

119894+ 120582(119909 minus 120585)

119894 and 120585 is a fixed point(reference point) There is no essential loss of generality if werestrict our considerations to 120585

119894= 0 The Dirac brackets can

now be determined and we simply note the only nontrivialDirac bracket involving the canonical variables that is

119860119894(119909) Π

119895(119910)

lowast

= 120575119895

119894120575(3)

(119909 minus 119910)

minus 120597119909

119894int

1

0

119889120582119909119894120575(3)

(120582119909 minus 119910)

(9)

In passing we also recall that the transition to a quantumtheory is made by the replacement of the Dirac brackets bythe operator commutation relations according to 119860 119861

lowastrarr

(119894ℏ)[119860 119861]With the foregoing information we can now proceed to

obtain the interaction energy As alreadymentioned in orderto accomplish this purpose we will calculate the expectationvalue of the energy operator119867 in the physical state |Φ⟩ wherethe physical states |Φ⟩ are gauge-invariant ones The physicalstate can be written as

|Φ⟩ equiv10038161003816100381610038161003816Ψ (y) Ψ (y1015840)⟩

= 120595 (y) exp(119894119902ℏint

y

y1015840119889119911

119894119860

119894(119911))120595 (y1015840) |0⟩

(10)

where |0⟩ is the physical vacuum state and the line integralappearing in the above expression is along a space-like pathstarting at y1015840 and ending at y on a fixed time slice The pointwe wish to emphasize however is that the physical fermion(Ψ(y)) is not the Lagrangian fermion (120595(y)) which is neithergauge-invariant nor associated with an electric field In factthe physical fermion is the Lagrangian fermion together witha cloud (or dressing) of gauge fields

Making use of the above Hamiltonian structure [18] wefind that

Π119894 (119909)

10038161003816100381610038161003816Ψ (y) Ψ (y1015840)⟩ = Ψ (y) Ψ (y1015840)Π

119894 (119909) |0⟩

+ 119902int

y1015840

y119889119911

119894120575(3)

(z minus x) |Φ⟩ (11)

With the aid of (11) and (7) the lowest-order modification in120572 of the interaction energy takes the form

⟨119867⟩Φ = ⟨119867⟩0 + 1198811+ 119881

2 (12)

4 Advances in High Energy Physics

where ⟨119867⟩0= ⟨0|119867|0⟩ The 119881

1 119881

2terms are given by

1198811=1199022

2int119889

3119909int

y

y1015840119889119911

119894120575(3)

(x minus z) inty

y1015840119889119911

1015840119894120575(3)

(x minus z1015840) (13)

1198812=1199022

2

120572

3120587int

infin

41198982

119889120591120588 (120591)

120591int 119889

3119909

sdot int

y1015840

y119889119911

1015840

119894120575(3)

(z1015840 minus x) nabla2

119909

1

120591 minus nabla2

119909

int

y1015840

y119889119911

119894120575(3)

(z minus x) (14)

where 120588(120591) = (1 + 21198982120591)radic1 minus 4119898

2120591

We note that term (13) may look peculiar but it is nothingbut the familiar Coulomb interaction plus a self-energy term[52] Now making use of the Green function 119866(z z1015840) =

(14120587)(119890minusradic120591|zminusz1015840|

|z minus z1015840|) the term (14) can be rewritten inthe form

1198812=1199022

2

120572

3120587int

infin

41198982

119889120591120588 (120591)

120591int

y1015840

y119889119911

1015840

119894nabla

2

1199111015840 int

y1015840

y119889119911

119894119866(z z1015840)

= minus120572

3120587

1199022

2int

infin

41198982

119889120591120588 (120591)

120591

119890minusradic120591|yminusy1015840|

1003816100381610038161003816y minus y10158401003816100381610038161003816

(15)

Since the second and third term on the right-hand sideof (12) are clearly dependent on the distance between theexternal static fields the potential for two opposite chargeslocated at y and y1015840 reads

119881 = minus1199022

4120587

1

119871(1 +

120572

3120587int

infin

41198982

119889120591120588 (120591)

120591119890minusradic120591119871

) (16)

where 119871 = |y minus y1015840| Accordingly one recovers the knownUehling potential which finds here an entirely differentderivation

Beforewe proceed further wewish to show that this resultcan be written alternatively in a more explicit form Makinguse of [53]

120594119899(119911) = int

infin

1

119889119905119890minus119905119911

119905119899

(1 +1

21199052)radic1 minus

1

1199052 (17)

we then get

119881 = minus1199022

4120587

1

119871(1 +

2120572

31205871205941 (2119898119871)) (18)

By the transformation 119905 = cosh 119906 [54] the functions 120594119899can

be reduced to the form [53]

120594119899(119911) = 119870119894

119899minus1(119911) minus

1

2119870119894

119899+1(119911) minus

1

2119870119894

119899+3(119911) (19)

where the functions 119870119894 denote Bessel function integralsHence we see that the interaction energy (with 119898 = 1)becomes

119881 = minus1199022

4120587

1

1198711 +

2120572

3120587[(1 +

1198712

3)119870

0 (2119871)

minus (5119871

3+2119871

3

3)119870

1(2119871)

+ (3119871

2+2119871

3

3)int

infin

2119871

1198891199051198700 (119905)]

(20)

where1198700(119911) and119870

1(119911) aremodified Bessel functions Finally

with the aid of asymptotic forms for Bessel functions it is asimple matter to find expressions for 119881 for large and small 119871

Before concluding this subsection we discuss an alterna-tive way of stating our previous result (16) which displayscertain distinctive features of our methodology We start byconsidering [52 55]

119881 equiv 119902 (A0(0) minusA

0(L)) (21)

where the physical scalar potential is given by

A0(119905 r) = int

1

0

119889120582119903119894119864

119894(119905 120582r) (22)

This follows from the vector gauge-invariant field expression

A120583 (119909) equiv 119860

120583 (119909) + 120597120583(minusint

119909

120585

119889119911120583119860

120583 (119911)) (23)

where the line integral is along a space-like path from thepoint 120585 to 119909 on a fixed slice time It is also important toobserve that the gauge-invariant variables (22) commute withthe sole first constraint (Gauss law) showing in this waythat these fields are physical variables In as much as we areinterested in estimating the lowest-order correction to theCoulomb energy we will retain only the leading term inexpression 119864

119894= (1 minus (1205723120587)ΔM)

minus1Π

119894 Making use of this lastexpression (22) gives

A0(119905 r)

= int

1

0

119889120582119903119894120597120582r119894(minus

1198690(120582r)nabla

2

120582r)

+120572

3120587int

infin

41198982

119889120591

120591120588 (120591) int

1

0

119889120582119903119894120597120582r119894(minus

1198690(120582r)

nabla2

120582r minus 120591)

(24)

to get the last line we used Gauss law for the present theorythat is 120597

119894Π

119894= 119869

0 (where we have included the externalcurrent 1198690 to represent the presence of two opposite charges)Accordingly for 1198690(119905 r) = 119902120575

(3)(r) the potential for a pair of

static point-like opposite charges located at 0 and L is givenby

119881 = minus1199022

4120587

1

119871(1 +

120572

3120587int

infin

41198982

119889120591120588 (120591)

120591119890minusradic120591119871

) (25)

after subtracting a self-energy term

Advances in High Energy Physics 5

3 Euler-Heisenberg-Like Model

Proceeding in the sameway aswe did in the foregoing sectionwe will now consider the interaction energy for Euler-Heisenberg-like electrodynamics Nevertheless in order toput our discussion into context it is useful to describe verybriefly the model under consideration In such a case theLagrangian density reads

L =1205732

21 minus [1 +

1

1205732F minus

1

12057321205742G

2]

119901

(26)

where we have included two parameters 120573 and 120574 As usualF = (14)119865

120583]119865120583] G = (14)119865

120583]119865120583] 119865

120583] = 120597120583119860] minus 120597]119860120583

and 119865

120583]= (12)120576

120583]120588120582119865120588120582 Let us also mention here that in

our previous paper [19] we have studied the domain 0 lt

119901 lt 1 Moreover it follows from (26) that when 119901 = 2

the model contains to orders O(11205732) and O(11205742

) a Euler-Heisenberg-likemodel with the appropriate identifications ofthe constants Interestingly we also observe that in the limit120574 rarr infin we obtain a Wichmann-Kroll model This remarkopens up the way to discuss the effect of these nonlinearcorrections on the interaction energy as we are going tostudy below In fact we will consider a massive Wichmann-Kroll systemThemotivation for this study comes from recentconsiderations in the context of dualities [56] where massiveBorn-Infeld systems play an important role

Having made these observations we can write immedi-ately the field equations for 119901 = 2

120597120583[Γ(119865

120583]minus

2

1205742G119865

120583])] = 0 (27)

while the Bianchi identities are given by

120597120583119865

120583]= 0 (28)

where

Γ = 1 +F

1205732minus

G2

12057321205742 (29)

Also it is straightforward to see that Gauss law becomes

nabla sdotD = 0 (30)

whereD is given by

D

= [1 minus

(E2minus B2

)

21205732

minus(E sdot B)2

12057321205742

](E +2

1205742(E sdot B)B)

(31)

Again from (30) for 1198690(119905 r) = 119890120575(3)(r) we findD = (119876119903

2)119903

where 119876 = 1198904120587 This then implies that for a point-likecharge 119890 at the origin the expression

119876

1199032= (1 minus

E2

21205732) |E| (32)

tells us that for 119903 rarr 0 the electrostatic field becomessingular at 119903 = 0 in contrast to the 0 lt 119901 lt 1 case wherethe electrostatic field is finite Even so in this theory thephenomenon of birefringence is present Before going intodetails we would like to recall that birefringence refers to theproperty that polarized light in a particular direction (opticalaxis) travels at a different velocity from that of light polarizedin a direction perpendicular to this axis Indeed due toquantum fluctuations the QED vacuum has this property aswe are going to show

To illustrate this important feature we introduce thevectorsD = 120597L120597E andH = minus120597L120597B

D = Γ(E + 2B (E sdot B)

1205742

)

H = Γ(B minus 2E (E sdot B)

1205742

)

(33)

where Γ = 1 + (121205732)(B2

minus E2) minus (1120573

21205742)(E sdot B)2 We thus

obtain the equations of motion

nabla sdotD = 0

120597D120597119905

minus nabla timesH = 0

nabla sdot B = 0

120597B120597119905

+ nabla times E = 0

(34)

With the aid from (33) we find the electric permittivity 120576119894119895

and the inverse magnetic permeability (120583minus1)119894119895 tensors of the

vacuum that is

120576119894119895= Γ(120575

119894119895+

2119861119894119861119895

1205742

)

(120583minus1)119894119895= Γ(120575

119894119895minus

2119864119894119864

119895

1205742

)

(35)

with119863119894= 120576

119894119895119864

119895and 119861

119894= 120583

119894119895119867

119895

In accordance with our previous procedure [18 19] wecan now linearize the above equations To do this it isadvantageous to introduce a weak electromagnetic wave(E

119901B

119901) propagating in the presence of a strong constant

external field (E0B

0) On these assumptions we readily find

that for the case of a purely magnetic field (E0= 0) the

vectorsD andH become

D = (1 +B2

0

21205732)[E

119901+

2

1205742(E

119901sdot B

0)B

0]

H = (1 +B2

0

21205732)

sdot [B119901+

1

1205732(1 + B2

02120573

2)(B

119901sdot B

0)B

0]

(36)

where we have to keep only linear terms in E119901B

119901 As

before we consider the 119911-axis as the direction of the external

6 Advances in High Energy Physics

magnetic field (B0= 119861

0e3) and assuming that the light wave

moves along the 119909-axis the decomposition into a plane wavefor the fields E

119901and B

119901can be written as

E119901 (x 119905) = E119890minus119894(119908119905minusksdotx)

B119901(x 119905) = B119890minus119894(119908119905minusksdotx)

(37)

In this case it clearly follows that

(1198962

1199082minus 120576

2212058333)119864

2= 0 (38)

(1198962

1199082minus 120576

3312058322)119864

3= 0 (39)

As a consequence we have two different situations Firstif E perp B

0(perpendicular polarization) from (39) 119864

3= 0 and

from (38) we get 1198962119908

2= 120576

2212058333 This then means that the

dispersion relation of the photon takes the form

119899perp= radic

1 + B2

02120573

2

1 + 3B2

02120573

2 (40)

Second if E B0(parallel polarization) from (38) 119864

2= 0

and from (39) we get 1198962119908

2= 120576

3312058322 This leads to

119899= radic1 +

2B2

0

1205742 (41)

Thus we verify that in the case of a generalized Euler-Heisenberg electrodynamics the phenomenon of birefrin-gence is present

We now pass to the calculation of the interaction energybetween static point-like sources for a massive Wichmann-Kroll-like model our analysis follows closely that of [18 19]The corresponding theory is governed by the Lagrangiandensity

L = minus1

4119865

2

120583] +1

321205732(119865

120583]119865120583])2

+119898

2

2119860

120583119860

120583 (42)

Next in order to handle the second term on the right hand in(42) we introduce an auxiliary field 120585 such that its equation ofmotion gives back the original theory This allows us to writethe Lagrangian density as

L = minus1

4119865120583]119865

120583]+

120585

321205732119865120583]119865

120583]minus

1

12812057321205852

+119898

2

2119860

120583119860

120583

(43)

With the redefinition 120578 = 1 minus 12058581205732 (43) becomes

L = minus1

4120578119865

120583]119865120583]minus1

2(1 minus 120578)

2+119898

2

2119860

120583119860

120583 (44)

Before we proceed to work out explicitly the interactionenergy we will first restore the gauge invariance in (44)

Following an earlier procedure we readily verify that thecanonical momenta read Π

120583= minus120578119865

0120583 which results in theusual primary constraint Π0

= 0 and Π119894= 120578119865

1198940 In this wayone obtains

119867119862= int119889

3119909Π

119894120597119894119860

0+

1

2120578Π

2+120578

2B2

minus119898

2

2119860

120583119860

120583

+1

2(1 minus 120578)

2

(45)

The consistency condition Π = 0 leads to the constraint Γ equiv

120597119894Π

119894+119898

2119860

0= 0 As a result both constraints are second-class

To convert the second-class system into first-class we willadopt the procedure described previously Thus we enlargethe original phase space by introducing a canonical pair offields 120579 and Π

120579 It follows therefore that a new set of first-

class constraints can be defined in this extended space Λ1equiv

Π0+ 119898

2120579 = 0 and Λ

2equiv Γ + Π

120579= 0 Notice that this new

120579-field is not to be confused with the usual noncommutativeparameter This then shows that the new constraints arefirst-class and therefore restore the gauge symmetry As iswell known this procedure reproduces the usual Stuckelbergformalism From this the new effective Lagrangian densityafter integrating out the 120579 fields becomes

L = minus1

4119865120583] (120578 +

1198982

Δ)119865

120583]minus1

2(1 minus 120578)

2 (46)

Now writing 120590 = 120578 + 1198982Δ expression (46) can be brought

to the form

L = minus1

4119865120583]120590119865

120583]minus

119896

128(1 minus 120590 +

1198982

Δ)

2

(47)

where 119896 = 641205732

We are now ready to compute the interaction energy Inthis case the canonical momenta are Π120583

= minus1205901198650120583 with the

usual primary constraint Π0= 0 and Π

119894= 120590119865

1198940 Hence thecanonical Hamiltonian is expressed as

119867119862= int119889

3119909Π

119894120597119894119860

0+

1

2120590Π

2+120590

2B2

+119896

128(1 minus 120590 +

1198982

Δ)

2

(48)

Time conservation of the primary constraint Π0 yields thesecondary constraint Γ

1equiv 120597

119894Π

119894= 0 Similarly the P

120590

constraint yields no further constraints and just determinesthe field 120590 In this case at leading order in 120573 the field 120590 isgiven by

120590 = (1 +119898

2

Δminus

B2

21205732)

sdot [1 minus3

21205732

1

(1 + 1198982Δ minus B2

21205732)3Π

2]

(49)

Advances in High Energy Physics 7

which will be used to eliminate 120590 As before the corre-sponding total (first-class) Hamiltonian that generates thetime evolution of the dynamical variables is 119867 = 119867

119862+

int1198893119909(119906

0(119909)Π

0(119909) + 119906

1(119909)Γ

1(119909)) where 119906

0(119909) and 119906

1(119909) are

the Lagrangemultiplier utilized to implement the constraintsIn the same way as was done in the previous subsection

the expectation value of the energy operator119867 in the physicalstate |Φ⟩ becomes

⟨119867⟩Φ = ⟨Φ|int1198893119909

1

119894(1 +

1198982

Δ)

minus1

Π119894+

15

81205732Π

4

minus15119898

2

21205732Π

2 1

ΔΠ

2 |Φ⟩

(50)

in this last line we have considered only quadratic terms in119898

2In such a case by employing (50) the lowest-order

modification in 1205732 and1198982 of the interaction energy takes theform

⟨119867⟩Φ = ⟨119867⟩0 + 1198811+ 119881

2+ 119881

3 (51)

where ⟨119867⟩0= ⟨0|119867|0⟩ The 119881

1 119881

2 and 119881

3terms are given by

1198811=1199022

2int119889

3119909int

y

y1015840119889119911

1015840119894120575(3)

(x minus z1015840)(1 minus 1198982

nabla2)

minus1

sdot int

y

y1015840119889119911

119894120575(3)

(x minus z)

1198812= minus

151199024

81205732int119889

3119909int

y

y1015840119889119911

119894120575(3)

(x minus z)

sdot int

y

y1015840119889119911

1015840119894120575(3)

(x minus z1015840)inty

y1015840119889119906

119896120575(3)

(x minus u)

sdot int

y

y1015840119889V

119896120575(3)

(x minus k)

1198813=15119898

21199024

21205732

int1198893119909int

y

y1015840119889119911

119894120575(3)

(x minus z)

sdot int

y

y1015840119889119911

1015840119894120575(3)

(x minus z1015840)inty

y1015840119889119906

119896120575(3)

(x minus u)

sdot int

y

y1015840119889V

119896120575(3)

(x minus k)

(52)

Finally with the aid of expressions (52) the potential fora pair of static point-like opposite charges located at 0 and Lis given by

119881 = minus1199022

4120587

119890minus119898119871

119871+

1199024

161205871205732(3

8120587

1

1198712minus 5119898

2)

1

1198713 (53)

observe that when 119898 = 0 profile (53) reduces to the knownWichmann-Kroll interaction energy On the other hand for119898 = 0 the key role played by the mass term in transformingthe Coulomb potential into the Yukawa one should be notedInterestingly enough an unexpected feature is found In factprofile (53) displays a new long-range 11198713 correction whereits strength is proportional to 119898

2 It is also important toobserve that an analogous correction has been found in Born-Infeld electrodynamics in the context of very special relativity[57] In this way we establish a new connection betweennonlinear effectives theories

Before we proceed further we should comment on ourresult In the case of QED (Euler-Heisenberg Lagrangiandensity) the parameter 11205732 is given by 11205732

= (1645)(1198904ℏ

1198984

1198901198887) where 119898

119890is the electron mass In this context we

also recall the currently accepted upper limit for the photonmass that is 119898

120574sim 2 times 10

minus16 eV Thus for the QED casefrom (53) it follows that the second term on the right handside would be detectable in long-range distances (sim109 m) Inother words we see that detectable corrections induced byvacuum polarization with a mass term would be present atlow energy scales

From (53) it clearly follows that the interaction energybetween heavy charged charges at leading order in 120573 is notfinite at the origin Motivated by this one may consider theabove calculation in a noncommutative geometry based onfindings of our previous studies [18 19] In such a case theelectric field at leading order in 120573

2 and1198982 takes the form

119864119894= [(1 +

1198982

Δ)

minus1

+3

21205732Π

2minus6119898

2

1205732Π

2 1

Δ]

sdot 120597119894(minus

119890120579nabla2

120575(3)

(x)nabla

2)

(54)

where it may be recalled that we are now replacing the source120575(3)(x minus y) by the smeared source 119890

120579nabla2

120575(3)(x minus y) with the

parameter 120579 being noncommutative Now making use of(22) we readily find that

A0(119905 r) = A

(1)

0(119905 r) +A

(2)

0(119905 r) +A

(3)

0(119905 r) (55)

The term A(1)

0(119905 r) was first calculated in [50] we can

therefore write only the result

A(1)

0(119905 r)

= 1199021198901198982120579

4120587

1

119903[119890

minus119898119903minus

1

radic120587int

infin

11990324120579

1198891199061

radic119906119890minus119906minus119898

211990324119906

]

minus 119902119898

41205871198901198982120579

(56)

8 Advances in High Energy Physics

Meanwhile the terms A(2)

0(119905 r) and A

(3)

0(119905 r) after some

manipulation can be brought to the form

A(2)

0(119905 r) =

121199023

120573212058732

119899119894int

119909

0

119889119906119894 1

11990661205743(3

21199062

4120579)

A(3)

0(119905 r) =

311989821199023

120573212058752

119899119894int

119909

0

119889119906119894 1

11990641205742(3

21199062

4120579)

sdot [4120579

1199062120574(

3

21199062

4120579) minus 120574(

1

21199062

4120579)]

(57)

where 120574(32 11990324120579) is the lower incomplete Gamma functiondefined by 120574(119886119887 119909) equiv int

119909

0(119889119906119906)119906

119886119887119890minus119906

Inserting these expressions in (21) we finally obtain thestatic potential for two opposite charges 119902 located at 0 and Las

119881 = minus119902

4120587

1198901198982120579

119871[119890

minus119898119871minus

1

radic120587int

infin

11987124120579

1198891199061

radic119906119890minus119906minus119898

211990624119906

]

minus12119902

4

120573212058732

119899119894int

119871

0

119889119906119894 1

11990661205743(3

21199062

4120579) minus

311989821199024

120573212058752

sdot 119899119894int

119909

0

119889119906119894 1

11990641205742(3

21199062

4120579)

sdot [4120579

1199062120574(

3

21199062

4120579) minus 120574(

1

21199062

4120579)]

(58)

which is finite for 119871 rarr 0 It is a simple matter to verify thatin the limit 120579 rarr 0 we recover our above result

4 Logarithmic Correction

We now want to extend what we have done to Euler-Heisenberg-like electrodynamics at strong fields As alreadymentioned such theories show a power behavior that istypical for critical phenomena [58] In such a case theLagrangian density reads

L = minus1

4119865120583]119865

120583]minus1198870

8119865120583]119865

120583] log(119865120583]119865

120583]

41205822

) (59)

where 1198870and 120582 are constants In fact by choosing 119887

0=

11989026120587

2 and 120582 = 1198982

1198901198883119890ℏ we recover the Euler-Heisenberg

electrodynamics at strong fields [58]In the same way as was done in the previous section one

can introduce an auxiliary field 120585 to handle the logarithm in(59) This leads to

L = minus1

41205721119865120583]119865

120583]minus 120572

2(119865

120583]119865120583])2

(60)

where 1205721= 1 minus (119887

02)(1 + log 120585) and 120572

2= 119887

012058532120582

2A similar procedure can be used to manipulate the

quadratic term in (60) Accordingly by introducing a secondauxiliary field 120578 one easily finds

L = minus1

4119865120583] (1205721

+ 41205722120578) 119865

120583]+1205782

41205722 (61)

By setting 120590 = 1205721+ 4120572

2120578 we then have

L = minus1

4120590119865

120583]119865120583]+

1

641205722

(120590 minus 1205721)2 (62)

It is once again straightforward to apply the gauge-invariant formalism discussed in the foregoing section Thecanonical momenta read Π

120583= minus120590119865

0120583 and at once werecognize the two primary constraints Π0

= 0 and P120590

equiv

120597119871120597 = 0The canonical Hamiltonian corresponding to (62)is

119867119862= int119889

3119909

sdot Π119894120597119894119860

0+

1

2120590Π

2+120590

2B2

minus1

641205722

(120590 minus 1205721)2

(63)

Requiring the primary constraint Π0 to be preserved intime one obtains the secondary constraint Γ

1= 120597

119894Π

119894= 0 In

the same way for the constraintP120590 we get the auxiliary field

120590 as

120590 = (1 minus1198870

2(1 + ln 120585) +

1198870B2

21205822120585)[1

+3119887

0B2

21205822

120585

(1 minus (11988702) (1 + ln 120585) + (119887

0B2

21205822) 120585)

3]

(64)

Hence we obtain

119867119862= int119889

3119909

sdot Π119894120597119894119860

0+1

2+1198870

4(1 + log 120585)Π2 minus

31198870120585

21205822Π

4

(65)

As before requiring the primary constraint P120585to be

preserved in time one obtains the auxiliary field 120585 In this case120585 = 1205826Π

2 Consequently we get

119867119862= int119889

3119909Π

119894120597119894119860

0+1

2(1 + 119887

0)Π

2minus6119887

0

1205822Π

4 (66)

Following the same steps that led to (50) we find that

⟨119867⟩(1)

Φ= ⟨Φ|int119889

3119909

1

2minus

3

81205732Π

4 |Φ⟩ (67)

It should be noted that this expression is similar to (50) in thelimit 119898 rarr 0 except by the changed sign in front of the Π4termHencewe see that the potential for two opposite chargesin 0 and L is given by

119881 = minus1199022

4120587

1

119871minus

1199024

604012057321205872

1

1198715 (68)

5 Final Remarks

Finally within the gauge-invariant but path-dependent vari-ables formalism we have considered the confinement versus

Advances in High Energy Physics 9

screening issue for both massive Euler-Heisenberg-like andEuler-Heisenberg electrodynamics in the approximation ofthe strong-field limit Once again a correct identificationof physical degrees of freedom has been fundamental forunderstanding the physics hidden in gauge theories Inter-estingly enough their noncommutative version displays anultraviolet finite static potential The analysis above revealsthe key role played by the new quantum of length in ouranalysis In a general perspective the benefit of consideringthe present approach is to provide a unification scenarioamong different models as well as exploiting the equivalencein explicit calculations as we have illustrated in the course ofthis work

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

It is a pleasure for the author to thank J A Helayel-Netofor helpful comments on the paper This work was partiallysupported by Fondecyt (Chile) Grant 1130426 and DGIP(UTFSM) internal project USM 111458

References

[1] G Breit and J A Wheeler ldquoCollision of two light quantardquoPhysical Review vol 46 no 12 pp 1087ndash1091 1934

[2] S L Adler ldquoPhoton splitting and photon dispersion in a strongmagnetic fieldrdquo Annals of Physics vol 67 no 2 pp 599ndash6471971

[3] V Costantini B De Tollis and G Pistoni ldquoNonlinear effects inquantum electrodynamicsrdquo Il Nuovo Cimento A vol 2 no 3pp 733ndash787 1971

[4] R Ruffini G Vereshchagin and S-S Xue ldquoElectron-positronpairs in physics and astrophysics from heavy nuclei to blackholesrdquo Physics Reports vol 487 no 1ndash4 pp 1ndash140 2010

[5] G V Dunne ldquoThe Heisenberg-Euler effective action 75 yearsonrdquo International Journal of Modern Physics Conference Seriesvol 14 pp 42ndash56 2012

[6] C Bamber S J Boege TKoffas et al ldquoStudies of nonlinearQEDin collisions of 466GeV electrons with intense laser pulsesrdquoPhysical Review D vol 60 no 9 Article ID 092004 1999

[7] D L Burke R C Field G Horton-Smith et al ldquoPositronproduction in multiphoton light-by-light scatteringrdquo PhysicalReview Letters vol 79 no 9 article 1626 1997

[8] O J Pike F MacKenroth E G Hill and S J Rose ldquoA photonndashphoton collider in a vacuum hohlraumrdquo Nature Photonics vol8 no 6 pp 434ndash436 2014

[9] D Tommasini A Ferrando H Michinel and M Seco ldquoPre-cision tests of QED and non-standard models by searchingphoton-photon scattering in vacuum with high power lasersrdquoJournal ofHigh Energy Physics vol 2009 no 11 article 043 2009

[10] D Tommasini A Ferrando HMichinel andM Seco ldquoDetect-ing photon-photon scattering in vacuum at exawatt lasersrdquoPhysical Review A vol 77 no 4 Article ID 042101 2008

[11] M Born and L Infeld ldquoFoundations of the new field theoryrdquoProceedings of the Royal Society of London Series A Containing

Papers of a Mathematical and Physical Character vol 144 no852 pp 425ndash451 1934

[12] H Gies J Jaeckel and A Ringwald ldquoPolarized light propagat-ing in a magnetic field as a probe for millicharged fermionsrdquoPhysical Review Letters vol 97 no 14 Article ID 140402 2006

[13] E Masso and R Toldra ldquoLight spinless particle coupled tophotonsrdquo Physical Review D vol 52 no 4 pp 1755ndash1763 1995

[14] P Gaete and E I Guendelman ldquoConfinement in the presenceof external fields and axionsrdquoModern Physics Letters A vol 20no 5 article 319 2005

[15] P Gaete and E Spallucci ldquoConfinement effects from interactingchromo-magnetic and axion fieldsrdquo Journal of Physics A Math-ematical and General vol 39 no 20 pp 6021ndash6029 2006

[16] E S Fradkin and A A Tseytlin ldquoNon-linear electrodynamicsfrom quantized stringsrdquo Physics Letters B vol 163 no 1ndash4 pp123ndash130 1985

[17] E Bergshoeff E Sezgin C N Pope and P K TownsendldquoThe Born-Infeld action from conformal invariance of the opensuperstringrdquo Physics Letters B vol 188 no 1 pp 70ndash74 1987

[18] P Gaete and J Helayel-Neto ldquoFinite field-energy and interpar-ticle potential in logarithmic electrodynamicsrdquo The EuropeanPhysical Journal C vol 74 no 3 article 2816 2014

[19] P Gaete and J Helayel-Neto ldquoRemarks on nonlinear electrody-namicsrdquo European Physical Journal C vol 74 no 11 article 31822014

[20] S H Hendi ldquoAsymptotic ReissnerndashNordstrom black holesrdquoAnnals of Physics vol 333 pp 282ndash289 2013

[21] Z Zhao Q Pan S Chen and J Jing ldquoNotes on holographicsuperconductor models with the nonlinear electrodynamicsrdquoNuclear Physics B vol 871 no 1 pp 98ndash110 2013

[22] O Miskovic and R Olea ldquoConserved charges for black holes inEinstein-Gauss-Bonnet gravity coupled to nonlinear electrody-namics in AdS spacerdquo Physical Review D vol 83 no 2 ArticleID 024011 2011

[23] S H Mazharimousavi and M Halilsoy ldquoBlack holes and theclassical model of a particle in Einstein non-linear electrody-namics theoryrdquo Physics Letters B vol 678 no 4 pp 407ndash4102009

[24] G Amelino-Camelia ldquoSpecial treatmentrdquo Nature vol 418 no6893 pp 34ndash35 2002

[25] T Jacobson S Liberati andDMattingly ldquoThreshold effects andPlanck scale Lorentz violation combined constraints from highenergy astrophysicsrdquo Physical Review D vol 67 no 12 ArticleID 124011 2003

[26] T J Konopka and S A Major ldquoObservational limits onquantum geometry effectsrdquo New Journal of Physics vol 4 pp571ndash5718 2002

[27] SHossenfelder ldquoInterpretation of quantumfield theories with aminimal length scalerdquo Physical Review D vol 73 no 10 ArticleID 105013 9 pages 2006

[28] P Nicolini ldquoNoncommutative black holes the final appeal toquantum gravity a reviewrdquo International Journal of ModernPhysics A vol 24 no 7 pp 1229ndash1308 2009

[29] S Pramanik S Ghosh and P Pal ldquoElectrodynamics of a gener-alized charged particle in doubly special relativity frameworkrdquoAnnals of Physics vol 346 pp 113ndash128 2014

[30] A F Ali S Das and E C Vagenas ldquoDiscreteness of space fromthe generalized uncertainty principlerdquo Physics Letters B vol678 no 5 pp 497ndash499 2009

[31] G Amelino-Camelia ldquoQuantum-spacetime phenomenologyrdquoLiving Reviews in Relativity vol 16 article 5 2013

10 Advances in High Energy Physics

[32] E Witten ldquoNoncommutative geometry and string field theoryrdquoNuclear Physics B vol 268 no 2 pp 253ndash294 1986

[33] N Seiberg and E Witten ldquoString theory and noncommutativegeometryrdquo Journal of High Energy Physics vol 1999 article 0321999

[34] M R Douglas and N A Nekrasov ldquoNoncommutative fieldtheoryrdquo Reviews of Modern Physics vol 73 no 4 pp 977ndash10292001

[35] R J Szabo ldquoQuantum field theory on noncommutative spacesrdquoPhysics Reports vol 378 no 4 pp 207ndash299 2003

[36] J Gomis K Kamimura and T Mateos ldquoGauge and BRST gen-erators for space-time non-commutative U(1) theoryrdquo Journalof High Energy Physics vol 2001 no 3 article 010 2001

[37] A A Bichl J M Grimstrup L Popp M Schweda and RWulkenhaar ldquoPerturbative analysis of the Seiberg-Witten maprdquoInternational Journal of Modern Physics A vol 17 no 16 pp2219ndash2231 2002

[38] A Smailagic and E Spallucci ldquoUV divergence-free QFT onnoncommutative planerdquo Journal of Physics AMathematical andGeneral vol 36 no 39 pp L517ndashL521 2003

[39] A Smailagic and E Spallucci ldquoFeynman path integral on thenon-commutative planerdquo Journal of Physics A Mathematicaland General vol 36 no 33 pp L467ndashL471 2003

[40] M Knezevic D Knezevic and D Spasojevic ldquoStatistics ofequally weighted random paths on a class of self-similarstructuresrdquo Journal of Physics A Mathematical and Generalvol 37 no 1 pp 1ndash8 2004 Erratum in Journal of Physics AMathematical and General vol 37 p 7169 2004

[41] E Spallucci A Smailagic and P Nicolini ldquoTrace anomaly ona quantum spacetime manifoldrdquo Physical Review D vol 73Article ID 084004 2006

[42] R Banerjee S Gangopadhyay and S K Modak ldquoVoros prod-uct noncommutative Schwarzschild black hole and correctedarea lawrdquo Physics Letters B vol 686 no 2-3 pp 181ndash187 2010

[43] A B Hammou M Lagraa and M M Sheikh-Jabbari ldquoCoher-ent state induced star product on 119877

3

120582and the fuzzy sphererdquo

Physical Review D vol 66 no 2 Article ID 025025 2002[44] L Modesto J W Moffat and P Nicolini ldquoBlack holes in an

ultraviolet complete quantum gravityrdquo Physics Letters B vol695 no 1ndash4 pp 397ndash400 2011

[45] P Nicolini ldquoNonlocal and generalizeduncertainty principleblack holesrdquo httparxivorgabs12022102

[46] P Nicolini A Smailagic and E Spallucci ldquoNoncommutativegeometry inspired Schwarzschild black holerdquo Physics Letters Bvol 632 no 4 pp 547ndash551 2006

[47] S Pramanik S Das and S Ghosh ldquoNoncommutative extensionof AdS-CFT and holographic superconductorsrdquo Physics LettersB vol 742 pp 266ndash273 2015

[48] W Dittrich and M Reuter Effective Lagrangians in QuantumElectrodynamics Springer Berlin Germany 1985

[49] M Ostrogradsky ldquoMemoires sur les equations differen-tielles relativrs au probleme des isoperimetersrdquo Memoires delrsquoAcademie Imperiale des Sciences de St Petersbourg vol 4 p385 1850

[50] P Gaete and E Spallucci ldquoFinite axionic electrodynamicsfrom a new non-commutative approachrdquo Journal of Physics AMathematical and Theoretical vol 45 no 6 Article ID 06540114 pages 2012

[51] P Gaete ldquoSome considerations about Podolsky-axionic electro-dynamicsrdquo International Journal of Modern Physics A vol 27no 11 Article ID 1250061 2012

[52] P Gaete ldquoRemarks on gauge-invariant variables and interactionenergy in QEDrdquo Physical Review D vol 59 no 12 Article ID127702 1999

[53] S Klarsfeld ldquoAnalytical expressions for the evaluation ofvacuum-polarization potentials in muonic atomsrdquo Physics Let-ters B vol 66 no 1 pp 86ndash88 1977

[54] A M Frolov and D M Wardlaw ldquoAnalytical formula for theUehling potentialrdquo The European Physical Journal B vol 85article 348 2012

[55] P Gaete ldquoOn gauge-invariant variables in QEDrdquo Zeitschrift furPhysik C Particles and Fields vol 76 no 2 pp 355ndash361 1997

[56] S Ferrara and A Sagnotti ldquoMassive Born-Infeld and other dualpairsrdquo Journal of High Energy Physics vol 2015 no 4 article 0322015

[57] R Bufalo ldquoBornndashinfeld electrodynamics in very special relativ-ityrdquo Physics Letters B vol 746 pp 251ndash256 2015

[58] HKleinert E Strobel and S-S Xue ldquoFractional effective actionat strong electromagnetic fieldsrdquo Physical Review D vol 88 no2 Article ID 025049 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

4 Advances in High Energy Physics

where ⟨119867⟩0= ⟨0|119867|0⟩ The 119881

1 119881

2terms are given by

1198811=1199022

2int119889

3119909int

y

y1015840119889119911

119894120575(3)

(x minus z) inty

y1015840119889119911

1015840119894120575(3)

(x minus z1015840) (13)

1198812=1199022

2

120572

3120587int

infin

41198982

119889120591120588 (120591)

120591int 119889

3119909

sdot int

y1015840

y119889119911

1015840

119894120575(3)

(z1015840 minus x) nabla2

119909

1

120591 minus nabla2

119909

int

y1015840

y119889119911

119894120575(3)

(z minus x) (14)

where 120588(120591) = (1 + 21198982120591)radic1 minus 4119898

2120591

We note that term (13) may look peculiar but it is nothingbut the familiar Coulomb interaction plus a self-energy term[52] Now making use of the Green function 119866(z z1015840) =

(14120587)(119890minusradic120591|zminusz1015840|

|z minus z1015840|) the term (14) can be rewritten inthe form

1198812=1199022

2

120572

3120587int

infin

41198982

119889120591120588 (120591)

120591int

y1015840

y119889119911

1015840

119894nabla

2

1199111015840 int

y1015840

y119889119911

119894119866(z z1015840)

= minus120572

3120587

1199022

2int

infin

41198982

119889120591120588 (120591)

120591

119890minusradic120591|yminusy1015840|

1003816100381610038161003816y minus y10158401003816100381610038161003816

(15)

Since the second and third term on the right-hand sideof (12) are clearly dependent on the distance between theexternal static fields the potential for two opposite chargeslocated at y and y1015840 reads

119881 = minus1199022

4120587

1

119871(1 +

120572

3120587int

infin

41198982

119889120591120588 (120591)

120591119890minusradic120591119871

) (16)

where 119871 = |y minus y1015840| Accordingly one recovers the knownUehling potential which finds here an entirely differentderivation

Beforewe proceed further wewish to show that this resultcan be written alternatively in a more explicit form Makinguse of [53]

120594119899(119911) = int

infin

1

119889119905119890minus119905119911

119905119899

(1 +1

21199052)radic1 minus

1

1199052 (17)

we then get

119881 = minus1199022

4120587

1

119871(1 +

2120572

31205871205941 (2119898119871)) (18)

By the transformation 119905 = cosh 119906 [54] the functions 120594119899can

be reduced to the form [53]

120594119899(119911) = 119870119894

119899minus1(119911) minus

1

2119870119894

119899+1(119911) minus

1

2119870119894

119899+3(119911) (19)

where the functions 119870119894 denote Bessel function integralsHence we see that the interaction energy (with 119898 = 1)becomes

119881 = minus1199022

4120587

1

1198711 +

2120572

3120587[(1 +

1198712

3)119870

0 (2119871)

minus (5119871

3+2119871

3

3)119870

1(2119871)

+ (3119871

2+2119871

3

3)int

infin

2119871

1198891199051198700 (119905)]

(20)

where1198700(119911) and119870

1(119911) aremodified Bessel functions Finally

with the aid of asymptotic forms for Bessel functions it is asimple matter to find expressions for 119881 for large and small 119871

Before concluding this subsection we discuss an alterna-tive way of stating our previous result (16) which displayscertain distinctive features of our methodology We start byconsidering [52 55]

119881 equiv 119902 (A0(0) minusA

0(L)) (21)

where the physical scalar potential is given by

A0(119905 r) = int

1

0

119889120582119903119894119864

119894(119905 120582r) (22)

This follows from the vector gauge-invariant field expression

A120583 (119909) equiv 119860

120583 (119909) + 120597120583(minusint

119909

120585

119889119911120583119860

120583 (119911)) (23)

where the line integral is along a space-like path from thepoint 120585 to 119909 on a fixed slice time It is also important toobserve that the gauge-invariant variables (22) commute withthe sole first constraint (Gauss law) showing in this waythat these fields are physical variables In as much as we areinterested in estimating the lowest-order correction to theCoulomb energy we will retain only the leading term inexpression 119864

119894= (1 minus (1205723120587)ΔM)

minus1Π

119894 Making use of this lastexpression (22) gives

A0(119905 r)

= int

1

0

119889120582119903119894120597120582r119894(minus

1198690(120582r)nabla

2

120582r)

+120572

3120587int

infin

41198982

119889120591

120591120588 (120591) int

1

0

119889120582119903119894120597120582r119894(minus

1198690(120582r)

nabla2

120582r minus 120591)

(24)

to get the last line we used Gauss law for the present theorythat is 120597

119894Π

119894= 119869

0 (where we have included the externalcurrent 1198690 to represent the presence of two opposite charges)Accordingly for 1198690(119905 r) = 119902120575

(3)(r) the potential for a pair of

static point-like opposite charges located at 0 and L is givenby

119881 = minus1199022

4120587

1

119871(1 +

120572

3120587int

infin

41198982

119889120591120588 (120591)

120591119890minusradic120591119871

) (25)

after subtracting a self-energy term

Advances in High Energy Physics 5

3 Euler-Heisenberg-Like Model

Proceeding in the sameway aswe did in the foregoing sectionwe will now consider the interaction energy for Euler-Heisenberg-like electrodynamics Nevertheless in order toput our discussion into context it is useful to describe verybriefly the model under consideration In such a case theLagrangian density reads

L =1205732

21 minus [1 +

1

1205732F minus

1

12057321205742G

2]

119901

(26)

where we have included two parameters 120573 and 120574 As usualF = (14)119865

120583]119865120583] G = (14)119865

120583]119865120583] 119865

120583] = 120597120583119860] minus 120597]119860120583

and 119865

120583]= (12)120576

120583]120588120582119865120588120582 Let us also mention here that in

our previous paper [19] we have studied the domain 0 lt

119901 lt 1 Moreover it follows from (26) that when 119901 = 2

the model contains to orders O(11205732) and O(11205742

) a Euler-Heisenberg-likemodel with the appropriate identifications ofthe constants Interestingly we also observe that in the limit120574 rarr infin we obtain a Wichmann-Kroll model This remarkopens up the way to discuss the effect of these nonlinearcorrections on the interaction energy as we are going tostudy below In fact we will consider a massive Wichmann-Kroll systemThemotivation for this study comes from recentconsiderations in the context of dualities [56] where massiveBorn-Infeld systems play an important role

Having made these observations we can write immedi-ately the field equations for 119901 = 2

120597120583[Γ(119865

120583]minus

2

1205742G119865

120583])] = 0 (27)

while the Bianchi identities are given by

120597120583119865

120583]= 0 (28)

where

Γ = 1 +F

1205732minus

G2

12057321205742 (29)

Also it is straightforward to see that Gauss law becomes

nabla sdotD = 0 (30)

whereD is given by

D

= [1 minus

(E2minus B2

)

21205732

minus(E sdot B)2

12057321205742

](E +2

1205742(E sdot B)B)

(31)

Again from (30) for 1198690(119905 r) = 119890120575(3)(r) we findD = (119876119903

2)119903

where 119876 = 1198904120587 This then implies that for a point-likecharge 119890 at the origin the expression

119876

1199032= (1 minus

E2

21205732) |E| (32)

tells us that for 119903 rarr 0 the electrostatic field becomessingular at 119903 = 0 in contrast to the 0 lt 119901 lt 1 case wherethe electrostatic field is finite Even so in this theory thephenomenon of birefringence is present Before going intodetails we would like to recall that birefringence refers to theproperty that polarized light in a particular direction (opticalaxis) travels at a different velocity from that of light polarizedin a direction perpendicular to this axis Indeed due toquantum fluctuations the QED vacuum has this property aswe are going to show

To illustrate this important feature we introduce thevectorsD = 120597L120597E andH = minus120597L120597B

D = Γ(E + 2B (E sdot B)

1205742

)

H = Γ(B minus 2E (E sdot B)

1205742

)

(33)

where Γ = 1 + (121205732)(B2

minus E2) minus (1120573

21205742)(E sdot B)2 We thus

obtain the equations of motion

nabla sdotD = 0

120597D120597119905

minus nabla timesH = 0

nabla sdot B = 0

120597B120597119905

+ nabla times E = 0

(34)

With the aid from (33) we find the electric permittivity 120576119894119895

and the inverse magnetic permeability (120583minus1)119894119895 tensors of the

vacuum that is

120576119894119895= Γ(120575

119894119895+

2119861119894119861119895

1205742

)

(120583minus1)119894119895= Γ(120575

119894119895minus

2119864119894119864

119895

1205742

)

(35)

with119863119894= 120576

119894119895119864

119895and 119861

119894= 120583

119894119895119867

119895

In accordance with our previous procedure [18 19] wecan now linearize the above equations To do this it isadvantageous to introduce a weak electromagnetic wave(E

119901B

119901) propagating in the presence of a strong constant

external field (E0B

0) On these assumptions we readily find

that for the case of a purely magnetic field (E0= 0) the

vectorsD andH become

D = (1 +B2

0

21205732)[E

119901+

2

1205742(E

119901sdot B

0)B

0]

H = (1 +B2

0

21205732)

sdot [B119901+

1

1205732(1 + B2

02120573

2)(B

119901sdot B

0)B

0]

(36)

where we have to keep only linear terms in E119901B

119901 As

before we consider the 119911-axis as the direction of the external

6 Advances in High Energy Physics

magnetic field (B0= 119861

0e3) and assuming that the light wave

moves along the 119909-axis the decomposition into a plane wavefor the fields E

119901and B

119901can be written as

E119901 (x 119905) = E119890minus119894(119908119905minusksdotx)

B119901(x 119905) = B119890minus119894(119908119905minusksdotx)

(37)

In this case it clearly follows that

(1198962

1199082minus 120576

2212058333)119864

2= 0 (38)

(1198962

1199082minus 120576

3312058322)119864

3= 0 (39)

As a consequence we have two different situations Firstif E perp B

0(perpendicular polarization) from (39) 119864

3= 0 and

from (38) we get 1198962119908

2= 120576

2212058333 This then means that the

dispersion relation of the photon takes the form

119899perp= radic

1 + B2

02120573

2

1 + 3B2

02120573

2 (40)

Second if E B0(parallel polarization) from (38) 119864

2= 0

and from (39) we get 1198962119908

2= 120576

3312058322 This leads to

119899= radic1 +

2B2

0

1205742 (41)

Thus we verify that in the case of a generalized Euler-Heisenberg electrodynamics the phenomenon of birefrin-gence is present

We now pass to the calculation of the interaction energybetween static point-like sources for a massive Wichmann-Kroll-like model our analysis follows closely that of [18 19]The corresponding theory is governed by the Lagrangiandensity

L = minus1

4119865

2

120583] +1

321205732(119865

120583]119865120583])2

+119898

2

2119860

120583119860

120583 (42)

Next in order to handle the second term on the right hand in(42) we introduce an auxiliary field 120585 such that its equation ofmotion gives back the original theory This allows us to writethe Lagrangian density as

L = minus1

4119865120583]119865

120583]+

120585

321205732119865120583]119865

120583]minus

1

12812057321205852

+119898

2

2119860

120583119860

120583

(43)

With the redefinition 120578 = 1 minus 12058581205732 (43) becomes

L = minus1

4120578119865

120583]119865120583]minus1

2(1 minus 120578)

2+119898

2

2119860

120583119860

120583 (44)

Before we proceed to work out explicitly the interactionenergy we will first restore the gauge invariance in (44)

Following an earlier procedure we readily verify that thecanonical momenta read Π

120583= minus120578119865

0120583 which results in theusual primary constraint Π0

= 0 and Π119894= 120578119865

1198940 In this wayone obtains

119867119862= int119889

3119909Π

119894120597119894119860

0+

1

2120578Π

2+120578

2B2

minus119898

2

2119860

120583119860

120583

+1

2(1 minus 120578)

2

(45)

The consistency condition Π = 0 leads to the constraint Γ equiv

120597119894Π

119894+119898

2119860

0= 0 As a result both constraints are second-class

To convert the second-class system into first-class we willadopt the procedure described previously Thus we enlargethe original phase space by introducing a canonical pair offields 120579 and Π

120579 It follows therefore that a new set of first-

class constraints can be defined in this extended space Λ1equiv

Π0+ 119898

2120579 = 0 and Λ

2equiv Γ + Π

120579= 0 Notice that this new

120579-field is not to be confused with the usual noncommutativeparameter This then shows that the new constraints arefirst-class and therefore restore the gauge symmetry As iswell known this procedure reproduces the usual Stuckelbergformalism From this the new effective Lagrangian densityafter integrating out the 120579 fields becomes

L = minus1

4119865120583] (120578 +

1198982

Δ)119865

120583]minus1

2(1 minus 120578)

2 (46)

Now writing 120590 = 120578 + 1198982Δ expression (46) can be brought

to the form

L = minus1

4119865120583]120590119865

120583]minus

119896

128(1 minus 120590 +

1198982

Δ)

2

(47)

where 119896 = 641205732

We are now ready to compute the interaction energy Inthis case the canonical momenta are Π120583

= minus1205901198650120583 with the

usual primary constraint Π0= 0 and Π

119894= 120590119865

1198940 Hence thecanonical Hamiltonian is expressed as

119867119862= int119889

3119909Π

119894120597119894119860

0+

1

2120590Π

2+120590

2B2

+119896

128(1 minus 120590 +

1198982

Δ)

2

(48)

Time conservation of the primary constraint Π0 yields thesecondary constraint Γ

1equiv 120597

119894Π

119894= 0 Similarly the P

120590

constraint yields no further constraints and just determinesthe field 120590 In this case at leading order in 120573 the field 120590 isgiven by

120590 = (1 +119898

2

Δminus

B2

21205732)

sdot [1 minus3

21205732

1

(1 + 1198982Δ minus B2

21205732)3Π

2]

(49)

Advances in High Energy Physics 7

which will be used to eliminate 120590 As before the corre-sponding total (first-class) Hamiltonian that generates thetime evolution of the dynamical variables is 119867 = 119867

119862+

int1198893119909(119906

0(119909)Π

0(119909) + 119906

1(119909)Γ

1(119909)) where 119906

0(119909) and 119906

1(119909) are

the Lagrangemultiplier utilized to implement the constraintsIn the same way as was done in the previous subsection

the expectation value of the energy operator119867 in the physicalstate |Φ⟩ becomes

⟨119867⟩Φ = ⟨Φ|int1198893119909

1

119894(1 +

1198982

Δ)

minus1

Π119894+

15

81205732Π

4

minus15119898

2

21205732Π

2 1

ΔΠ

2 |Φ⟩

(50)

in this last line we have considered only quadratic terms in119898

2In such a case by employing (50) the lowest-order

modification in 1205732 and1198982 of the interaction energy takes theform

⟨119867⟩Φ = ⟨119867⟩0 + 1198811+ 119881

2+ 119881

3 (51)

where ⟨119867⟩0= ⟨0|119867|0⟩ The 119881

1 119881

2 and 119881

3terms are given by

1198811=1199022

2int119889

3119909int

y

y1015840119889119911

1015840119894120575(3)

(x minus z1015840)(1 minus 1198982

nabla2)

minus1

sdot int

y

y1015840119889119911

119894120575(3)

(x minus z)

1198812= minus

151199024

81205732int119889

3119909int

y

y1015840119889119911

119894120575(3)

(x minus z)

sdot int

y

y1015840119889119911

1015840119894120575(3)

(x minus z1015840)inty

y1015840119889119906

119896120575(3)

(x minus u)

sdot int

y

y1015840119889V

119896120575(3)

(x minus k)

1198813=15119898

21199024

21205732

int1198893119909int

y

y1015840119889119911

119894120575(3)

(x minus z)

sdot int

y

y1015840119889119911

1015840119894120575(3)

(x minus z1015840)inty

y1015840119889119906

119896120575(3)

(x minus u)

sdot int

y

y1015840119889V

119896120575(3)

(x minus k)

(52)

Finally with the aid of expressions (52) the potential fora pair of static point-like opposite charges located at 0 and Lis given by

119881 = minus1199022

4120587

119890minus119898119871

119871+

1199024

161205871205732(3

8120587

1

1198712minus 5119898

2)

1

1198713 (53)

observe that when 119898 = 0 profile (53) reduces to the knownWichmann-Kroll interaction energy On the other hand for119898 = 0 the key role played by the mass term in transformingthe Coulomb potential into the Yukawa one should be notedInterestingly enough an unexpected feature is found In factprofile (53) displays a new long-range 11198713 correction whereits strength is proportional to 119898

2 It is also important toobserve that an analogous correction has been found in Born-Infeld electrodynamics in the context of very special relativity[57] In this way we establish a new connection betweennonlinear effectives theories

Before we proceed further we should comment on ourresult In the case of QED (Euler-Heisenberg Lagrangiandensity) the parameter 11205732 is given by 11205732

= (1645)(1198904ℏ

1198984

1198901198887) where 119898

119890is the electron mass In this context we

also recall the currently accepted upper limit for the photonmass that is 119898

120574sim 2 times 10

minus16 eV Thus for the QED casefrom (53) it follows that the second term on the right handside would be detectable in long-range distances (sim109 m) Inother words we see that detectable corrections induced byvacuum polarization with a mass term would be present atlow energy scales

From (53) it clearly follows that the interaction energybetween heavy charged charges at leading order in 120573 is notfinite at the origin Motivated by this one may consider theabove calculation in a noncommutative geometry based onfindings of our previous studies [18 19] In such a case theelectric field at leading order in 120573

2 and1198982 takes the form

119864119894= [(1 +

1198982

Δ)

minus1

+3

21205732Π

2minus6119898

2

1205732Π

2 1

Δ]

sdot 120597119894(minus

119890120579nabla2

120575(3)

(x)nabla

2)

(54)

where it may be recalled that we are now replacing the source120575(3)(x minus y) by the smeared source 119890

120579nabla2

120575(3)(x minus y) with the

parameter 120579 being noncommutative Now making use of(22) we readily find that

A0(119905 r) = A

(1)

0(119905 r) +A

(2)

0(119905 r) +A

(3)

0(119905 r) (55)

The term A(1)

0(119905 r) was first calculated in [50] we can

therefore write only the result

A(1)

0(119905 r)

= 1199021198901198982120579

4120587

1

119903[119890

minus119898119903minus

1

radic120587int

infin

11990324120579

1198891199061

radic119906119890minus119906minus119898

211990324119906

]

minus 119902119898

41205871198901198982120579

(56)

8 Advances in High Energy Physics

Meanwhile the terms A(2)

0(119905 r) and A

(3)

0(119905 r) after some

manipulation can be brought to the form

A(2)

0(119905 r) =

121199023

120573212058732

119899119894int

119909

0

119889119906119894 1

11990661205743(3

21199062

4120579)

A(3)

0(119905 r) =

311989821199023

120573212058752

119899119894int

119909

0

119889119906119894 1

11990641205742(3

21199062

4120579)

sdot [4120579

1199062120574(

3

21199062

4120579) minus 120574(

1

21199062

4120579)]

(57)

where 120574(32 11990324120579) is the lower incomplete Gamma functiondefined by 120574(119886119887 119909) equiv int

119909

0(119889119906119906)119906

119886119887119890minus119906

Inserting these expressions in (21) we finally obtain thestatic potential for two opposite charges 119902 located at 0 and Las

119881 = minus119902

4120587

1198901198982120579

119871[119890

minus119898119871minus

1

radic120587int

infin

11987124120579

1198891199061

radic119906119890minus119906minus119898

211990624119906

]

minus12119902

4

120573212058732

119899119894int

119871

0

119889119906119894 1

11990661205743(3

21199062

4120579) minus

311989821199024

120573212058752

sdot 119899119894int

119909

0

119889119906119894 1

11990641205742(3

21199062

4120579)

sdot [4120579

1199062120574(

3

21199062

4120579) minus 120574(

1

21199062

4120579)]

(58)

which is finite for 119871 rarr 0 It is a simple matter to verify thatin the limit 120579 rarr 0 we recover our above result

4 Logarithmic Correction

We now want to extend what we have done to Euler-Heisenberg-like electrodynamics at strong fields As alreadymentioned such theories show a power behavior that istypical for critical phenomena [58] In such a case theLagrangian density reads

L = minus1

4119865120583]119865

120583]minus1198870

8119865120583]119865

120583] log(119865120583]119865

120583]

41205822

) (59)

where 1198870and 120582 are constants In fact by choosing 119887

0=

11989026120587

2 and 120582 = 1198982

1198901198883119890ℏ we recover the Euler-Heisenberg

electrodynamics at strong fields [58]In the same way as was done in the previous section one

can introduce an auxiliary field 120585 to handle the logarithm in(59) This leads to

L = minus1

41205721119865120583]119865

120583]minus 120572

2(119865

120583]119865120583])2

(60)

where 1205721= 1 minus (119887

02)(1 + log 120585) and 120572

2= 119887

012058532120582

2A similar procedure can be used to manipulate the

quadratic term in (60) Accordingly by introducing a secondauxiliary field 120578 one easily finds

L = minus1

4119865120583] (1205721

+ 41205722120578) 119865

120583]+1205782

41205722 (61)

By setting 120590 = 1205721+ 4120572

2120578 we then have

L = minus1

4120590119865

120583]119865120583]+

1

641205722

(120590 minus 1205721)2 (62)

It is once again straightforward to apply the gauge-invariant formalism discussed in the foregoing section Thecanonical momenta read Π

120583= minus120590119865

0120583 and at once werecognize the two primary constraints Π0

= 0 and P120590

equiv

120597119871120597 = 0The canonical Hamiltonian corresponding to (62)is

119867119862= int119889

3119909

sdot Π119894120597119894119860

0+

1

2120590Π

2+120590

2B2

minus1

641205722

(120590 minus 1205721)2

(63)

Requiring the primary constraint Π0 to be preserved intime one obtains the secondary constraint Γ

1= 120597

119894Π

119894= 0 In

the same way for the constraintP120590 we get the auxiliary field

120590 as

120590 = (1 minus1198870

2(1 + ln 120585) +

1198870B2

21205822120585)[1

+3119887

0B2

21205822

120585

(1 minus (11988702) (1 + ln 120585) + (119887

0B2

21205822) 120585)

3]

(64)

Hence we obtain

119867119862= int119889

3119909

sdot Π119894120597119894119860

0+1

2+1198870

4(1 + log 120585)Π2 minus

31198870120585

21205822Π

4

(65)

As before requiring the primary constraint P120585to be

preserved in time one obtains the auxiliary field 120585 In this case120585 = 1205826Π

2 Consequently we get

119867119862= int119889

3119909Π

119894120597119894119860

0+1

2(1 + 119887

0)Π

2minus6119887

0

1205822Π

4 (66)

Following the same steps that led to (50) we find that

⟨119867⟩(1)

Φ= ⟨Φ|int119889

3119909

1

2minus

3

81205732Π

4 |Φ⟩ (67)

It should be noted that this expression is similar to (50) in thelimit 119898 rarr 0 except by the changed sign in front of the Π4termHencewe see that the potential for two opposite chargesin 0 and L is given by

119881 = minus1199022

4120587

1

119871minus

1199024

604012057321205872

1

1198715 (68)

5 Final Remarks

Finally within the gauge-invariant but path-dependent vari-ables formalism we have considered the confinement versus

Advances in High Energy Physics 9

screening issue for both massive Euler-Heisenberg-like andEuler-Heisenberg electrodynamics in the approximation ofthe strong-field limit Once again a correct identificationof physical degrees of freedom has been fundamental forunderstanding the physics hidden in gauge theories Inter-estingly enough their noncommutative version displays anultraviolet finite static potential The analysis above revealsthe key role played by the new quantum of length in ouranalysis In a general perspective the benefit of consideringthe present approach is to provide a unification scenarioamong different models as well as exploiting the equivalencein explicit calculations as we have illustrated in the course ofthis work

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

It is a pleasure for the author to thank J A Helayel-Netofor helpful comments on the paper This work was partiallysupported by Fondecyt (Chile) Grant 1130426 and DGIP(UTFSM) internal project USM 111458

References

[1] G Breit and J A Wheeler ldquoCollision of two light quantardquoPhysical Review vol 46 no 12 pp 1087ndash1091 1934

[2] S L Adler ldquoPhoton splitting and photon dispersion in a strongmagnetic fieldrdquo Annals of Physics vol 67 no 2 pp 599ndash6471971

[3] V Costantini B De Tollis and G Pistoni ldquoNonlinear effects inquantum electrodynamicsrdquo Il Nuovo Cimento A vol 2 no 3pp 733ndash787 1971

[4] R Ruffini G Vereshchagin and S-S Xue ldquoElectron-positronpairs in physics and astrophysics from heavy nuclei to blackholesrdquo Physics Reports vol 487 no 1ndash4 pp 1ndash140 2010

[5] G V Dunne ldquoThe Heisenberg-Euler effective action 75 yearsonrdquo International Journal of Modern Physics Conference Seriesvol 14 pp 42ndash56 2012

[6] C Bamber S J Boege TKoffas et al ldquoStudies of nonlinearQEDin collisions of 466GeV electrons with intense laser pulsesrdquoPhysical Review D vol 60 no 9 Article ID 092004 1999

[7] D L Burke R C Field G Horton-Smith et al ldquoPositronproduction in multiphoton light-by-light scatteringrdquo PhysicalReview Letters vol 79 no 9 article 1626 1997

[8] O J Pike F MacKenroth E G Hill and S J Rose ldquoA photonndashphoton collider in a vacuum hohlraumrdquo Nature Photonics vol8 no 6 pp 434ndash436 2014

[9] D Tommasini A Ferrando H Michinel and M Seco ldquoPre-cision tests of QED and non-standard models by searchingphoton-photon scattering in vacuum with high power lasersrdquoJournal ofHigh Energy Physics vol 2009 no 11 article 043 2009

[10] D Tommasini A Ferrando HMichinel andM Seco ldquoDetect-ing photon-photon scattering in vacuum at exawatt lasersrdquoPhysical Review A vol 77 no 4 Article ID 042101 2008

[11] M Born and L Infeld ldquoFoundations of the new field theoryrdquoProceedings of the Royal Society of London Series A Containing

Papers of a Mathematical and Physical Character vol 144 no852 pp 425ndash451 1934

[12] H Gies J Jaeckel and A Ringwald ldquoPolarized light propagat-ing in a magnetic field as a probe for millicharged fermionsrdquoPhysical Review Letters vol 97 no 14 Article ID 140402 2006

[13] E Masso and R Toldra ldquoLight spinless particle coupled tophotonsrdquo Physical Review D vol 52 no 4 pp 1755ndash1763 1995

[14] P Gaete and E I Guendelman ldquoConfinement in the presenceof external fields and axionsrdquoModern Physics Letters A vol 20no 5 article 319 2005

[15] P Gaete and E Spallucci ldquoConfinement effects from interactingchromo-magnetic and axion fieldsrdquo Journal of Physics A Math-ematical and General vol 39 no 20 pp 6021ndash6029 2006

[16] E S Fradkin and A A Tseytlin ldquoNon-linear electrodynamicsfrom quantized stringsrdquo Physics Letters B vol 163 no 1ndash4 pp123ndash130 1985

[17] E Bergshoeff E Sezgin C N Pope and P K TownsendldquoThe Born-Infeld action from conformal invariance of the opensuperstringrdquo Physics Letters B vol 188 no 1 pp 70ndash74 1987

[18] P Gaete and J Helayel-Neto ldquoFinite field-energy and interpar-ticle potential in logarithmic electrodynamicsrdquo The EuropeanPhysical Journal C vol 74 no 3 article 2816 2014

[19] P Gaete and J Helayel-Neto ldquoRemarks on nonlinear electrody-namicsrdquo European Physical Journal C vol 74 no 11 article 31822014

[20] S H Hendi ldquoAsymptotic ReissnerndashNordstrom black holesrdquoAnnals of Physics vol 333 pp 282ndash289 2013

[21] Z Zhao Q Pan S Chen and J Jing ldquoNotes on holographicsuperconductor models with the nonlinear electrodynamicsrdquoNuclear Physics B vol 871 no 1 pp 98ndash110 2013

[22] O Miskovic and R Olea ldquoConserved charges for black holes inEinstein-Gauss-Bonnet gravity coupled to nonlinear electrody-namics in AdS spacerdquo Physical Review D vol 83 no 2 ArticleID 024011 2011

[23] S H Mazharimousavi and M Halilsoy ldquoBlack holes and theclassical model of a particle in Einstein non-linear electrody-namics theoryrdquo Physics Letters B vol 678 no 4 pp 407ndash4102009

[24] G Amelino-Camelia ldquoSpecial treatmentrdquo Nature vol 418 no6893 pp 34ndash35 2002

[25] T Jacobson S Liberati andDMattingly ldquoThreshold effects andPlanck scale Lorentz violation combined constraints from highenergy astrophysicsrdquo Physical Review D vol 67 no 12 ArticleID 124011 2003

[26] T J Konopka and S A Major ldquoObservational limits onquantum geometry effectsrdquo New Journal of Physics vol 4 pp571ndash5718 2002

[27] SHossenfelder ldquoInterpretation of quantumfield theories with aminimal length scalerdquo Physical Review D vol 73 no 10 ArticleID 105013 9 pages 2006

[28] P Nicolini ldquoNoncommutative black holes the final appeal toquantum gravity a reviewrdquo International Journal of ModernPhysics A vol 24 no 7 pp 1229ndash1308 2009

[29] S Pramanik S Ghosh and P Pal ldquoElectrodynamics of a gener-alized charged particle in doubly special relativity frameworkrdquoAnnals of Physics vol 346 pp 113ndash128 2014

[30] A F Ali S Das and E C Vagenas ldquoDiscreteness of space fromthe generalized uncertainty principlerdquo Physics Letters B vol678 no 5 pp 497ndash499 2009

[31] G Amelino-Camelia ldquoQuantum-spacetime phenomenologyrdquoLiving Reviews in Relativity vol 16 article 5 2013

10 Advances in High Energy Physics

[32] E Witten ldquoNoncommutative geometry and string field theoryrdquoNuclear Physics B vol 268 no 2 pp 253ndash294 1986

[33] N Seiberg and E Witten ldquoString theory and noncommutativegeometryrdquo Journal of High Energy Physics vol 1999 article 0321999

[34] M R Douglas and N A Nekrasov ldquoNoncommutative fieldtheoryrdquo Reviews of Modern Physics vol 73 no 4 pp 977ndash10292001

[35] R J Szabo ldquoQuantum field theory on noncommutative spacesrdquoPhysics Reports vol 378 no 4 pp 207ndash299 2003

[36] J Gomis K Kamimura and T Mateos ldquoGauge and BRST gen-erators for space-time non-commutative U(1) theoryrdquo Journalof High Energy Physics vol 2001 no 3 article 010 2001

[37] A A Bichl J M Grimstrup L Popp M Schweda and RWulkenhaar ldquoPerturbative analysis of the Seiberg-Witten maprdquoInternational Journal of Modern Physics A vol 17 no 16 pp2219ndash2231 2002

[38] A Smailagic and E Spallucci ldquoUV divergence-free QFT onnoncommutative planerdquo Journal of Physics AMathematical andGeneral vol 36 no 39 pp L517ndashL521 2003

[39] A Smailagic and E Spallucci ldquoFeynman path integral on thenon-commutative planerdquo Journal of Physics A Mathematicaland General vol 36 no 33 pp L467ndashL471 2003

[40] M Knezevic D Knezevic and D Spasojevic ldquoStatistics ofequally weighted random paths on a class of self-similarstructuresrdquo Journal of Physics A Mathematical and Generalvol 37 no 1 pp 1ndash8 2004 Erratum in Journal of Physics AMathematical and General vol 37 p 7169 2004

[41] E Spallucci A Smailagic and P Nicolini ldquoTrace anomaly ona quantum spacetime manifoldrdquo Physical Review D vol 73Article ID 084004 2006

[42] R Banerjee S Gangopadhyay and S K Modak ldquoVoros prod-uct noncommutative Schwarzschild black hole and correctedarea lawrdquo Physics Letters B vol 686 no 2-3 pp 181ndash187 2010

[43] A B Hammou M Lagraa and M M Sheikh-Jabbari ldquoCoher-ent state induced star product on 119877

3

120582and the fuzzy sphererdquo

Physical Review D vol 66 no 2 Article ID 025025 2002[44] L Modesto J W Moffat and P Nicolini ldquoBlack holes in an

ultraviolet complete quantum gravityrdquo Physics Letters B vol695 no 1ndash4 pp 397ndash400 2011

[45] P Nicolini ldquoNonlocal and generalizeduncertainty principleblack holesrdquo httparxivorgabs12022102

[46] P Nicolini A Smailagic and E Spallucci ldquoNoncommutativegeometry inspired Schwarzschild black holerdquo Physics Letters Bvol 632 no 4 pp 547ndash551 2006

[47] S Pramanik S Das and S Ghosh ldquoNoncommutative extensionof AdS-CFT and holographic superconductorsrdquo Physics LettersB vol 742 pp 266ndash273 2015

[48] W Dittrich and M Reuter Effective Lagrangians in QuantumElectrodynamics Springer Berlin Germany 1985

[49] M Ostrogradsky ldquoMemoires sur les equations differen-tielles relativrs au probleme des isoperimetersrdquo Memoires delrsquoAcademie Imperiale des Sciences de St Petersbourg vol 4 p385 1850

[50] P Gaete and E Spallucci ldquoFinite axionic electrodynamicsfrom a new non-commutative approachrdquo Journal of Physics AMathematical and Theoretical vol 45 no 6 Article ID 06540114 pages 2012

[51] P Gaete ldquoSome considerations about Podolsky-axionic electro-dynamicsrdquo International Journal of Modern Physics A vol 27no 11 Article ID 1250061 2012

[52] P Gaete ldquoRemarks on gauge-invariant variables and interactionenergy in QEDrdquo Physical Review D vol 59 no 12 Article ID127702 1999

[53] S Klarsfeld ldquoAnalytical expressions for the evaluation ofvacuum-polarization potentials in muonic atomsrdquo Physics Let-ters B vol 66 no 1 pp 86ndash88 1977

[54] A M Frolov and D M Wardlaw ldquoAnalytical formula for theUehling potentialrdquo The European Physical Journal B vol 85article 348 2012

[55] P Gaete ldquoOn gauge-invariant variables in QEDrdquo Zeitschrift furPhysik C Particles and Fields vol 76 no 2 pp 355ndash361 1997

[56] S Ferrara and A Sagnotti ldquoMassive Born-Infeld and other dualpairsrdquo Journal of High Energy Physics vol 2015 no 4 article 0322015

[57] R Bufalo ldquoBornndashinfeld electrodynamics in very special relativ-ityrdquo Physics Letters B vol 746 pp 251ndash256 2015

[58] HKleinert E Strobel and S-S Xue ldquoFractional effective actionat strong electromagnetic fieldsrdquo Physical Review D vol 88 no2 Article ID 025049 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

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FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

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Statistical MechanicsInternational Journal of

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GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

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Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

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Soft MatterJournal of

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PhotonicsJournal of

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Journal of

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ThermodynamicsJournal of

Advances in High Energy Physics 5

3 Euler-Heisenberg-Like Model

Proceeding in the sameway aswe did in the foregoing sectionwe will now consider the interaction energy for Euler-Heisenberg-like electrodynamics Nevertheless in order toput our discussion into context it is useful to describe verybriefly the model under consideration In such a case theLagrangian density reads

L =1205732

21 minus [1 +

1

1205732F minus

1

12057321205742G

2]

119901

(26)

where we have included two parameters 120573 and 120574 As usualF = (14)119865

120583]119865120583] G = (14)119865

120583]119865120583] 119865

120583] = 120597120583119860] minus 120597]119860120583

and 119865

120583]= (12)120576

120583]120588120582119865120588120582 Let us also mention here that in

our previous paper [19] we have studied the domain 0 lt

119901 lt 1 Moreover it follows from (26) that when 119901 = 2

the model contains to orders O(11205732) and O(11205742

) a Euler-Heisenberg-likemodel with the appropriate identifications ofthe constants Interestingly we also observe that in the limit120574 rarr infin we obtain a Wichmann-Kroll model This remarkopens up the way to discuss the effect of these nonlinearcorrections on the interaction energy as we are going tostudy below In fact we will consider a massive Wichmann-Kroll systemThemotivation for this study comes from recentconsiderations in the context of dualities [56] where massiveBorn-Infeld systems play an important role

Having made these observations we can write immedi-ately the field equations for 119901 = 2

120597120583[Γ(119865

120583]minus

2

1205742G119865

120583])] = 0 (27)

while the Bianchi identities are given by

120597120583119865

120583]= 0 (28)

where

Γ = 1 +F

1205732minus

G2

12057321205742 (29)

Also it is straightforward to see that Gauss law becomes

nabla sdotD = 0 (30)

whereD is given by

D

= [1 minus

(E2minus B2

)

21205732

minus(E sdot B)2

12057321205742

](E +2

1205742(E sdot B)B)

(31)

Again from (30) for 1198690(119905 r) = 119890120575(3)(r) we findD = (119876119903

2)119903

where 119876 = 1198904120587 This then implies that for a point-likecharge 119890 at the origin the expression

119876

1199032= (1 minus

E2

21205732) |E| (32)

tells us that for 119903 rarr 0 the electrostatic field becomessingular at 119903 = 0 in contrast to the 0 lt 119901 lt 1 case wherethe electrostatic field is finite Even so in this theory thephenomenon of birefringence is present Before going intodetails we would like to recall that birefringence refers to theproperty that polarized light in a particular direction (opticalaxis) travels at a different velocity from that of light polarizedin a direction perpendicular to this axis Indeed due toquantum fluctuations the QED vacuum has this property aswe are going to show

To illustrate this important feature we introduce thevectorsD = 120597L120597E andH = minus120597L120597B

D = Γ(E + 2B (E sdot B)

1205742

)

H = Γ(B minus 2E (E sdot B)

1205742

)

(33)

where Γ = 1 + (121205732)(B2

minus E2) minus (1120573

21205742)(E sdot B)2 We thus

obtain the equations of motion

nabla sdotD = 0

120597D120597119905

minus nabla timesH = 0

nabla sdot B = 0

120597B120597119905

+ nabla times E = 0

(34)

With the aid from (33) we find the electric permittivity 120576119894119895

and the inverse magnetic permeability (120583minus1)119894119895 tensors of the

vacuum that is

120576119894119895= Γ(120575

119894119895+

2119861119894119861119895

1205742

)

(120583minus1)119894119895= Γ(120575

119894119895minus

2119864119894119864

119895

1205742

)

(35)

with119863119894= 120576

119894119895119864

119895and 119861

119894= 120583

119894119895119867

119895

In accordance with our previous procedure [18 19] wecan now linearize the above equations To do this it isadvantageous to introduce a weak electromagnetic wave(E

119901B

119901) propagating in the presence of a strong constant

external field (E0B

0) On these assumptions we readily find

that for the case of a purely magnetic field (E0= 0) the

vectorsD andH become

D = (1 +B2

0

21205732)[E

119901+

2

1205742(E

119901sdot B

0)B

0]

H = (1 +B2

0

21205732)

sdot [B119901+

1

1205732(1 + B2

02120573

2)(B

119901sdot B

0)B

0]

(36)

where we have to keep only linear terms in E119901B

119901 As

before we consider the 119911-axis as the direction of the external

6 Advances in High Energy Physics

magnetic field (B0= 119861

0e3) and assuming that the light wave

moves along the 119909-axis the decomposition into a plane wavefor the fields E

119901and B

119901can be written as

E119901 (x 119905) = E119890minus119894(119908119905minusksdotx)

B119901(x 119905) = B119890minus119894(119908119905minusksdotx)

(37)

In this case it clearly follows that

(1198962

1199082minus 120576

2212058333)119864

2= 0 (38)

(1198962

1199082minus 120576

3312058322)119864

3= 0 (39)

As a consequence we have two different situations Firstif E perp B

0(perpendicular polarization) from (39) 119864

3= 0 and

from (38) we get 1198962119908

2= 120576

2212058333 This then means that the

dispersion relation of the photon takes the form

119899perp= radic

1 + B2

02120573

2

1 + 3B2

02120573

2 (40)

Second if E B0(parallel polarization) from (38) 119864

2= 0

and from (39) we get 1198962119908

2= 120576

3312058322 This leads to

119899= radic1 +

2B2

0

1205742 (41)

Thus we verify that in the case of a generalized Euler-Heisenberg electrodynamics the phenomenon of birefrin-gence is present

We now pass to the calculation of the interaction energybetween static point-like sources for a massive Wichmann-Kroll-like model our analysis follows closely that of [18 19]The corresponding theory is governed by the Lagrangiandensity

L = minus1

4119865

2

120583] +1

321205732(119865

120583]119865120583])2

+119898

2

2119860

120583119860

120583 (42)

Next in order to handle the second term on the right hand in(42) we introduce an auxiliary field 120585 such that its equation ofmotion gives back the original theory This allows us to writethe Lagrangian density as

L = minus1

4119865120583]119865

120583]+

120585

321205732119865120583]119865

120583]minus

1

12812057321205852

+119898

2

2119860

120583119860

120583

(43)

With the redefinition 120578 = 1 minus 12058581205732 (43) becomes

L = minus1

4120578119865

120583]119865120583]minus1

2(1 minus 120578)

2+119898

2

2119860

120583119860

120583 (44)

Before we proceed to work out explicitly the interactionenergy we will first restore the gauge invariance in (44)

Following an earlier procedure we readily verify that thecanonical momenta read Π

120583= minus120578119865

0120583 which results in theusual primary constraint Π0

= 0 and Π119894= 120578119865

1198940 In this wayone obtains

119867119862= int119889

3119909Π

119894120597119894119860

0+

1

2120578Π

2+120578

2B2

minus119898

2

2119860

120583119860

120583

+1

2(1 minus 120578)

2

(45)

The consistency condition Π = 0 leads to the constraint Γ equiv

120597119894Π

119894+119898

2119860

0= 0 As a result both constraints are second-class

To convert the second-class system into first-class we willadopt the procedure described previously Thus we enlargethe original phase space by introducing a canonical pair offields 120579 and Π

120579 It follows therefore that a new set of first-

class constraints can be defined in this extended space Λ1equiv

Π0+ 119898

2120579 = 0 and Λ

2equiv Γ + Π

120579= 0 Notice that this new

120579-field is not to be confused with the usual noncommutativeparameter This then shows that the new constraints arefirst-class and therefore restore the gauge symmetry As iswell known this procedure reproduces the usual Stuckelbergformalism From this the new effective Lagrangian densityafter integrating out the 120579 fields becomes

L = minus1

4119865120583] (120578 +

1198982

Δ)119865

120583]minus1

2(1 minus 120578)

2 (46)

Now writing 120590 = 120578 + 1198982Δ expression (46) can be brought

to the form

L = minus1

4119865120583]120590119865

120583]minus

119896

128(1 minus 120590 +

1198982

Δ)

2

(47)

where 119896 = 641205732

We are now ready to compute the interaction energy Inthis case the canonical momenta are Π120583

= minus1205901198650120583 with the

usual primary constraint Π0= 0 and Π

119894= 120590119865

1198940 Hence thecanonical Hamiltonian is expressed as

119867119862= int119889

3119909Π

119894120597119894119860

0+

1

2120590Π

2+120590

2B2

+119896

128(1 minus 120590 +

1198982

Δ)

2

(48)

Time conservation of the primary constraint Π0 yields thesecondary constraint Γ

1equiv 120597

119894Π

119894= 0 Similarly the P

120590

constraint yields no further constraints and just determinesthe field 120590 In this case at leading order in 120573 the field 120590 isgiven by

120590 = (1 +119898

2

Δminus

B2

21205732)

sdot [1 minus3

21205732

1

(1 + 1198982Δ minus B2

21205732)3Π

2]

(49)

Advances in High Energy Physics 7

which will be used to eliminate 120590 As before the corre-sponding total (first-class) Hamiltonian that generates thetime evolution of the dynamical variables is 119867 = 119867

119862+

int1198893119909(119906

0(119909)Π

0(119909) + 119906

1(119909)Γ

1(119909)) where 119906

0(119909) and 119906

1(119909) are

the Lagrangemultiplier utilized to implement the constraintsIn the same way as was done in the previous subsection

the expectation value of the energy operator119867 in the physicalstate |Φ⟩ becomes

⟨119867⟩Φ = ⟨Φ|int1198893119909

1

119894(1 +

1198982

Δ)

minus1

Π119894+

15

81205732Π

4

minus15119898

2

21205732Π

2 1

ΔΠ

2 |Φ⟩

(50)

in this last line we have considered only quadratic terms in119898

2In such a case by employing (50) the lowest-order

modification in 1205732 and1198982 of the interaction energy takes theform

⟨119867⟩Φ = ⟨119867⟩0 + 1198811+ 119881

2+ 119881

3 (51)

where ⟨119867⟩0= ⟨0|119867|0⟩ The 119881

1 119881

2 and 119881

3terms are given by

1198811=1199022

2int119889

3119909int

y

y1015840119889119911

1015840119894120575(3)

(x minus z1015840)(1 minus 1198982

nabla2)

minus1

sdot int

y

y1015840119889119911

119894120575(3)

(x minus z)

1198812= minus

151199024

81205732int119889

3119909int

y

y1015840119889119911

119894120575(3)

(x minus z)

sdot int

y

y1015840119889119911

1015840119894120575(3)

(x minus z1015840)inty

y1015840119889119906

119896120575(3)

(x minus u)

sdot int

y

y1015840119889V

119896120575(3)

(x minus k)

1198813=15119898

21199024

21205732

int1198893119909int

y

y1015840119889119911

119894120575(3)

(x minus z)

sdot int

y

y1015840119889119911

1015840119894120575(3)

(x minus z1015840)inty

y1015840119889119906

119896120575(3)

(x minus u)

sdot int

y

y1015840119889V

119896120575(3)

(x minus k)

(52)

Finally with the aid of expressions (52) the potential fora pair of static point-like opposite charges located at 0 and Lis given by

119881 = minus1199022

4120587

119890minus119898119871

119871+

1199024

161205871205732(3

8120587

1

1198712minus 5119898

2)

1

1198713 (53)

observe that when 119898 = 0 profile (53) reduces to the knownWichmann-Kroll interaction energy On the other hand for119898 = 0 the key role played by the mass term in transformingthe Coulomb potential into the Yukawa one should be notedInterestingly enough an unexpected feature is found In factprofile (53) displays a new long-range 11198713 correction whereits strength is proportional to 119898

2 It is also important toobserve that an analogous correction has been found in Born-Infeld electrodynamics in the context of very special relativity[57] In this way we establish a new connection betweennonlinear effectives theories

Before we proceed further we should comment on ourresult In the case of QED (Euler-Heisenberg Lagrangiandensity) the parameter 11205732 is given by 11205732

= (1645)(1198904ℏ

1198984

1198901198887) where 119898

119890is the electron mass In this context we

also recall the currently accepted upper limit for the photonmass that is 119898

120574sim 2 times 10

minus16 eV Thus for the QED casefrom (53) it follows that the second term on the right handside would be detectable in long-range distances (sim109 m) Inother words we see that detectable corrections induced byvacuum polarization with a mass term would be present atlow energy scales

From (53) it clearly follows that the interaction energybetween heavy charged charges at leading order in 120573 is notfinite at the origin Motivated by this one may consider theabove calculation in a noncommutative geometry based onfindings of our previous studies [18 19] In such a case theelectric field at leading order in 120573

2 and1198982 takes the form

119864119894= [(1 +

1198982

Δ)

minus1

+3

21205732Π

2minus6119898

2

1205732Π

2 1

Δ]

sdot 120597119894(minus

119890120579nabla2

120575(3)

(x)nabla

2)

(54)

where it may be recalled that we are now replacing the source120575(3)(x minus y) by the smeared source 119890

120579nabla2

120575(3)(x minus y) with the

parameter 120579 being noncommutative Now making use of(22) we readily find that

A0(119905 r) = A

(1)

0(119905 r) +A

(2)

0(119905 r) +A

(3)

0(119905 r) (55)

The term A(1)

0(119905 r) was first calculated in [50] we can

therefore write only the result

A(1)

0(119905 r)

= 1199021198901198982120579

4120587

1

119903[119890

minus119898119903minus

1

radic120587int

infin

11990324120579

1198891199061

radic119906119890minus119906minus119898

211990324119906

]

minus 119902119898

41205871198901198982120579

(56)

8 Advances in High Energy Physics

Meanwhile the terms A(2)

0(119905 r) and A

(3)

0(119905 r) after some

manipulation can be brought to the form

A(2)

0(119905 r) =

121199023

120573212058732

119899119894int

119909

0

119889119906119894 1

11990661205743(3

21199062

4120579)

A(3)

0(119905 r) =

311989821199023

120573212058752

119899119894int

119909

0

119889119906119894 1

11990641205742(3

21199062

4120579)

sdot [4120579

1199062120574(

3

21199062

4120579) minus 120574(

1

21199062

4120579)]

(57)

where 120574(32 11990324120579) is the lower incomplete Gamma functiondefined by 120574(119886119887 119909) equiv int

119909

0(119889119906119906)119906

119886119887119890minus119906

Inserting these expressions in (21) we finally obtain thestatic potential for two opposite charges 119902 located at 0 and Las

119881 = minus119902

4120587

1198901198982120579

119871[119890

minus119898119871minus

1

radic120587int

infin

11987124120579

1198891199061

radic119906119890minus119906minus119898

211990624119906

]

minus12119902

4

120573212058732

119899119894int

119871

0

119889119906119894 1

11990661205743(3

21199062

4120579) minus

311989821199024

120573212058752

sdot 119899119894int

119909

0

119889119906119894 1

11990641205742(3

21199062

4120579)

sdot [4120579

1199062120574(

3

21199062

4120579) minus 120574(

1

21199062

4120579)]

(58)

which is finite for 119871 rarr 0 It is a simple matter to verify thatin the limit 120579 rarr 0 we recover our above result

4 Logarithmic Correction

We now want to extend what we have done to Euler-Heisenberg-like electrodynamics at strong fields As alreadymentioned such theories show a power behavior that istypical for critical phenomena [58] In such a case theLagrangian density reads

L = minus1

4119865120583]119865

120583]minus1198870

8119865120583]119865

120583] log(119865120583]119865

120583]

41205822

) (59)

where 1198870and 120582 are constants In fact by choosing 119887

0=

11989026120587

2 and 120582 = 1198982

1198901198883119890ℏ we recover the Euler-Heisenberg

electrodynamics at strong fields [58]In the same way as was done in the previous section one

can introduce an auxiliary field 120585 to handle the logarithm in(59) This leads to

L = minus1

41205721119865120583]119865

120583]minus 120572

2(119865

120583]119865120583])2

(60)

where 1205721= 1 minus (119887

02)(1 + log 120585) and 120572

2= 119887

012058532120582

2A similar procedure can be used to manipulate the

quadratic term in (60) Accordingly by introducing a secondauxiliary field 120578 one easily finds

L = minus1

4119865120583] (1205721

+ 41205722120578) 119865

120583]+1205782

41205722 (61)

By setting 120590 = 1205721+ 4120572

2120578 we then have

L = minus1

4120590119865

120583]119865120583]+

1

641205722

(120590 minus 1205721)2 (62)

It is once again straightforward to apply the gauge-invariant formalism discussed in the foregoing section Thecanonical momenta read Π

120583= minus120590119865

0120583 and at once werecognize the two primary constraints Π0

= 0 and P120590

equiv

120597119871120597 = 0The canonical Hamiltonian corresponding to (62)is

119867119862= int119889

3119909

sdot Π119894120597119894119860

0+

1

2120590Π

2+120590

2B2

minus1

641205722

(120590 minus 1205721)2

(63)

Requiring the primary constraint Π0 to be preserved intime one obtains the secondary constraint Γ

1= 120597

119894Π

119894= 0 In

the same way for the constraintP120590 we get the auxiliary field

120590 as

120590 = (1 minus1198870

2(1 + ln 120585) +

1198870B2

21205822120585)[1

+3119887

0B2

21205822

120585

(1 minus (11988702) (1 + ln 120585) + (119887

0B2

21205822) 120585)

3]

(64)

Hence we obtain

119867119862= int119889

3119909

sdot Π119894120597119894119860

0+1

2+1198870

4(1 + log 120585)Π2 minus

31198870120585

21205822Π

4

(65)

As before requiring the primary constraint P120585to be

preserved in time one obtains the auxiliary field 120585 In this case120585 = 1205826Π

2 Consequently we get

119867119862= int119889

3119909Π

119894120597119894119860

0+1

2(1 + 119887

0)Π

2minus6119887

0

1205822Π

4 (66)

Following the same steps that led to (50) we find that

⟨119867⟩(1)

Φ= ⟨Φ|int119889

3119909

1

2minus

3

81205732Π

4 |Φ⟩ (67)

It should be noted that this expression is similar to (50) in thelimit 119898 rarr 0 except by the changed sign in front of the Π4termHencewe see that the potential for two opposite chargesin 0 and L is given by

119881 = minus1199022

4120587

1

119871minus

1199024

604012057321205872

1

1198715 (68)

5 Final Remarks

Finally within the gauge-invariant but path-dependent vari-ables formalism we have considered the confinement versus

Advances in High Energy Physics 9

screening issue for both massive Euler-Heisenberg-like andEuler-Heisenberg electrodynamics in the approximation ofthe strong-field limit Once again a correct identificationof physical degrees of freedom has been fundamental forunderstanding the physics hidden in gauge theories Inter-estingly enough their noncommutative version displays anultraviolet finite static potential The analysis above revealsthe key role played by the new quantum of length in ouranalysis In a general perspective the benefit of consideringthe present approach is to provide a unification scenarioamong different models as well as exploiting the equivalencein explicit calculations as we have illustrated in the course ofthis work

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

It is a pleasure for the author to thank J A Helayel-Netofor helpful comments on the paper This work was partiallysupported by Fondecyt (Chile) Grant 1130426 and DGIP(UTFSM) internal project USM 111458

References

[1] G Breit and J A Wheeler ldquoCollision of two light quantardquoPhysical Review vol 46 no 12 pp 1087ndash1091 1934

[2] S L Adler ldquoPhoton splitting and photon dispersion in a strongmagnetic fieldrdquo Annals of Physics vol 67 no 2 pp 599ndash6471971

[3] V Costantini B De Tollis and G Pistoni ldquoNonlinear effects inquantum electrodynamicsrdquo Il Nuovo Cimento A vol 2 no 3pp 733ndash787 1971

[4] R Ruffini G Vereshchagin and S-S Xue ldquoElectron-positronpairs in physics and astrophysics from heavy nuclei to blackholesrdquo Physics Reports vol 487 no 1ndash4 pp 1ndash140 2010

[5] G V Dunne ldquoThe Heisenberg-Euler effective action 75 yearsonrdquo International Journal of Modern Physics Conference Seriesvol 14 pp 42ndash56 2012

[6] C Bamber S J Boege TKoffas et al ldquoStudies of nonlinearQEDin collisions of 466GeV electrons with intense laser pulsesrdquoPhysical Review D vol 60 no 9 Article ID 092004 1999

[7] D L Burke R C Field G Horton-Smith et al ldquoPositronproduction in multiphoton light-by-light scatteringrdquo PhysicalReview Letters vol 79 no 9 article 1626 1997

[8] O J Pike F MacKenroth E G Hill and S J Rose ldquoA photonndashphoton collider in a vacuum hohlraumrdquo Nature Photonics vol8 no 6 pp 434ndash436 2014

[9] D Tommasini A Ferrando H Michinel and M Seco ldquoPre-cision tests of QED and non-standard models by searchingphoton-photon scattering in vacuum with high power lasersrdquoJournal ofHigh Energy Physics vol 2009 no 11 article 043 2009

[10] D Tommasini A Ferrando HMichinel andM Seco ldquoDetect-ing photon-photon scattering in vacuum at exawatt lasersrdquoPhysical Review A vol 77 no 4 Article ID 042101 2008

[11] M Born and L Infeld ldquoFoundations of the new field theoryrdquoProceedings of the Royal Society of London Series A Containing

Papers of a Mathematical and Physical Character vol 144 no852 pp 425ndash451 1934

[12] H Gies J Jaeckel and A Ringwald ldquoPolarized light propagat-ing in a magnetic field as a probe for millicharged fermionsrdquoPhysical Review Letters vol 97 no 14 Article ID 140402 2006

[13] E Masso and R Toldra ldquoLight spinless particle coupled tophotonsrdquo Physical Review D vol 52 no 4 pp 1755ndash1763 1995

[14] P Gaete and E I Guendelman ldquoConfinement in the presenceof external fields and axionsrdquoModern Physics Letters A vol 20no 5 article 319 2005

[15] P Gaete and E Spallucci ldquoConfinement effects from interactingchromo-magnetic and axion fieldsrdquo Journal of Physics A Math-ematical and General vol 39 no 20 pp 6021ndash6029 2006

[16] E S Fradkin and A A Tseytlin ldquoNon-linear electrodynamicsfrom quantized stringsrdquo Physics Letters B vol 163 no 1ndash4 pp123ndash130 1985

[17] E Bergshoeff E Sezgin C N Pope and P K TownsendldquoThe Born-Infeld action from conformal invariance of the opensuperstringrdquo Physics Letters B vol 188 no 1 pp 70ndash74 1987

[18] P Gaete and J Helayel-Neto ldquoFinite field-energy and interpar-ticle potential in logarithmic electrodynamicsrdquo The EuropeanPhysical Journal C vol 74 no 3 article 2816 2014

[19] P Gaete and J Helayel-Neto ldquoRemarks on nonlinear electrody-namicsrdquo European Physical Journal C vol 74 no 11 article 31822014

[20] S H Hendi ldquoAsymptotic ReissnerndashNordstrom black holesrdquoAnnals of Physics vol 333 pp 282ndash289 2013

[21] Z Zhao Q Pan S Chen and J Jing ldquoNotes on holographicsuperconductor models with the nonlinear electrodynamicsrdquoNuclear Physics B vol 871 no 1 pp 98ndash110 2013

[22] O Miskovic and R Olea ldquoConserved charges for black holes inEinstein-Gauss-Bonnet gravity coupled to nonlinear electrody-namics in AdS spacerdquo Physical Review D vol 83 no 2 ArticleID 024011 2011

[23] S H Mazharimousavi and M Halilsoy ldquoBlack holes and theclassical model of a particle in Einstein non-linear electrody-namics theoryrdquo Physics Letters B vol 678 no 4 pp 407ndash4102009

[24] G Amelino-Camelia ldquoSpecial treatmentrdquo Nature vol 418 no6893 pp 34ndash35 2002

[25] T Jacobson S Liberati andDMattingly ldquoThreshold effects andPlanck scale Lorentz violation combined constraints from highenergy astrophysicsrdquo Physical Review D vol 67 no 12 ArticleID 124011 2003

[26] T J Konopka and S A Major ldquoObservational limits onquantum geometry effectsrdquo New Journal of Physics vol 4 pp571ndash5718 2002

[27] SHossenfelder ldquoInterpretation of quantumfield theories with aminimal length scalerdquo Physical Review D vol 73 no 10 ArticleID 105013 9 pages 2006

[28] P Nicolini ldquoNoncommutative black holes the final appeal toquantum gravity a reviewrdquo International Journal of ModernPhysics A vol 24 no 7 pp 1229ndash1308 2009

[29] S Pramanik S Ghosh and P Pal ldquoElectrodynamics of a gener-alized charged particle in doubly special relativity frameworkrdquoAnnals of Physics vol 346 pp 113ndash128 2014

[30] A F Ali S Das and E C Vagenas ldquoDiscreteness of space fromthe generalized uncertainty principlerdquo Physics Letters B vol678 no 5 pp 497ndash499 2009

[31] G Amelino-Camelia ldquoQuantum-spacetime phenomenologyrdquoLiving Reviews in Relativity vol 16 article 5 2013

10 Advances in High Energy Physics

[32] E Witten ldquoNoncommutative geometry and string field theoryrdquoNuclear Physics B vol 268 no 2 pp 253ndash294 1986

[33] N Seiberg and E Witten ldquoString theory and noncommutativegeometryrdquo Journal of High Energy Physics vol 1999 article 0321999

[34] M R Douglas and N A Nekrasov ldquoNoncommutative fieldtheoryrdquo Reviews of Modern Physics vol 73 no 4 pp 977ndash10292001

[35] R J Szabo ldquoQuantum field theory on noncommutative spacesrdquoPhysics Reports vol 378 no 4 pp 207ndash299 2003

[36] J Gomis K Kamimura and T Mateos ldquoGauge and BRST gen-erators for space-time non-commutative U(1) theoryrdquo Journalof High Energy Physics vol 2001 no 3 article 010 2001

[37] A A Bichl J M Grimstrup L Popp M Schweda and RWulkenhaar ldquoPerturbative analysis of the Seiberg-Witten maprdquoInternational Journal of Modern Physics A vol 17 no 16 pp2219ndash2231 2002

[38] A Smailagic and E Spallucci ldquoUV divergence-free QFT onnoncommutative planerdquo Journal of Physics AMathematical andGeneral vol 36 no 39 pp L517ndashL521 2003

[39] A Smailagic and E Spallucci ldquoFeynman path integral on thenon-commutative planerdquo Journal of Physics A Mathematicaland General vol 36 no 33 pp L467ndashL471 2003

[40] M Knezevic D Knezevic and D Spasojevic ldquoStatistics ofequally weighted random paths on a class of self-similarstructuresrdquo Journal of Physics A Mathematical and Generalvol 37 no 1 pp 1ndash8 2004 Erratum in Journal of Physics AMathematical and General vol 37 p 7169 2004

[41] E Spallucci A Smailagic and P Nicolini ldquoTrace anomaly ona quantum spacetime manifoldrdquo Physical Review D vol 73Article ID 084004 2006

[42] R Banerjee S Gangopadhyay and S K Modak ldquoVoros prod-uct noncommutative Schwarzschild black hole and correctedarea lawrdquo Physics Letters B vol 686 no 2-3 pp 181ndash187 2010

[43] A B Hammou M Lagraa and M M Sheikh-Jabbari ldquoCoher-ent state induced star product on 119877

3

120582and the fuzzy sphererdquo

Physical Review D vol 66 no 2 Article ID 025025 2002[44] L Modesto J W Moffat and P Nicolini ldquoBlack holes in an

ultraviolet complete quantum gravityrdquo Physics Letters B vol695 no 1ndash4 pp 397ndash400 2011

[45] P Nicolini ldquoNonlocal and generalizeduncertainty principleblack holesrdquo httparxivorgabs12022102

[46] P Nicolini A Smailagic and E Spallucci ldquoNoncommutativegeometry inspired Schwarzschild black holerdquo Physics Letters Bvol 632 no 4 pp 547ndash551 2006

[47] S Pramanik S Das and S Ghosh ldquoNoncommutative extensionof AdS-CFT and holographic superconductorsrdquo Physics LettersB vol 742 pp 266ndash273 2015

[48] W Dittrich and M Reuter Effective Lagrangians in QuantumElectrodynamics Springer Berlin Germany 1985

[49] M Ostrogradsky ldquoMemoires sur les equations differen-tielles relativrs au probleme des isoperimetersrdquo Memoires delrsquoAcademie Imperiale des Sciences de St Petersbourg vol 4 p385 1850

[50] P Gaete and E Spallucci ldquoFinite axionic electrodynamicsfrom a new non-commutative approachrdquo Journal of Physics AMathematical and Theoretical vol 45 no 6 Article ID 06540114 pages 2012

[51] P Gaete ldquoSome considerations about Podolsky-axionic electro-dynamicsrdquo International Journal of Modern Physics A vol 27no 11 Article ID 1250061 2012

[52] P Gaete ldquoRemarks on gauge-invariant variables and interactionenergy in QEDrdquo Physical Review D vol 59 no 12 Article ID127702 1999

[53] S Klarsfeld ldquoAnalytical expressions for the evaluation ofvacuum-polarization potentials in muonic atomsrdquo Physics Let-ters B vol 66 no 1 pp 86ndash88 1977

[54] A M Frolov and D M Wardlaw ldquoAnalytical formula for theUehling potentialrdquo The European Physical Journal B vol 85article 348 2012

[55] P Gaete ldquoOn gauge-invariant variables in QEDrdquo Zeitschrift furPhysik C Particles and Fields vol 76 no 2 pp 355ndash361 1997

[56] S Ferrara and A Sagnotti ldquoMassive Born-Infeld and other dualpairsrdquo Journal of High Energy Physics vol 2015 no 4 article 0322015

[57] R Bufalo ldquoBornndashinfeld electrodynamics in very special relativ-ityrdquo Physics Letters B vol 746 pp 251ndash256 2015

[58] HKleinert E Strobel and S-S Xue ldquoFractional effective actionat strong electromagnetic fieldsrdquo Physical Review D vol 88 no2 Article ID 025049 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

6 Advances in High Energy Physics

magnetic field (B0= 119861

0e3) and assuming that the light wave

moves along the 119909-axis the decomposition into a plane wavefor the fields E

119901and B

119901can be written as

E119901 (x 119905) = E119890minus119894(119908119905minusksdotx)

B119901(x 119905) = B119890minus119894(119908119905minusksdotx)

(37)

In this case it clearly follows that

(1198962

1199082minus 120576

2212058333)119864

2= 0 (38)

(1198962

1199082minus 120576

3312058322)119864

3= 0 (39)

As a consequence we have two different situations Firstif E perp B

0(perpendicular polarization) from (39) 119864

3= 0 and

from (38) we get 1198962119908

2= 120576

2212058333 This then means that the

dispersion relation of the photon takes the form

119899perp= radic

1 + B2

02120573

2

1 + 3B2

02120573

2 (40)

Second if E B0(parallel polarization) from (38) 119864

2= 0

and from (39) we get 1198962119908

2= 120576

3312058322 This leads to

119899= radic1 +

2B2

0

1205742 (41)

Thus we verify that in the case of a generalized Euler-Heisenberg electrodynamics the phenomenon of birefrin-gence is present

We now pass to the calculation of the interaction energybetween static point-like sources for a massive Wichmann-Kroll-like model our analysis follows closely that of [18 19]The corresponding theory is governed by the Lagrangiandensity

L = minus1

4119865

2

120583] +1

321205732(119865

120583]119865120583])2

+119898

2

2119860

120583119860

120583 (42)

Next in order to handle the second term on the right hand in(42) we introduce an auxiliary field 120585 such that its equation ofmotion gives back the original theory This allows us to writethe Lagrangian density as

L = minus1

4119865120583]119865

120583]+

120585

321205732119865120583]119865

120583]minus

1

12812057321205852

+119898

2

2119860

120583119860

120583

(43)

With the redefinition 120578 = 1 minus 12058581205732 (43) becomes

L = minus1

4120578119865

120583]119865120583]minus1

2(1 minus 120578)

2+119898

2

2119860

120583119860

120583 (44)

Before we proceed to work out explicitly the interactionenergy we will first restore the gauge invariance in (44)

Following an earlier procedure we readily verify that thecanonical momenta read Π

120583= minus120578119865

0120583 which results in theusual primary constraint Π0

= 0 and Π119894= 120578119865

1198940 In this wayone obtains

119867119862= int119889

3119909Π

119894120597119894119860

0+

1

2120578Π

2+120578

2B2

minus119898

2

2119860

120583119860

120583

+1

2(1 minus 120578)

2

(45)

The consistency condition Π = 0 leads to the constraint Γ equiv

120597119894Π

119894+119898

2119860

0= 0 As a result both constraints are second-class

To convert the second-class system into first-class we willadopt the procedure described previously Thus we enlargethe original phase space by introducing a canonical pair offields 120579 and Π

120579 It follows therefore that a new set of first-

class constraints can be defined in this extended space Λ1equiv

Π0+ 119898

2120579 = 0 and Λ

2equiv Γ + Π

120579= 0 Notice that this new

120579-field is not to be confused with the usual noncommutativeparameter This then shows that the new constraints arefirst-class and therefore restore the gauge symmetry As iswell known this procedure reproduces the usual Stuckelbergformalism From this the new effective Lagrangian densityafter integrating out the 120579 fields becomes

L = minus1

4119865120583] (120578 +

1198982

Δ)119865

120583]minus1

2(1 minus 120578)

2 (46)

Now writing 120590 = 120578 + 1198982Δ expression (46) can be brought

to the form

L = minus1

4119865120583]120590119865

120583]minus

119896

128(1 minus 120590 +

1198982

Δ)

2

(47)

where 119896 = 641205732

We are now ready to compute the interaction energy Inthis case the canonical momenta are Π120583

= minus1205901198650120583 with the

usual primary constraint Π0= 0 and Π

119894= 120590119865

1198940 Hence thecanonical Hamiltonian is expressed as

119867119862= int119889

3119909Π

119894120597119894119860

0+

1

2120590Π

2+120590

2B2

+119896

128(1 minus 120590 +

1198982

Δ)

2

(48)

Time conservation of the primary constraint Π0 yields thesecondary constraint Γ

1equiv 120597

119894Π

119894= 0 Similarly the P

120590

constraint yields no further constraints and just determinesthe field 120590 In this case at leading order in 120573 the field 120590 isgiven by

120590 = (1 +119898

2

Δminus

B2

21205732)

sdot [1 minus3

21205732

1

(1 + 1198982Δ minus B2

21205732)3Π

2]

(49)

Advances in High Energy Physics 7

which will be used to eliminate 120590 As before the corre-sponding total (first-class) Hamiltonian that generates thetime evolution of the dynamical variables is 119867 = 119867

119862+

int1198893119909(119906

0(119909)Π

0(119909) + 119906

1(119909)Γ

1(119909)) where 119906

0(119909) and 119906

1(119909) are

the Lagrangemultiplier utilized to implement the constraintsIn the same way as was done in the previous subsection

the expectation value of the energy operator119867 in the physicalstate |Φ⟩ becomes

⟨119867⟩Φ = ⟨Φ|int1198893119909

1

119894(1 +

1198982

Δ)

minus1

Π119894+

15

81205732Π

4

minus15119898

2

21205732Π

2 1

ΔΠ

2 |Φ⟩

(50)

in this last line we have considered only quadratic terms in119898

2In such a case by employing (50) the lowest-order

modification in 1205732 and1198982 of the interaction energy takes theform

⟨119867⟩Φ = ⟨119867⟩0 + 1198811+ 119881

2+ 119881

3 (51)

where ⟨119867⟩0= ⟨0|119867|0⟩ The 119881

1 119881

2 and 119881

3terms are given by

1198811=1199022

2int119889

3119909int

y

y1015840119889119911

1015840119894120575(3)

(x minus z1015840)(1 minus 1198982

nabla2)

minus1

sdot int

y

y1015840119889119911

119894120575(3)

(x minus z)

1198812= minus

151199024

81205732int119889

3119909int

y

y1015840119889119911

119894120575(3)

(x minus z)

sdot int

y

y1015840119889119911

1015840119894120575(3)

(x minus z1015840)inty

y1015840119889119906

119896120575(3)

(x minus u)

sdot int

y

y1015840119889V

119896120575(3)

(x minus k)

1198813=15119898

21199024

21205732

int1198893119909int

y

y1015840119889119911

119894120575(3)

(x minus z)

sdot int

y

y1015840119889119911

1015840119894120575(3)

(x minus z1015840)inty

y1015840119889119906

119896120575(3)

(x minus u)

sdot int

y

y1015840119889V

119896120575(3)

(x minus k)

(52)

Finally with the aid of expressions (52) the potential fora pair of static point-like opposite charges located at 0 and Lis given by

119881 = minus1199022

4120587

119890minus119898119871

119871+

1199024

161205871205732(3

8120587

1

1198712minus 5119898

2)

1

1198713 (53)

observe that when 119898 = 0 profile (53) reduces to the knownWichmann-Kroll interaction energy On the other hand for119898 = 0 the key role played by the mass term in transformingthe Coulomb potential into the Yukawa one should be notedInterestingly enough an unexpected feature is found In factprofile (53) displays a new long-range 11198713 correction whereits strength is proportional to 119898

2 It is also important toobserve that an analogous correction has been found in Born-Infeld electrodynamics in the context of very special relativity[57] In this way we establish a new connection betweennonlinear effectives theories

Before we proceed further we should comment on ourresult In the case of QED (Euler-Heisenberg Lagrangiandensity) the parameter 11205732 is given by 11205732

= (1645)(1198904ℏ

1198984

1198901198887) where 119898

119890is the electron mass In this context we

also recall the currently accepted upper limit for the photonmass that is 119898

120574sim 2 times 10

minus16 eV Thus for the QED casefrom (53) it follows that the second term on the right handside would be detectable in long-range distances (sim109 m) Inother words we see that detectable corrections induced byvacuum polarization with a mass term would be present atlow energy scales

From (53) it clearly follows that the interaction energybetween heavy charged charges at leading order in 120573 is notfinite at the origin Motivated by this one may consider theabove calculation in a noncommutative geometry based onfindings of our previous studies [18 19] In such a case theelectric field at leading order in 120573

2 and1198982 takes the form

119864119894= [(1 +

1198982

Δ)

minus1

+3

21205732Π

2minus6119898

2

1205732Π

2 1

Δ]

sdot 120597119894(minus

119890120579nabla2

120575(3)

(x)nabla

2)

(54)

where it may be recalled that we are now replacing the source120575(3)(x minus y) by the smeared source 119890

120579nabla2

120575(3)(x minus y) with the

parameter 120579 being noncommutative Now making use of(22) we readily find that

A0(119905 r) = A

(1)

0(119905 r) +A

(2)

0(119905 r) +A

(3)

0(119905 r) (55)

The term A(1)

0(119905 r) was first calculated in [50] we can

therefore write only the result

A(1)

0(119905 r)

= 1199021198901198982120579

4120587

1

119903[119890

minus119898119903minus

1

radic120587int

infin

11990324120579

1198891199061

radic119906119890minus119906minus119898

211990324119906

]

minus 119902119898

41205871198901198982120579

(56)

8 Advances in High Energy Physics

Meanwhile the terms A(2)

0(119905 r) and A

(3)

0(119905 r) after some

manipulation can be brought to the form

A(2)

0(119905 r) =

121199023

120573212058732

119899119894int

119909

0

119889119906119894 1

11990661205743(3

21199062

4120579)

A(3)

0(119905 r) =

311989821199023

120573212058752

119899119894int

119909

0

119889119906119894 1

11990641205742(3

21199062

4120579)

sdot [4120579

1199062120574(

3

21199062

4120579) minus 120574(

1

21199062

4120579)]

(57)

where 120574(32 11990324120579) is the lower incomplete Gamma functiondefined by 120574(119886119887 119909) equiv int

119909

0(119889119906119906)119906

119886119887119890minus119906

Inserting these expressions in (21) we finally obtain thestatic potential for two opposite charges 119902 located at 0 and Las

119881 = minus119902

4120587

1198901198982120579

119871[119890

minus119898119871minus

1

radic120587int

infin

11987124120579

1198891199061

radic119906119890minus119906minus119898

211990624119906

]

minus12119902

4

120573212058732

119899119894int

119871

0

119889119906119894 1

11990661205743(3

21199062

4120579) minus

311989821199024

120573212058752

sdot 119899119894int

119909

0

119889119906119894 1

11990641205742(3

21199062

4120579)

sdot [4120579

1199062120574(

3

21199062

4120579) minus 120574(

1

21199062

4120579)]

(58)

which is finite for 119871 rarr 0 It is a simple matter to verify thatin the limit 120579 rarr 0 we recover our above result

4 Logarithmic Correction

We now want to extend what we have done to Euler-Heisenberg-like electrodynamics at strong fields As alreadymentioned such theories show a power behavior that istypical for critical phenomena [58] In such a case theLagrangian density reads

L = minus1

4119865120583]119865

120583]minus1198870

8119865120583]119865

120583] log(119865120583]119865

120583]

41205822

) (59)

where 1198870and 120582 are constants In fact by choosing 119887

0=

11989026120587

2 and 120582 = 1198982

1198901198883119890ℏ we recover the Euler-Heisenberg

electrodynamics at strong fields [58]In the same way as was done in the previous section one

can introduce an auxiliary field 120585 to handle the logarithm in(59) This leads to

L = minus1

41205721119865120583]119865

120583]minus 120572

2(119865

120583]119865120583])2

(60)

where 1205721= 1 minus (119887

02)(1 + log 120585) and 120572

2= 119887

012058532120582

2A similar procedure can be used to manipulate the

quadratic term in (60) Accordingly by introducing a secondauxiliary field 120578 one easily finds

L = minus1

4119865120583] (1205721

+ 41205722120578) 119865

120583]+1205782

41205722 (61)

By setting 120590 = 1205721+ 4120572

2120578 we then have

L = minus1

4120590119865

120583]119865120583]+

1

641205722

(120590 minus 1205721)2 (62)

It is once again straightforward to apply the gauge-invariant formalism discussed in the foregoing section Thecanonical momenta read Π

120583= minus120590119865

0120583 and at once werecognize the two primary constraints Π0

= 0 and P120590

equiv

120597119871120597 = 0The canonical Hamiltonian corresponding to (62)is

119867119862= int119889

3119909

sdot Π119894120597119894119860

0+

1

2120590Π

2+120590

2B2

minus1

641205722

(120590 minus 1205721)2

(63)

Requiring the primary constraint Π0 to be preserved intime one obtains the secondary constraint Γ

1= 120597

119894Π

119894= 0 In

the same way for the constraintP120590 we get the auxiliary field

120590 as

120590 = (1 minus1198870

2(1 + ln 120585) +

1198870B2

21205822120585)[1

+3119887

0B2

21205822

120585

(1 minus (11988702) (1 + ln 120585) + (119887

0B2

21205822) 120585)

3]

(64)

Hence we obtain

119867119862= int119889

3119909

sdot Π119894120597119894119860

0+1

2+1198870

4(1 + log 120585)Π2 minus

31198870120585

21205822Π

4

(65)

As before requiring the primary constraint P120585to be

preserved in time one obtains the auxiliary field 120585 In this case120585 = 1205826Π

2 Consequently we get

119867119862= int119889

3119909Π

119894120597119894119860

0+1

2(1 + 119887

0)Π

2minus6119887

0

1205822Π

4 (66)

Following the same steps that led to (50) we find that

⟨119867⟩(1)

Φ= ⟨Φ|int119889

3119909

1

2minus

3

81205732Π

4 |Φ⟩ (67)

It should be noted that this expression is similar to (50) in thelimit 119898 rarr 0 except by the changed sign in front of the Π4termHencewe see that the potential for two opposite chargesin 0 and L is given by

119881 = minus1199022

4120587

1

119871minus

1199024

604012057321205872

1

1198715 (68)

5 Final Remarks

Finally within the gauge-invariant but path-dependent vari-ables formalism we have considered the confinement versus

Advances in High Energy Physics 9

screening issue for both massive Euler-Heisenberg-like andEuler-Heisenberg electrodynamics in the approximation ofthe strong-field limit Once again a correct identificationof physical degrees of freedom has been fundamental forunderstanding the physics hidden in gauge theories Inter-estingly enough their noncommutative version displays anultraviolet finite static potential The analysis above revealsthe key role played by the new quantum of length in ouranalysis In a general perspective the benefit of consideringthe present approach is to provide a unification scenarioamong different models as well as exploiting the equivalencein explicit calculations as we have illustrated in the course ofthis work

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

It is a pleasure for the author to thank J A Helayel-Netofor helpful comments on the paper This work was partiallysupported by Fondecyt (Chile) Grant 1130426 and DGIP(UTFSM) internal project USM 111458

References

[1] G Breit and J A Wheeler ldquoCollision of two light quantardquoPhysical Review vol 46 no 12 pp 1087ndash1091 1934

[2] S L Adler ldquoPhoton splitting and photon dispersion in a strongmagnetic fieldrdquo Annals of Physics vol 67 no 2 pp 599ndash6471971

[3] V Costantini B De Tollis and G Pistoni ldquoNonlinear effects inquantum electrodynamicsrdquo Il Nuovo Cimento A vol 2 no 3pp 733ndash787 1971

[4] R Ruffini G Vereshchagin and S-S Xue ldquoElectron-positronpairs in physics and astrophysics from heavy nuclei to blackholesrdquo Physics Reports vol 487 no 1ndash4 pp 1ndash140 2010

[5] G V Dunne ldquoThe Heisenberg-Euler effective action 75 yearsonrdquo International Journal of Modern Physics Conference Seriesvol 14 pp 42ndash56 2012

[6] C Bamber S J Boege TKoffas et al ldquoStudies of nonlinearQEDin collisions of 466GeV electrons with intense laser pulsesrdquoPhysical Review D vol 60 no 9 Article ID 092004 1999

[7] D L Burke R C Field G Horton-Smith et al ldquoPositronproduction in multiphoton light-by-light scatteringrdquo PhysicalReview Letters vol 79 no 9 article 1626 1997

[8] O J Pike F MacKenroth E G Hill and S J Rose ldquoA photonndashphoton collider in a vacuum hohlraumrdquo Nature Photonics vol8 no 6 pp 434ndash436 2014

[9] D Tommasini A Ferrando H Michinel and M Seco ldquoPre-cision tests of QED and non-standard models by searchingphoton-photon scattering in vacuum with high power lasersrdquoJournal ofHigh Energy Physics vol 2009 no 11 article 043 2009

[10] D Tommasini A Ferrando HMichinel andM Seco ldquoDetect-ing photon-photon scattering in vacuum at exawatt lasersrdquoPhysical Review A vol 77 no 4 Article ID 042101 2008

[11] M Born and L Infeld ldquoFoundations of the new field theoryrdquoProceedings of the Royal Society of London Series A Containing

Papers of a Mathematical and Physical Character vol 144 no852 pp 425ndash451 1934

[12] H Gies J Jaeckel and A Ringwald ldquoPolarized light propagat-ing in a magnetic field as a probe for millicharged fermionsrdquoPhysical Review Letters vol 97 no 14 Article ID 140402 2006

[13] E Masso and R Toldra ldquoLight spinless particle coupled tophotonsrdquo Physical Review D vol 52 no 4 pp 1755ndash1763 1995

[14] P Gaete and E I Guendelman ldquoConfinement in the presenceof external fields and axionsrdquoModern Physics Letters A vol 20no 5 article 319 2005

[15] P Gaete and E Spallucci ldquoConfinement effects from interactingchromo-magnetic and axion fieldsrdquo Journal of Physics A Math-ematical and General vol 39 no 20 pp 6021ndash6029 2006

[16] E S Fradkin and A A Tseytlin ldquoNon-linear electrodynamicsfrom quantized stringsrdquo Physics Letters B vol 163 no 1ndash4 pp123ndash130 1985

[17] E Bergshoeff E Sezgin C N Pope and P K TownsendldquoThe Born-Infeld action from conformal invariance of the opensuperstringrdquo Physics Letters B vol 188 no 1 pp 70ndash74 1987

[18] P Gaete and J Helayel-Neto ldquoFinite field-energy and interpar-ticle potential in logarithmic electrodynamicsrdquo The EuropeanPhysical Journal C vol 74 no 3 article 2816 2014

[19] P Gaete and J Helayel-Neto ldquoRemarks on nonlinear electrody-namicsrdquo European Physical Journal C vol 74 no 11 article 31822014

[20] S H Hendi ldquoAsymptotic ReissnerndashNordstrom black holesrdquoAnnals of Physics vol 333 pp 282ndash289 2013

[21] Z Zhao Q Pan S Chen and J Jing ldquoNotes on holographicsuperconductor models with the nonlinear electrodynamicsrdquoNuclear Physics B vol 871 no 1 pp 98ndash110 2013

[22] O Miskovic and R Olea ldquoConserved charges for black holes inEinstein-Gauss-Bonnet gravity coupled to nonlinear electrody-namics in AdS spacerdquo Physical Review D vol 83 no 2 ArticleID 024011 2011

[23] S H Mazharimousavi and M Halilsoy ldquoBlack holes and theclassical model of a particle in Einstein non-linear electrody-namics theoryrdquo Physics Letters B vol 678 no 4 pp 407ndash4102009

[24] G Amelino-Camelia ldquoSpecial treatmentrdquo Nature vol 418 no6893 pp 34ndash35 2002

[25] T Jacobson S Liberati andDMattingly ldquoThreshold effects andPlanck scale Lorentz violation combined constraints from highenergy astrophysicsrdquo Physical Review D vol 67 no 12 ArticleID 124011 2003

[26] T J Konopka and S A Major ldquoObservational limits onquantum geometry effectsrdquo New Journal of Physics vol 4 pp571ndash5718 2002

[27] SHossenfelder ldquoInterpretation of quantumfield theories with aminimal length scalerdquo Physical Review D vol 73 no 10 ArticleID 105013 9 pages 2006

[28] P Nicolini ldquoNoncommutative black holes the final appeal toquantum gravity a reviewrdquo International Journal of ModernPhysics A vol 24 no 7 pp 1229ndash1308 2009

[29] S Pramanik S Ghosh and P Pal ldquoElectrodynamics of a gener-alized charged particle in doubly special relativity frameworkrdquoAnnals of Physics vol 346 pp 113ndash128 2014

[30] A F Ali S Das and E C Vagenas ldquoDiscreteness of space fromthe generalized uncertainty principlerdquo Physics Letters B vol678 no 5 pp 497ndash499 2009

[31] G Amelino-Camelia ldquoQuantum-spacetime phenomenologyrdquoLiving Reviews in Relativity vol 16 article 5 2013

10 Advances in High Energy Physics

[32] E Witten ldquoNoncommutative geometry and string field theoryrdquoNuclear Physics B vol 268 no 2 pp 253ndash294 1986

[33] N Seiberg and E Witten ldquoString theory and noncommutativegeometryrdquo Journal of High Energy Physics vol 1999 article 0321999

[34] M R Douglas and N A Nekrasov ldquoNoncommutative fieldtheoryrdquo Reviews of Modern Physics vol 73 no 4 pp 977ndash10292001

[35] R J Szabo ldquoQuantum field theory on noncommutative spacesrdquoPhysics Reports vol 378 no 4 pp 207ndash299 2003

[36] J Gomis K Kamimura and T Mateos ldquoGauge and BRST gen-erators for space-time non-commutative U(1) theoryrdquo Journalof High Energy Physics vol 2001 no 3 article 010 2001

[37] A A Bichl J M Grimstrup L Popp M Schweda and RWulkenhaar ldquoPerturbative analysis of the Seiberg-Witten maprdquoInternational Journal of Modern Physics A vol 17 no 16 pp2219ndash2231 2002

[38] A Smailagic and E Spallucci ldquoUV divergence-free QFT onnoncommutative planerdquo Journal of Physics AMathematical andGeneral vol 36 no 39 pp L517ndashL521 2003

[39] A Smailagic and E Spallucci ldquoFeynman path integral on thenon-commutative planerdquo Journal of Physics A Mathematicaland General vol 36 no 33 pp L467ndashL471 2003

[40] M Knezevic D Knezevic and D Spasojevic ldquoStatistics ofequally weighted random paths on a class of self-similarstructuresrdquo Journal of Physics A Mathematical and Generalvol 37 no 1 pp 1ndash8 2004 Erratum in Journal of Physics AMathematical and General vol 37 p 7169 2004

[41] E Spallucci A Smailagic and P Nicolini ldquoTrace anomaly ona quantum spacetime manifoldrdquo Physical Review D vol 73Article ID 084004 2006

[42] R Banerjee S Gangopadhyay and S K Modak ldquoVoros prod-uct noncommutative Schwarzschild black hole and correctedarea lawrdquo Physics Letters B vol 686 no 2-3 pp 181ndash187 2010

[43] A B Hammou M Lagraa and M M Sheikh-Jabbari ldquoCoher-ent state induced star product on 119877

3

120582and the fuzzy sphererdquo

Physical Review D vol 66 no 2 Article ID 025025 2002[44] L Modesto J W Moffat and P Nicolini ldquoBlack holes in an

ultraviolet complete quantum gravityrdquo Physics Letters B vol695 no 1ndash4 pp 397ndash400 2011

[45] P Nicolini ldquoNonlocal and generalizeduncertainty principleblack holesrdquo httparxivorgabs12022102

[46] P Nicolini A Smailagic and E Spallucci ldquoNoncommutativegeometry inspired Schwarzschild black holerdquo Physics Letters Bvol 632 no 4 pp 547ndash551 2006

[47] S Pramanik S Das and S Ghosh ldquoNoncommutative extensionof AdS-CFT and holographic superconductorsrdquo Physics LettersB vol 742 pp 266ndash273 2015

[48] W Dittrich and M Reuter Effective Lagrangians in QuantumElectrodynamics Springer Berlin Germany 1985

[49] M Ostrogradsky ldquoMemoires sur les equations differen-tielles relativrs au probleme des isoperimetersrdquo Memoires delrsquoAcademie Imperiale des Sciences de St Petersbourg vol 4 p385 1850

[50] P Gaete and E Spallucci ldquoFinite axionic electrodynamicsfrom a new non-commutative approachrdquo Journal of Physics AMathematical and Theoretical vol 45 no 6 Article ID 06540114 pages 2012

[51] P Gaete ldquoSome considerations about Podolsky-axionic electro-dynamicsrdquo International Journal of Modern Physics A vol 27no 11 Article ID 1250061 2012

[52] P Gaete ldquoRemarks on gauge-invariant variables and interactionenergy in QEDrdquo Physical Review D vol 59 no 12 Article ID127702 1999

[53] S Klarsfeld ldquoAnalytical expressions for the evaluation ofvacuum-polarization potentials in muonic atomsrdquo Physics Let-ters B vol 66 no 1 pp 86ndash88 1977

[54] A M Frolov and D M Wardlaw ldquoAnalytical formula for theUehling potentialrdquo The European Physical Journal B vol 85article 348 2012

[55] P Gaete ldquoOn gauge-invariant variables in QEDrdquo Zeitschrift furPhysik C Particles and Fields vol 76 no 2 pp 355ndash361 1997

[56] S Ferrara and A Sagnotti ldquoMassive Born-Infeld and other dualpairsrdquo Journal of High Energy Physics vol 2015 no 4 article 0322015

[57] R Bufalo ldquoBornndashinfeld electrodynamics in very special relativ-ityrdquo Physics Letters B vol 746 pp 251ndash256 2015

[58] HKleinert E Strobel and S-S Xue ldquoFractional effective actionat strong electromagnetic fieldsrdquo Physical Review D vol 88 no2 Article ID 025049 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Advances in High Energy Physics 7

which will be used to eliminate 120590 As before the corre-sponding total (first-class) Hamiltonian that generates thetime evolution of the dynamical variables is 119867 = 119867

119862+

int1198893119909(119906

0(119909)Π

0(119909) + 119906

1(119909)Γ

1(119909)) where 119906

0(119909) and 119906

1(119909) are

the Lagrangemultiplier utilized to implement the constraintsIn the same way as was done in the previous subsection

the expectation value of the energy operator119867 in the physicalstate |Φ⟩ becomes

⟨119867⟩Φ = ⟨Φ|int1198893119909

1

119894(1 +

1198982

Δ)

minus1

Π119894+

15

81205732Π

4

minus15119898

2

21205732Π

2 1

ΔΠ

2 |Φ⟩

(50)

in this last line we have considered only quadratic terms in119898

2In such a case by employing (50) the lowest-order

modification in 1205732 and1198982 of the interaction energy takes theform

⟨119867⟩Φ = ⟨119867⟩0 + 1198811+ 119881

2+ 119881

3 (51)

where ⟨119867⟩0= ⟨0|119867|0⟩ The 119881

1 119881

2 and 119881

3terms are given by

1198811=1199022

2int119889

3119909int

y

y1015840119889119911

1015840119894120575(3)

(x minus z1015840)(1 minus 1198982

nabla2)

minus1

sdot int

y

y1015840119889119911

119894120575(3)

(x minus z)

1198812= minus

151199024

81205732int119889

3119909int

y

y1015840119889119911

119894120575(3)

(x minus z)

sdot int

y

y1015840119889119911

1015840119894120575(3)

(x minus z1015840)inty

y1015840119889119906

119896120575(3)

(x minus u)

sdot int

y

y1015840119889V

119896120575(3)

(x minus k)

1198813=15119898

21199024

21205732

int1198893119909int

y

y1015840119889119911

119894120575(3)

(x minus z)

sdot int

y

y1015840119889119911

1015840119894120575(3)

(x minus z1015840)inty

y1015840119889119906

119896120575(3)

(x minus u)

sdot int

y

y1015840119889V

119896120575(3)

(x minus k)

(52)

Finally with the aid of expressions (52) the potential fora pair of static point-like opposite charges located at 0 and Lis given by

119881 = minus1199022

4120587

119890minus119898119871

119871+

1199024

161205871205732(3

8120587

1

1198712minus 5119898

2)

1

1198713 (53)

observe that when 119898 = 0 profile (53) reduces to the knownWichmann-Kroll interaction energy On the other hand for119898 = 0 the key role played by the mass term in transformingthe Coulomb potential into the Yukawa one should be notedInterestingly enough an unexpected feature is found In factprofile (53) displays a new long-range 11198713 correction whereits strength is proportional to 119898

2 It is also important toobserve that an analogous correction has been found in Born-Infeld electrodynamics in the context of very special relativity[57] In this way we establish a new connection betweennonlinear effectives theories

Before we proceed further we should comment on ourresult In the case of QED (Euler-Heisenberg Lagrangiandensity) the parameter 11205732 is given by 11205732

= (1645)(1198904ℏ

1198984

1198901198887) where 119898

119890is the electron mass In this context we

also recall the currently accepted upper limit for the photonmass that is 119898

120574sim 2 times 10

minus16 eV Thus for the QED casefrom (53) it follows that the second term on the right handside would be detectable in long-range distances (sim109 m) Inother words we see that detectable corrections induced byvacuum polarization with a mass term would be present atlow energy scales

From (53) it clearly follows that the interaction energybetween heavy charged charges at leading order in 120573 is notfinite at the origin Motivated by this one may consider theabove calculation in a noncommutative geometry based onfindings of our previous studies [18 19] In such a case theelectric field at leading order in 120573

2 and1198982 takes the form

119864119894= [(1 +

1198982

Δ)

minus1

+3

21205732Π

2minus6119898

2

1205732Π

2 1

Δ]

sdot 120597119894(minus

119890120579nabla2

120575(3)

(x)nabla

2)

(54)

where it may be recalled that we are now replacing the source120575(3)(x minus y) by the smeared source 119890

120579nabla2

120575(3)(x minus y) with the

parameter 120579 being noncommutative Now making use of(22) we readily find that

A0(119905 r) = A

(1)

0(119905 r) +A

(2)

0(119905 r) +A

(3)

0(119905 r) (55)

The term A(1)

0(119905 r) was first calculated in [50] we can

therefore write only the result

A(1)

0(119905 r)

= 1199021198901198982120579

4120587

1

119903[119890

minus119898119903minus

1

radic120587int

infin

11990324120579

1198891199061

radic119906119890minus119906minus119898

211990324119906

]

minus 119902119898

41205871198901198982120579

(56)

8 Advances in High Energy Physics

Meanwhile the terms A(2)

0(119905 r) and A

(3)

0(119905 r) after some

manipulation can be brought to the form

A(2)

0(119905 r) =

121199023

120573212058732

119899119894int

119909

0

119889119906119894 1

11990661205743(3

21199062

4120579)

A(3)

0(119905 r) =

311989821199023

120573212058752

119899119894int

119909

0

119889119906119894 1

11990641205742(3

21199062

4120579)

sdot [4120579

1199062120574(

3

21199062

4120579) minus 120574(

1

21199062

4120579)]

(57)

where 120574(32 11990324120579) is the lower incomplete Gamma functiondefined by 120574(119886119887 119909) equiv int

119909

0(119889119906119906)119906

119886119887119890minus119906

Inserting these expressions in (21) we finally obtain thestatic potential for two opposite charges 119902 located at 0 and Las

119881 = minus119902

4120587

1198901198982120579

119871[119890

minus119898119871minus

1

radic120587int

infin

11987124120579

1198891199061

radic119906119890minus119906minus119898

211990624119906

]

minus12119902

4

120573212058732

119899119894int

119871

0

119889119906119894 1

11990661205743(3

21199062

4120579) minus

311989821199024

120573212058752

sdot 119899119894int

119909

0

119889119906119894 1

11990641205742(3

21199062

4120579)

sdot [4120579

1199062120574(

3

21199062

4120579) minus 120574(

1

21199062

4120579)]

(58)

which is finite for 119871 rarr 0 It is a simple matter to verify thatin the limit 120579 rarr 0 we recover our above result

4 Logarithmic Correction

We now want to extend what we have done to Euler-Heisenberg-like electrodynamics at strong fields As alreadymentioned such theories show a power behavior that istypical for critical phenomena [58] In such a case theLagrangian density reads

L = minus1

4119865120583]119865

120583]minus1198870

8119865120583]119865

120583] log(119865120583]119865

120583]

41205822

) (59)

where 1198870and 120582 are constants In fact by choosing 119887

0=

11989026120587

2 and 120582 = 1198982

1198901198883119890ℏ we recover the Euler-Heisenberg

electrodynamics at strong fields [58]In the same way as was done in the previous section one

can introduce an auxiliary field 120585 to handle the logarithm in(59) This leads to

L = minus1

41205721119865120583]119865

120583]minus 120572

2(119865

120583]119865120583])2

(60)

where 1205721= 1 minus (119887

02)(1 + log 120585) and 120572

2= 119887

012058532120582

2A similar procedure can be used to manipulate the

quadratic term in (60) Accordingly by introducing a secondauxiliary field 120578 one easily finds

L = minus1

4119865120583] (1205721

+ 41205722120578) 119865

120583]+1205782

41205722 (61)

By setting 120590 = 1205721+ 4120572

2120578 we then have

L = minus1

4120590119865

120583]119865120583]+

1

641205722

(120590 minus 1205721)2 (62)

It is once again straightforward to apply the gauge-invariant formalism discussed in the foregoing section Thecanonical momenta read Π

120583= minus120590119865

0120583 and at once werecognize the two primary constraints Π0

= 0 and P120590

equiv

120597119871120597 = 0The canonical Hamiltonian corresponding to (62)is

119867119862= int119889

3119909

sdot Π119894120597119894119860

0+

1

2120590Π

2+120590

2B2

minus1

641205722

(120590 minus 1205721)2

(63)

Requiring the primary constraint Π0 to be preserved intime one obtains the secondary constraint Γ

1= 120597

119894Π

119894= 0 In

the same way for the constraintP120590 we get the auxiliary field

120590 as

120590 = (1 minus1198870

2(1 + ln 120585) +

1198870B2

21205822120585)[1

+3119887

0B2

21205822

120585

(1 minus (11988702) (1 + ln 120585) + (119887

0B2

21205822) 120585)

3]

(64)

Hence we obtain

119867119862= int119889

3119909

sdot Π119894120597119894119860

0+1

2+1198870

4(1 + log 120585)Π2 minus

31198870120585

21205822Π

4

(65)

As before requiring the primary constraint P120585to be

preserved in time one obtains the auxiliary field 120585 In this case120585 = 1205826Π

2 Consequently we get

119867119862= int119889

3119909Π

119894120597119894119860

0+1

2(1 + 119887

0)Π

2minus6119887

0

1205822Π

4 (66)

Following the same steps that led to (50) we find that

⟨119867⟩(1)

Φ= ⟨Φ|int119889

3119909

1

2minus

3

81205732Π

4 |Φ⟩ (67)

It should be noted that this expression is similar to (50) in thelimit 119898 rarr 0 except by the changed sign in front of the Π4termHencewe see that the potential for two opposite chargesin 0 and L is given by

119881 = minus1199022

4120587

1

119871minus

1199024

604012057321205872

1

1198715 (68)

5 Final Remarks

Finally within the gauge-invariant but path-dependent vari-ables formalism we have considered the confinement versus

Advances in High Energy Physics 9

screening issue for both massive Euler-Heisenberg-like andEuler-Heisenberg electrodynamics in the approximation ofthe strong-field limit Once again a correct identificationof physical degrees of freedom has been fundamental forunderstanding the physics hidden in gauge theories Inter-estingly enough their noncommutative version displays anultraviolet finite static potential The analysis above revealsthe key role played by the new quantum of length in ouranalysis In a general perspective the benefit of consideringthe present approach is to provide a unification scenarioamong different models as well as exploiting the equivalencein explicit calculations as we have illustrated in the course ofthis work

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

It is a pleasure for the author to thank J A Helayel-Netofor helpful comments on the paper This work was partiallysupported by Fondecyt (Chile) Grant 1130426 and DGIP(UTFSM) internal project USM 111458

References

[1] G Breit and J A Wheeler ldquoCollision of two light quantardquoPhysical Review vol 46 no 12 pp 1087ndash1091 1934

[2] S L Adler ldquoPhoton splitting and photon dispersion in a strongmagnetic fieldrdquo Annals of Physics vol 67 no 2 pp 599ndash6471971

[3] V Costantini B De Tollis and G Pistoni ldquoNonlinear effects inquantum electrodynamicsrdquo Il Nuovo Cimento A vol 2 no 3pp 733ndash787 1971

[4] R Ruffini G Vereshchagin and S-S Xue ldquoElectron-positronpairs in physics and astrophysics from heavy nuclei to blackholesrdquo Physics Reports vol 487 no 1ndash4 pp 1ndash140 2010

[5] G V Dunne ldquoThe Heisenberg-Euler effective action 75 yearsonrdquo International Journal of Modern Physics Conference Seriesvol 14 pp 42ndash56 2012

[6] C Bamber S J Boege TKoffas et al ldquoStudies of nonlinearQEDin collisions of 466GeV electrons with intense laser pulsesrdquoPhysical Review D vol 60 no 9 Article ID 092004 1999

[7] D L Burke R C Field G Horton-Smith et al ldquoPositronproduction in multiphoton light-by-light scatteringrdquo PhysicalReview Letters vol 79 no 9 article 1626 1997

[8] O J Pike F MacKenroth E G Hill and S J Rose ldquoA photonndashphoton collider in a vacuum hohlraumrdquo Nature Photonics vol8 no 6 pp 434ndash436 2014

[9] D Tommasini A Ferrando H Michinel and M Seco ldquoPre-cision tests of QED and non-standard models by searchingphoton-photon scattering in vacuum with high power lasersrdquoJournal ofHigh Energy Physics vol 2009 no 11 article 043 2009

[10] D Tommasini A Ferrando HMichinel andM Seco ldquoDetect-ing photon-photon scattering in vacuum at exawatt lasersrdquoPhysical Review A vol 77 no 4 Article ID 042101 2008

[11] M Born and L Infeld ldquoFoundations of the new field theoryrdquoProceedings of the Royal Society of London Series A Containing

Papers of a Mathematical and Physical Character vol 144 no852 pp 425ndash451 1934

[12] H Gies J Jaeckel and A Ringwald ldquoPolarized light propagat-ing in a magnetic field as a probe for millicharged fermionsrdquoPhysical Review Letters vol 97 no 14 Article ID 140402 2006

[13] E Masso and R Toldra ldquoLight spinless particle coupled tophotonsrdquo Physical Review D vol 52 no 4 pp 1755ndash1763 1995

[14] P Gaete and E I Guendelman ldquoConfinement in the presenceof external fields and axionsrdquoModern Physics Letters A vol 20no 5 article 319 2005

[15] P Gaete and E Spallucci ldquoConfinement effects from interactingchromo-magnetic and axion fieldsrdquo Journal of Physics A Math-ematical and General vol 39 no 20 pp 6021ndash6029 2006

[16] E S Fradkin and A A Tseytlin ldquoNon-linear electrodynamicsfrom quantized stringsrdquo Physics Letters B vol 163 no 1ndash4 pp123ndash130 1985

[17] E Bergshoeff E Sezgin C N Pope and P K TownsendldquoThe Born-Infeld action from conformal invariance of the opensuperstringrdquo Physics Letters B vol 188 no 1 pp 70ndash74 1987

[18] P Gaete and J Helayel-Neto ldquoFinite field-energy and interpar-ticle potential in logarithmic electrodynamicsrdquo The EuropeanPhysical Journal C vol 74 no 3 article 2816 2014

[19] P Gaete and J Helayel-Neto ldquoRemarks on nonlinear electrody-namicsrdquo European Physical Journal C vol 74 no 11 article 31822014

[20] S H Hendi ldquoAsymptotic ReissnerndashNordstrom black holesrdquoAnnals of Physics vol 333 pp 282ndash289 2013

[21] Z Zhao Q Pan S Chen and J Jing ldquoNotes on holographicsuperconductor models with the nonlinear electrodynamicsrdquoNuclear Physics B vol 871 no 1 pp 98ndash110 2013

[22] O Miskovic and R Olea ldquoConserved charges for black holes inEinstein-Gauss-Bonnet gravity coupled to nonlinear electrody-namics in AdS spacerdquo Physical Review D vol 83 no 2 ArticleID 024011 2011

[23] S H Mazharimousavi and M Halilsoy ldquoBlack holes and theclassical model of a particle in Einstein non-linear electrody-namics theoryrdquo Physics Letters B vol 678 no 4 pp 407ndash4102009

[24] G Amelino-Camelia ldquoSpecial treatmentrdquo Nature vol 418 no6893 pp 34ndash35 2002

[25] T Jacobson S Liberati andDMattingly ldquoThreshold effects andPlanck scale Lorentz violation combined constraints from highenergy astrophysicsrdquo Physical Review D vol 67 no 12 ArticleID 124011 2003

[26] T J Konopka and S A Major ldquoObservational limits onquantum geometry effectsrdquo New Journal of Physics vol 4 pp571ndash5718 2002

[27] SHossenfelder ldquoInterpretation of quantumfield theories with aminimal length scalerdquo Physical Review D vol 73 no 10 ArticleID 105013 9 pages 2006

[28] P Nicolini ldquoNoncommutative black holes the final appeal toquantum gravity a reviewrdquo International Journal of ModernPhysics A vol 24 no 7 pp 1229ndash1308 2009

[29] S Pramanik S Ghosh and P Pal ldquoElectrodynamics of a gener-alized charged particle in doubly special relativity frameworkrdquoAnnals of Physics vol 346 pp 113ndash128 2014

[30] A F Ali S Das and E C Vagenas ldquoDiscreteness of space fromthe generalized uncertainty principlerdquo Physics Letters B vol678 no 5 pp 497ndash499 2009

[31] G Amelino-Camelia ldquoQuantum-spacetime phenomenologyrdquoLiving Reviews in Relativity vol 16 article 5 2013

10 Advances in High Energy Physics

[32] E Witten ldquoNoncommutative geometry and string field theoryrdquoNuclear Physics B vol 268 no 2 pp 253ndash294 1986

[33] N Seiberg and E Witten ldquoString theory and noncommutativegeometryrdquo Journal of High Energy Physics vol 1999 article 0321999

[34] M R Douglas and N A Nekrasov ldquoNoncommutative fieldtheoryrdquo Reviews of Modern Physics vol 73 no 4 pp 977ndash10292001

[35] R J Szabo ldquoQuantum field theory on noncommutative spacesrdquoPhysics Reports vol 378 no 4 pp 207ndash299 2003

[36] J Gomis K Kamimura and T Mateos ldquoGauge and BRST gen-erators for space-time non-commutative U(1) theoryrdquo Journalof High Energy Physics vol 2001 no 3 article 010 2001

[37] A A Bichl J M Grimstrup L Popp M Schweda and RWulkenhaar ldquoPerturbative analysis of the Seiberg-Witten maprdquoInternational Journal of Modern Physics A vol 17 no 16 pp2219ndash2231 2002

[38] A Smailagic and E Spallucci ldquoUV divergence-free QFT onnoncommutative planerdquo Journal of Physics AMathematical andGeneral vol 36 no 39 pp L517ndashL521 2003

[39] A Smailagic and E Spallucci ldquoFeynman path integral on thenon-commutative planerdquo Journal of Physics A Mathematicaland General vol 36 no 33 pp L467ndashL471 2003

[40] M Knezevic D Knezevic and D Spasojevic ldquoStatistics ofequally weighted random paths on a class of self-similarstructuresrdquo Journal of Physics A Mathematical and Generalvol 37 no 1 pp 1ndash8 2004 Erratum in Journal of Physics AMathematical and General vol 37 p 7169 2004

[41] E Spallucci A Smailagic and P Nicolini ldquoTrace anomaly ona quantum spacetime manifoldrdquo Physical Review D vol 73Article ID 084004 2006

[42] R Banerjee S Gangopadhyay and S K Modak ldquoVoros prod-uct noncommutative Schwarzschild black hole and correctedarea lawrdquo Physics Letters B vol 686 no 2-3 pp 181ndash187 2010

[43] A B Hammou M Lagraa and M M Sheikh-Jabbari ldquoCoher-ent state induced star product on 119877

3

120582and the fuzzy sphererdquo

Physical Review D vol 66 no 2 Article ID 025025 2002[44] L Modesto J W Moffat and P Nicolini ldquoBlack holes in an

ultraviolet complete quantum gravityrdquo Physics Letters B vol695 no 1ndash4 pp 397ndash400 2011

[45] P Nicolini ldquoNonlocal and generalizeduncertainty principleblack holesrdquo httparxivorgabs12022102

[46] P Nicolini A Smailagic and E Spallucci ldquoNoncommutativegeometry inspired Schwarzschild black holerdquo Physics Letters Bvol 632 no 4 pp 547ndash551 2006

[47] S Pramanik S Das and S Ghosh ldquoNoncommutative extensionof AdS-CFT and holographic superconductorsrdquo Physics LettersB vol 742 pp 266ndash273 2015

[48] W Dittrich and M Reuter Effective Lagrangians in QuantumElectrodynamics Springer Berlin Germany 1985

[49] M Ostrogradsky ldquoMemoires sur les equations differen-tielles relativrs au probleme des isoperimetersrdquo Memoires delrsquoAcademie Imperiale des Sciences de St Petersbourg vol 4 p385 1850

[50] P Gaete and E Spallucci ldquoFinite axionic electrodynamicsfrom a new non-commutative approachrdquo Journal of Physics AMathematical and Theoretical vol 45 no 6 Article ID 06540114 pages 2012

[51] P Gaete ldquoSome considerations about Podolsky-axionic electro-dynamicsrdquo International Journal of Modern Physics A vol 27no 11 Article ID 1250061 2012

[52] P Gaete ldquoRemarks on gauge-invariant variables and interactionenergy in QEDrdquo Physical Review D vol 59 no 12 Article ID127702 1999

[53] S Klarsfeld ldquoAnalytical expressions for the evaluation ofvacuum-polarization potentials in muonic atomsrdquo Physics Let-ters B vol 66 no 1 pp 86ndash88 1977

[54] A M Frolov and D M Wardlaw ldquoAnalytical formula for theUehling potentialrdquo The European Physical Journal B vol 85article 348 2012

[55] P Gaete ldquoOn gauge-invariant variables in QEDrdquo Zeitschrift furPhysik C Particles and Fields vol 76 no 2 pp 355ndash361 1997

[56] S Ferrara and A Sagnotti ldquoMassive Born-Infeld and other dualpairsrdquo Journal of High Energy Physics vol 2015 no 4 article 0322015

[57] R Bufalo ldquoBornndashinfeld electrodynamics in very special relativ-ityrdquo Physics Letters B vol 746 pp 251ndash256 2015

[58] HKleinert E Strobel and S-S Xue ldquoFractional effective actionat strong electromagnetic fieldsrdquo Physical Review D vol 88 no2 Article ID 025049 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

8 Advances in High Energy Physics

Meanwhile the terms A(2)

0(119905 r) and A

(3)

0(119905 r) after some

manipulation can be brought to the form

A(2)

0(119905 r) =

121199023

120573212058732

119899119894int

119909

0

119889119906119894 1

11990661205743(3

21199062

4120579)

A(3)

0(119905 r) =

311989821199023

120573212058752

119899119894int

119909

0

119889119906119894 1

11990641205742(3

21199062

4120579)

sdot [4120579

1199062120574(

3

21199062

4120579) minus 120574(

1

21199062

4120579)]

(57)

where 120574(32 11990324120579) is the lower incomplete Gamma functiondefined by 120574(119886119887 119909) equiv int

119909

0(119889119906119906)119906

119886119887119890minus119906

Inserting these expressions in (21) we finally obtain thestatic potential for two opposite charges 119902 located at 0 and Las

119881 = minus119902

4120587

1198901198982120579

119871[119890

minus119898119871minus

1

radic120587int

infin

11987124120579

1198891199061

radic119906119890minus119906minus119898

211990624119906

]

minus12119902

4

120573212058732

119899119894int

119871

0

119889119906119894 1

11990661205743(3

21199062

4120579) minus

311989821199024

120573212058752

sdot 119899119894int

119909

0

119889119906119894 1

11990641205742(3

21199062

4120579)

sdot [4120579

1199062120574(

3

21199062

4120579) minus 120574(

1

21199062

4120579)]

(58)

which is finite for 119871 rarr 0 It is a simple matter to verify thatin the limit 120579 rarr 0 we recover our above result

4 Logarithmic Correction

We now want to extend what we have done to Euler-Heisenberg-like electrodynamics at strong fields As alreadymentioned such theories show a power behavior that istypical for critical phenomena [58] In such a case theLagrangian density reads

L = minus1

4119865120583]119865

120583]minus1198870

8119865120583]119865

120583] log(119865120583]119865

120583]

41205822

) (59)

where 1198870and 120582 are constants In fact by choosing 119887

0=

11989026120587

2 and 120582 = 1198982

1198901198883119890ℏ we recover the Euler-Heisenberg

electrodynamics at strong fields [58]In the same way as was done in the previous section one

can introduce an auxiliary field 120585 to handle the logarithm in(59) This leads to

L = minus1

41205721119865120583]119865

120583]minus 120572

2(119865

120583]119865120583])2

(60)

where 1205721= 1 minus (119887

02)(1 + log 120585) and 120572

2= 119887

012058532120582

2A similar procedure can be used to manipulate the

quadratic term in (60) Accordingly by introducing a secondauxiliary field 120578 one easily finds

L = minus1

4119865120583] (1205721

+ 41205722120578) 119865

120583]+1205782

41205722 (61)

By setting 120590 = 1205721+ 4120572

2120578 we then have

L = minus1

4120590119865

120583]119865120583]+

1

641205722

(120590 minus 1205721)2 (62)

It is once again straightforward to apply the gauge-invariant formalism discussed in the foregoing section Thecanonical momenta read Π

120583= minus120590119865

0120583 and at once werecognize the two primary constraints Π0

= 0 and P120590

equiv

120597119871120597 = 0The canonical Hamiltonian corresponding to (62)is

119867119862= int119889

3119909

sdot Π119894120597119894119860

0+

1

2120590Π

2+120590

2B2

minus1

641205722

(120590 minus 1205721)2

(63)

Requiring the primary constraint Π0 to be preserved intime one obtains the secondary constraint Γ

1= 120597

119894Π

119894= 0 In

the same way for the constraintP120590 we get the auxiliary field

120590 as

120590 = (1 minus1198870

2(1 + ln 120585) +

1198870B2

21205822120585)[1

+3119887

0B2

21205822

120585

(1 minus (11988702) (1 + ln 120585) + (119887

0B2

21205822) 120585)

3]

(64)

Hence we obtain

119867119862= int119889

3119909

sdot Π119894120597119894119860

0+1

2+1198870

4(1 + log 120585)Π2 minus

31198870120585

21205822Π

4

(65)

As before requiring the primary constraint P120585to be

preserved in time one obtains the auxiliary field 120585 In this case120585 = 1205826Π

2 Consequently we get

119867119862= int119889

3119909Π

119894120597119894119860

0+1

2(1 + 119887

0)Π

2minus6119887

0

1205822Π

4 (66)

Following the same steps that led to (50) we find that

⟨119867⟩(1)

Φ= ⟨Φ|int119889

3119909

1

2minus

3

81205732Π

4 |Φ⟩ (67)

It should be noted that this expression is similar to (50) in thelimit 119898 rarr 0 except by the changed sign in front of the Π4termHencewe see that the potential for two opposite chargesin 0 and L is given by

119881 = minus1199022

4120587

1

119871minus

1199024

604012057321205872

1

1198715 (68)

5 Final Remarks

Finally within the gauge-invariant but path-dependent vari-ables formalism we have considered the confinement versus

Advances in High Energy Physics 9

screening issue for both massive Euler-Heisenberg-like andEuler-Heisenberg electrodynamics in the approximation ofthe strong-field limit Once again a correct identificationof physical degrees of freedom has been fundamental forunderstanding the physics hidden in gauge theories Inter-estingly enough their noncommutative version displays anultraviolet finite static potential The analysis above revealsthe key role played by the new quantum of length in ouranalysis In a general perspective the benefit of consideringthe present approach is to provide a unification scenarioamong different models as well as exploiting the equivalencein explicit calculations as we have illustrated in the course ofthis work

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

It is a pleasure for the author to thank J A Helayel-Netofor helpful comments on the paper This work was partiallysupported by Fondecyt (Chile) Grant 1130426 and DGIP(UTFSM) internal project USM 111458

References

[1] G Breit and J A Wheeler ldquoCollision of two light quantardquoPhysical Review vol 46 no 12 pp 1087ndash1091 1934

[2] S L Adler ldquoPhoton splitting and photon dispersion in a strongmagnetic fieldrdquo Annals of Physics vol 67 no 2 pp 599ndash6471971

[3] V Costantini B De Tollis and G Pistoni ldquoNonlinear effects inquantum electrodynamicsrdquo Il Nuovo Cimento A vol 2 no 3pp 733ndash787 1971

[4] R Ruffini G Vereshchagin and S-S Xue ldquoElectron-positronpairs in physics and astrophysics from heavy nuclei to blackholesrdquo Physics Reports vol 487 no 1ndash4 pp 1ndash140 2010

[5] G V Dunne ldquoThe Heisenberg-Euler effective action 75 yearsonrdquo International Journal of Modern Physics Conference Seriesvol 14 pp 42ndash56 2012

[6] C Bamber S J Boege TKoffas et al ldquoStudies of nonlinearQEDin collisions of 466GeV electrons with intense laser pulsesrdquoPhysical Review D vol 60 no 9 Article ID 092004 1999

[7] D L Burke R C Field G Horton-Smith et al ldquoPositronproduction in multiphoton light-by-light scatteringrdquo PhysicalReview Letters vol 79 no 9 article 1626 1997

[8] O J Pike F MacKenroth E G Hill and S J Rose ldquoA photonndashphoton collider in a vacuum hohlraumrdquo Nature Photonics vol8 no 6 pp 434ndash436 2014

[9] D Tommasini A Ferrando H Michinel and M Seco ldquoPre-cision tests of QED and non-standard models by searchingphoton-photon scattering in vacuum with high power lasersrdquoJournal ofHigh Energy Physics vol 2009 no 11 article 043 2009

[10] D Tommasini A Ferrando HMichinel andM Seco ldquoDetect-ing photon-photon scattering in vacuum at exawatt lasersrdquoPhysical Review A vol 77 no 4 Article ID 042101 2008

[11] M Born and L Infeld ldquoFoundations of the new field theoryrdquoProceedings of the Royal Society of London Series A Containing

Papers of a Mathematical and Physical Character vol 144 no852 pp 425ndash451 1934

[12] H Gies J Jaeckel and A Ringwald ldquoPolarized light propagat-ing in a magnetic field as a probe for millicharged fermionsrdquoPhysical Review Letters vol 97 no 14 Article ID 140402 2006

[13] E Masso and R Toldra ldquoLight spinless particle coupled tophotonsrdquo Physical Review D vol 52 no 4 pp 1755ndash1763 1995

[14] P Gaete and E I Guendelman ldquoConfinement in the presenceof external fields and axionsrdquoModern Physics Letters A vol 20no 5 article 319 2005

[15] P Gaete and E Spallucci ldquoConfinement effects from interactingchromo-magnetic and axion fieldsrdquo Journal of Physics A Math-ematical and General vol 39 no 20 pp 6021ndash6029 2006

[16] E S Fradkin and A A Tseytlin ldquoNon-linear electrodynamicsfrom quantized stringsrdquo Physics Letters B vol 163 no 1ndash4 pp123ndash130 1985

[17] E Bergshoeff E Sezgin C N Pope and P K TownsendldquoThe Born-Infeld action from conformal invariance of the opensuperstringrdquo Physics Letters B vol 188 no 1 pp 70ndash74 1987

[18] P Gaete and J Helayel-Neto ldquoFinite field-energy and interpar-ticle potential in logarithmic electrodynamicsrdquo The EuropeanPhysical Journal C vol 74 no 3 article 2816 2014

[19] P Gaete and J Helayel-Neto ldquoRemarks on nonlinear electrody-namicsrdquo European Physical Journal C vol 74 no 11 article 31822014

[20] S H Hendi ldquoAsymptotic ReissnerndashNordstrom black holesrdquoAnnals of Physics vol 333 pp 282ndash289 2013

[21] Z Zhao Q Pan S Chen and J Jing ldquoNotes on holographicsuperconductor models with the nonlinear electrodynamicsrdquoNuclear Physics B vol 871 no 1 pp 98ndash110 2013

[22] O Miskovic and R Olea ldquoConserved charges for black holes inEinstein-Gauss-Bonnet gravity coupled to nonlinear electrody-namics in AdS spacerdquo Physical Review D vol 83 no 2 ArticleID 024011 2011

[23] S H Mazharimousavi and M Halilsoy ldquoBlack holes and theclassical model of a particle in Einstein non-linear electrody-namics theoryrdquo Physics Letters B vol 678 no 4 pp 407ndash4102009

[24] G Amelino-Camelia ldquoSpecial treatmentrdquo Nature vol 418 no6893 pp 34ndash35 2002

[25] T Jacobson S Liberati andDMattingly ldquoThreshold effects andPlanck scale Lorentz violation combined constraints from highenergy astrophysicsrdquo Physical Review D vol 67 no 12 ArticleID 124011 2003

[26] T J Konopka and S A Major ldquoObservational limits onquantum geometry effectsrdquo New Journal of Physics vol 4 pp571ndash5718 2002

[27] SHossenfelder ldquoInterpretation of quantumfield theories with aminimal length scalerdquo Physical Review D vol 73 no 10 ArticleID 105013 9 pages 2006

[28] P Nicolini ldquoNoncommutative black holes the final appeal toquantum gravity a reviewrdquo International Journal of ModernPhysics A vol 24 no 7 pp 1229ndash1308 2009

[29] S Pramanik S Ghosh and P Pal ldquoElectrodynamics of a gener-alized charged particle in doubly special relativity frameworkrdquoAnnals of Physics vol 346 pp 113ndash128 2014

[30] A F Ali S Das and E C Vagenas ldquoDiscreteness of space fromthe generalized uncertainty principlerdquo Physics Letters B vol678 no 5 pp 497ndash499 2009

[31] G Amelino-Camelia ldquoQuantum-spacetime phenomenologyrdquoLiving Reviews in Relativity vol 16 article 5 2013

10 Advances in High Energy Physics

[32] E Witten ldquoNoncommutative geometry and string field theoryrdquoNuclear Physics B vol 268 no 2 pp 253ndash294 1986

[33] N Seiberg and E Witten ldquoString theory and noncommutativegeometryrdquo Journal of High Energy Physics vol 1999 article 0321999

[34] M R Douglas and N A Nekrasov ldquoNoncommutative fieldtheoryrdquo Reviews of Modern Physics vol 73 no 4 pp 977ndash10292001

[35] R J Szabo ldquoQuantum field theory on noncommutative spacesrdquoPhysics Reports vol 378 no 4 pp 207ndash299 2003

[36] J Gomis K Kamimura and T Mateos ldquoGauge and BRST gen-erators for space-time non-commutative U(1) theoryrdquo Journalof High Energy Physics vol 2001 no 3 article 010 2001

[37] A A Bichl J M Grimstrup L Popp M Schweda and RWulkenhaar ldquoPerturbative analysis of the Seiberg-Witten maprdquoInternational Journal of Modern Physics A vol 17 no 16 pp2219ndash2231 2002

[38] A Smailagic and E Spallucci ldquoUV divergence-free QFT onnoncommutative planerdquo Journal of Physics AMathematical andGeneral vol 36 no 39 pp L517ndashL521 2003

[39] A Smailagic and E Spallucci ldquoFeynman path integral on thenon-commutative planerdquo Journal of Physics A Mathematicaland General vol 36 no 33 pp L467ndashL471 2003

[40] M Knezevic D Knezevic and D Spasojevic ldquoStatistics ofequally weighted random paths on a class of self-similarstructuresrdquo Journal of Physics A Mathematical and Generalvol 37 no 1 pp 1ndash8 2004 Erratum in Journal of Physics AMathematical and General vol 37 p 7169 2004

[41] E Spallucci A Smailagic and P Nicolini ldquoTrace anomaly ona quantum spacetime manifoldrdquo Physical Review D vol 73Article ID 084004 2006

[42] R Banerjee S Gangopadhyay and S K Modak ldquoVoros prod-uct noncommutative Schwarzschild black hole and correctedarea lawrdquo Physics Letters B vol 686 no 2-3 pp 181ndash187 2010

[43] A B Hammou M Lagraa and M M Sheikh-Jabbari ldquoCoher-ent state induced star product on 119877

3

120582and the fuzzy sphererdquo

Physical Review D vol 66 no 2 Article ID 025025 2002[44] L Modesto J W Moffat and P Nicolini ldquoBlack holes in an

ultraviolet complete quantum gravityrdquo Physics Letters B vol695 no 1ndash4 pp 397ndash400 2011

[45] P Nicolini ldquoNonlocal and generalizeduncertainty principleblack holesrdquo httparxivorgabs12022102

[46] P Nicolini A Smailagic and E Spallucci ldquoNoncommutativegeometry inspired Schwarzschild black holerdquo Physics Letters Bvol 632 no 4 pp 547ndash551 2006

[47] S Pramanik S Das and S Ghosh ldquoNoncommutative extensionof AdS-CFT and holographic superconductorsrdquo Physics LettersB vol 742 pp 266ndash273 2015

[48] W Dittrich and M Reuter Effective Lagrangians in QuantumElectrodynamics Springer Berlin Germany 1985

[49] M Ostrogradsky ldquoMemoires sur les equations differen-tielles relativrs au probleme des isoperimetersrdquo Memoires delrsquoAcademie Imperiale des Sciences de St Petersbourg vol 4 p385 1850

[50] P Gaete and E Spallucci ldquoFinite axionic electrodynamicsfrom a new non-commutative approachrdquo Journal of Physics AMathematical and Theoretical vol 45 no 6 Article ID 06540114 pages 2012

[51] P Gaete ldquoSome considerations about Podolsky-axionic electro-dynamicsrdquo International Journal of Modern Physics A vol 27no 11 Article ID 1250061 2012

[52] P Gaete ldquoRemarks on gauge-invariant variables and interactionenergy in QEDrdquo Physical Review D vol 59 no 12 Article ID127702 1999

[53] S Klarsfeld ldquoAnalytical expressions for the evaluation ofvacuum-polarization potentials in muonic atomsrdquo Physics Let-ters B vol 66 no 1 pp 86ndash88 1977

[54] A M Frolov and D M Wardlaw ldquoAnalytical formula for theUehling potentialrdquo The European Physical Journal B vol 85article 348 2012

[55] P Gaete ldquoOn gauge-invariant variables in QEDrdquo Zeitschrift furPhysik C Particles and Fields vol 76 no 2 pp 355ndash361 1997

[56] S Ferrara and A Sagnotti ldquoMassive Born-Infeld and other dualpairsrdquo Journal of High Energy Physics vol 2015 no 4 article 0322015

[57] R Bufalo ldquoBornndashinfeld electrodynamics in very special relativ-ityrdquo Physics Letters B vol 746 pp 251ndash256 2015

[58] HKleinert E Strobel and S-S Xue ldquoFractional effective actionat strong electromagnetic fieldsrdquo Physical Review D vol 88 no2 Article ID 025049 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Advances in High Energy Physics 9

screening issue for both massive Euler-Heisenberg-like andEuler-Heisenberg electrodynamics in the approximation ofthe strong-field limit Once again a correct identificationof physical degrees of freedom has been fundamental forunderstanding the physics hidden in gauge theories Inter-estingly enough their noncommutative version displays anultraviolet finite static potential The analysis above revealsthe key role played by the new quantum of length in ouranalysis In a general perspective the benefit of consideringthe present approach is to provide a unification scenarioamong different models as well as exploiting the equivalencein explicit calculations as we have illustrated in the course ofthis work

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

It is a pleasure for the author to thank J A Helayel-Netofor helpful comments on the paper This work was partiallysupported by Fondecyt (Chile) Grant 1130426 and DGIP(UTFSM) internal project USM 111458

References

[1] G Breit and J A Wheeler ldquoCollision of two light quantardquoPhysical Review vol 46 no 12 pp 1087ndash1091 1934

[2] S L Adler ldquoPhoton splitting and photon dispersion in a strongmagnetic fieldrdquo Annals of Physics vol 67 no 2 pp 599ndash6471971

[3] V Costantini B De Tollis and G Pistoni ldquoNonlinear effects inquantum electrodynamicsrdquo Il Nuovo Cimento A vol 2 no 3pp 733ndash787 1971

[4] R Ruffini G Vereshchagin and S-S Xue ldquoElectron-positronpairs in physics and astrophysics from heavy nuclei to blackholesrdquo Physics Reports vol 487 no 1ndash4 pp 1ndash140 2010

[5] G V Dunne ldquoThe Heisenberg-Euler effective action 75 yearsonrdquo International Journal of Modern Physics Conference Seriesvol 14 pp 42ndash56 2012

[6] C Bamber S J Boege TKoffas et al ldquoStudies of nonlinearQEDin collisions of 466GeV electrons with intense laser pulsesrdquoPhysical Review D vol 60 no 9 Article ID 092004 1999

[7] D L Burke R C Field G Horton-Smith et al ldquoPositronproduction in multiphoton light-by-light scatteringrdquo PhysicalReview Letters vol 79 no 9 article 1626 1997

[8] O J Pike F MacKenroth E G Hill and S J Rose ldquoA photonndashphoton collider in a vacuum hohlraumrdquo Nature Photonics vol8 no 6 pp 434ndash436 2014

[9] D Tommasini A Ferrando H Michinel and M Seco ldquoPre-cision tests of QED and non-standard models by searchingphoton-photon scattering in vacuum with high power lasersrdquoJournal ofHigh Energy Physics vol 2009 no 11 article 043 2009

[10] D Tommasini A Ferrando HMichinel andM Seco ldquoDetect-ing photon-photon scattering in vacuum at exawatt lasersrdquoPhysical Review A vol 77 no 4 Article ID 042101 2008

[11] M Born and L Infeld ldquoFoundations of the new field theoryrdquoProceedings of the Royal Society of London Series A Containing

Papers of a Mathematical and Physical Character vol 144 no852 pp 425ndash451 1934

[12] H Gies J Jaeckel and A Ringwald ldquoPolarized light propagat-ing in a magnetic field as a probe for millicharged fermionsrdquoPhysical Review Letters vol 97 no 14 Article ID 140402 2006

[13] E Masso and R Toldra ldquoLight spinless particle coupled tophotonsrdquo Physical Review D vol 52 no 4 pp 1755ndash1763 1995

[14] P Gaete and E I Guendelman ldquoConfinement in the presenceof external fields and axionsrdquoModern Physics Letters A vol 20no 5 article 319 2005

[15] P Gaete and E Spallucci ldquoConfinement effects from interactingchromo-magnetic and axion fieldsrdquo Journal of Physics A Math-ematical and General vol 39 no 20 pp 6021ndash6029 2006

[16] E S Fradkin and A A Tseytlin ldquoNon-linear electrodynamicsfrom quantized stringsrdquo Physics Letters B vol 163 no 1ndash4 pp123ndash130 1985

[17] E Bergshoeff E Sezgin C N Pope and P K TownsendldquoThe Born-Infeld action from conformal invariance of the opensuperstringrdquo Physics Letters B vol 188 no 1 pp 70ndash74 1987

[18] P Gaete and J Helayel-Neto ldquoFinite field-energy and interpar-ticle potential in logarithmic electrodynamicsrdquo The EuropeanPhysical Journal C vol 74 no 3 article 2816 2014

[19] P Gaete and J Helayel-Neto ldquoRemarks on nonlinear electrody-namicsrdquo European Physical Journal C vol 74 no 11 article 31822014

[20] S H Hendi ldquoAsymptotic ReissnerndashNordstrom black holesrdquoAnnals of Physics vol 333 pp 282ndash289 2013

[21] Z Zhao Q Pan S Chen and J Jing ldquoNotes on holographicsuperconductor models with the nonlinear electrodynamicsrdquoNuclear Physics B vol 871 no 1 pp 98ndash110 2013

[22] O Miskovic and R Olea ldquoConserved charges for black holes inEinstein-Gauss-Bonnet gravity coupled to nonlinear electrody-namics in AdS spacerdquo Physical Review D vol 83 no 2 ArticleID 024011 2011

[23] S H Mazharimousavi and M Halilsoy ldquoBlack holes and theclassical model of a particle in Einstein non-linear electrody-namics theoryrdquo Physics Letters B vol 678 no 4 pp 407ndash4102009

[24] G Amelino-Camelia ldquoSpecial treatmentrdquo Nature vol 418 no6893 pp 34ndash35 2002

[25] T Jacobson S Liberati andDMattingly ldquoThreshold effects andPlanck scale Lorentz violation combined constraints from highenergy astrophysicsrdquo Physical Review D vol 67 no 12 ArticleID 124011 2003

[26] T J Konopka and S A Major ldquoObservational limits onquantum geometry effectsrdquo New Journal of Physics vol 4 pp571ndash5718 2002

[27] SHossenfelder ldquoInterpretation of quantumfield theories with aminimal length scalerdquo Physical Review D vol 73 no 10 ArticleID 105013 9 pages 2006

[28] P Nicolini ldquoNoncommutative black holes the final appeal toquantum gravity a reviewrdquo International Journal of ModernPhysics A vol 24 no 7 pp 1229ndash1308 2009

[29] S Pramanik S Ghosh and P Pal ldquoElectrodynamics of a gener-alized charged particle in doubly special relativity frameworkrdquoAnnals of Physics vol 346 pp 113ndash128 2014

[30] A F Ali S Das and E C Vagenas ldquoDiscreteness of space fromthe generalized uncertainty principlerdquo Physics Letters B vol678 no 5 pp 497ndash499 2009

[31] G Amelino-Camelia ldquoQuantum-spacetime phenomenologyrdquoLiving Reviews in Relativity vol 16 article 5 2013

10 Advances in High Energy Physics

[32] E Witten ldquoNoncommutative geometry and string field theoryrdquoNuclear Physics B vol 268 no 2 pp 253ndash294 1986

[33] N Seiberg and E Witten ldquoString theory and noncommutativegeometryrdquo Journal of High Energy Physics vol 1999 article 0321999

[34] M R Douglas and N A Nekrasov ldquoNoncommutative fieldtheoryrdquo Reviews of Modern Physics vol 73 no 4 pp 977ndash10292001

[35] R J Szabo ldquoQuantum field theory on noncommutative spacesrdquoPhysics Reports vol 378 no 4 pp 207ndash299 2003

[36] J Gomis K Kamimura and T Mateos ldquoGauge and BRST gen-erators for space-time non-commutative U(1) theoryrdquo Journalof High Energy Physics vol 2001 no 3 article 010 2001

[37] A A Bichl J M Grimstrup L Popp M Schweda and RWulkenhaar ldquoPerturbative analysis of the Seiberg-Witten maprdquoInternational Journal of Modern Physics A vol 17 no 16 pp2219ndash2231 2002

[38] A Smailagic and E Spallucci ldquoUV divergence-free QFT onnoncommutative planerdquo Journal of Physics AMathematical andGeneral vol 36 no 39 pp L517ndashL521 2003

[39] A Smailagic and E Spallucci ldquoFeynman path integral on thenon-commutative planerdquo Journal of Physics A Mathematicaland General vol 36 no 33 pp L467ndashL471 2003

[40] M Knezevic D Knezevic and D Spasojevic ldquoStatistics ofequally weighted random paths on a class of self-similarstructuresrdquo Journal of Physics A Mathematical and Generalvol 37 no 1 pp 1ndash8 2004 Erratum in Journal of Physics AMathematical and General vol 37 p 7169 2004

[41] E Spallucci A Smailagic and P Nicolini ldquoTrace anomaly ona quantum spacetime manifoldrdquo Physical Review D vol 73Article ID 084004 2006

[42] R Banerjee S Gangopadhyay and S K Modak ldquoVoros prod-uct noncommutative Schwarzschild black hole and correctedarea lawrdquo Physics Letters B vol 686 no 2-3 pp 181ndash187 2010

[43] A B Hammou M Lagraa and M M Sheikh-Jabbari ldquoCoher-ent state induced star product on 119877

3

120582and the fuzzy sphererdquo

Physical Review D vol 66 no 2 Article ID 025025 2002[44] L Modesto J W Moffat and P Nicolini ldquoBlack holes in an

ultraviolet complete quantum gravityrdquo Physics Letters B vol695 no 1ndash4 pp 397ndash400 2011

[45] P Nicolini ldquoNonlocal and generalizeduncertainty principleblack holesrdquo httparxivorgabs12022102

[46] P Nicolini A Smailagic and E Spallucci ldquoNoncommutativegeometry inspired Schwarzschild black holerdquo Physics Letters Bvol 632 no 4 pp 547ndash551 2006

[47] S Pramanik S Das and S Ghosh ldquoNoncommutative extensionof AdS-CFT and holographic superconductorsrdquo Physics LettersB vol 742 pp 266ndash273 2015

[48] W Dittrich and M Reuter Effective Lagrangians in QuantumElectrodynamics Springer Berlin Germany 1985

[49] M Ostrogradsky ldquoMemoires sur les equations differen-tielles relativrs au probleme des isoperimetersrdquo Memoires delrsquoAcademie Imperiale des Sciences de St Petersbourg vol 4 p385 1850

[50] P Gaete and E Spallucci ldquoFinite axionic electrodynamicsfrom a new non-commutative approachrdquo Journal of Physics AMathematical and Theoretical vol 45 no 6 Article ID 06540114 pages 2012

[51] P Gaete ldquoSome considerations about Podolsky-axionic electro-dynamicsrdquo International Journal of Modern Physics A vol 27no 11 Article ID 1250061 2012

[52] P Gaete ldquoRemarks on gauge-invariant variables and interactionenergy in QEDrdquo Physical Review D vol 59 no 12 Article ID127702 1999

[53] S Klarsfeld ldquoAnalytical expressions for the evaluation ofvacuum-polarization potentials in muonic atomsrdquo Physics Let-ters B vol 66 no 1 pp 86ndash88 1977

[54] A M Frolov and D M Wardlaw ldquoAnalytical formula for theUehling potentialrdquo The European Physical Journal B vol 85article 348 2012

[55] P Gaete ldquoOn gauge-invariant variables in QEDrdquo Zeitschrift furPhysik C Particles and Fields vol 76 no 2 pp 355ndash361 1997

[56] S Ferrara and A Sagnotti ldquoMassive Born-Infeld and other dualpairsrdquo Journal of High Energy Physics vol 2015 no 4 article 0322015

[57] R Bufalo ldquoBornndashinfeld electrodynamics in very special relativ-ityrdquo Physics Letters B vol 746 pp 251ndash256 2015

[58] HKleinert E Strobel and S-S Xue ldquoFractional effective actionat strong electromagnetic fieldsrdquo Physical Review D vol 88 no2 Article ID 025049 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

10 Advances in High Energy Physics

[32] E Witten ldquoNoncommutative geometry and string field theoryrdquoNuclear Physics B vol 268 no 2 pp 253ndash294 1986

[33] N Seiberg and E Witten ldquoString theory and noncommutativegeometryrdquo Journal of High Energy Physics vol 1999 article 0321999

[34] M R Douglas and N A Nekrasov ldquoNoncommutative fieldtheoryrdquo Reviews of Modern Physics vol 73 no 4 pp 977ndash10292001

[35] R J Szabo ldquoQuantum field theory on noncommutative spacesrdquoPhysics Reports vol 378 no 4 pp 207ndash299 2003

[36] J Gomis K Kamimura and T Mateos ldquoGauge and BRST gen-erators for space-time non-commutative U(1) theoryrdquo Journalof High Energy Physics vol 2001 no 3 article 010 2001

[37] A A Bichl J M Grimstrup L Popp M Schweda and RWulkenhaar ldquoPerturbative analysis of the Seiberg-Witten maprdquoInternational Journal of Modern Physics A vol 17 no 16 pp2219ndash2231 2002

[38] A Smailagic and E Spallucci ldquoUV divergence-free QFT onnoncommutative planerdquo Journal of Physics AMathematical andGeneral vol 36 no 39 pp L517ndashL521 2003

[39] A Smailagic and E Spallucci ldquoFeynman path integral on thenon-commutative planerdquo Journal of Physics A Mathematicaland General vol 36 no 33 pp L467ndashL471 2003

[40] M Knezevic D Knezevic and D Spasojevic ldquoStatistics ofequally weighted random paths on a class of self-similarstructuresrdquo Journal of Physics A Mathematical and Generalvol 37 no 1 pp 1ndash8 2004 Erratum in Journal of Physics AMathematical and General vol 37 p 7169 2004

[41] E Spallucci A Smailagic and P Nicolini ldquoTrace anomaly ona quantum spacetime manifoldrdquo Physical Review D vol 73Article ID 084004 2006

[42] R Banerjee S Gangopadhyay and S K Modak ldquoVoros prod-uct noncommutative Schwarzschild black hole and correctedarea lawrdquo Physics Letters B vol 686 no 2-3 pp 181ndash187 2010

[43] A B Hammou M Lagraa and M M Sheikh-Jabbari ldquoCoher-ent state induced star product on 119877

3

120582and the fuzzy sphererdquo

Physical Review D vol 66 no 2 Article ID 025025 2002[44] L Modesto J W Moffat and P Nicolini ldquoBlack holes in an

ultraviolet complete quantum gravityrdquo Physics Letters B vol695 no 1ndash4 pp 397ndash400 2011

[45] P Nicolini ldquoNonlocal and generalizeduncertainty principleblack holesrdquo httparxivorgabs12022102

[46] P Nicolini A Smailagic and E Spallucci ldquoNoncommutativegeometry inspired Schwarzschild black holerdquo Physics Letters Bvol 632 no 4 pp 547ndash551 2006

[47] S Pramanik S Das and S Ghosh ldquoNoncommutative extensionof AdS-CFT and holographic superconductorsrdquo Physics LettersB vol 742 pp 266ndash273 2015

[48] W Dittrich and M Reuter Effective Lagrangians in QuantumElectrodynamics Springer Berlin Germany 1985

[49] M Ostrogradsky ldquoMemoires sur les equations differen-tielles relativrs au probleme des isoperimetersrdquo Memoires delrsquoAcademie Imperiale des Sciences de St Petersbourg vol 4 p385 1850

[50] P Gaete and E Spallucci ldquoFinite axionic electrodynamicsfrom a new non-commutative approachrdquo Journal of Physics AMathematical and Theoretical vol 45 no 6 Article ID 06540114 pages 2012

[51] P Gaete ldquoSome considerations about Podolsky-axionic electro-dynamicsrdquo International Journal of Modern Physics A vol 27no 11 Article ID 1250061 2012

[52] P Gaete ldquoRemarks on gauge-invariant variables and interactionenergy in QEDrdquo Physical Review D vol 59 no 12 Article ID127702 1999

[53] S Klarsfeld ldquoAnalytical expressions for the evaluation ofvacuum-polarization potentials in muonic atomsrdquo Physics Let-ters B vol 66 no 1 pp 86ndash88 1977

[54] A M Frolov and D M Wardlaw ldquoAnalytical formula for theUehling potentialrdquo The European Physical Journal B vol 85article 348 2012

[55] P Gaete ldquoOn gauge-invariant variables in QEDrdquo Zeitschrift furPhysik C Particles and Fields vol 76 no 2 pp 355ndash361 1997

[56] S Ferrara and A Sagnotti ldquoMassive Born-Infeld and other dualpairsrdquo Journal of High Energy Physics vol 2015 no 4 article 0322015

[57] R Bufalo ldquoBornndashinfeld electrodynamics in very special relativ-ityrdquo Physics Letters B vol 746 pp 251ndash256 2015

[58] HKleinert E Strobel and S-S Xue ldquoFractional effective actionat strong electromagnetic fieldsrdquo Physical Review D vol 88 no2 Article ID 025049 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of