On the Emergence of the Coulomb Forces in Quantum

Research ArticleOn the Emergence of the Coulomb Forces inQuantum Electrodynamics

Jan Naudts

Physics Department University of Antwerp Universiteitsplein 1 2610 Antwerpen Belgium

Correspondence should be addressed to Jan Naudts jannaudtsuantwerpenbe

Received 6 April 2017 Accepted 16 May 2017 Published 6 June 2017

Academic Editor Thierry Grandou

Copyright copy 2017 Jan Naudts This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited Thepublication of this article was funded by SCOAP3

A simple transformation of field variables eliminates Coulomb forces from the theory of quantum electrodynamics This suggeststhat Coulomb forces may be an emergent phenomenon rather than being fundamental This possibility is investigated in thecontext of reducible quantum electrodynamics It is shown that states exist which bind free photon and free electron fields Thebinding energy peaks in the long-wavelength limit This makes it plausible that Coulomb forces result from the interaction of theelectronpositron field with long-wavelength transversely polarized photons

1 Introduction

Quantum electrodynamics (QED) although very successfulis not a mathematically rigorous theory Several difficultieshave to be resolved or circumvented during the search fora consistent theory One of them is the excess number ofdegrees of freedom in the description of the electromagneticfields In this context the approach of Gupta [1] and Bleuler[2] is well known An alternative is found in the work ofCreutz [3] He uses a unitary transformation which yieldswave functions which do not undergo Coulomb forces Inquantum chromodynamics the light cone gauge is usedbecause then only transverse gluons remain [4] In the presentwork the assumption is that only transverse photons existand that the number of degrees of freedom of the electromag-netic field is two The argument of Creutz is used in oppositedirection to define field operators which satisfy the Maxwellequations in presence of Coulomb forces

Recently Verlinde [5 6] developed a cosmological theoryin which gravity forces are emergent forces in the sensethat they are produced by other more fundamental forcesof nature A similar statement is investigated here for therole of Coulomb forces in QED Not only are the degrees offreedom of the electromagnetic field limited to two but alsothe Hamiltonian does not contain a Coulomb potential

Further mathematical problems of QED disappear if oneabandons the axiom that representations of the canonicalcommutation and anticommutation relations must be irre-ducibleThis reducibleQED is studied in thework of Czachorand collaborators (see [7ndash9] and references given in thesepapers) This formalism can be simplified along the linesworked out by the author in a series of papers [10ndash12]

The reduction of the representation involves an integra-tion over three-dimensional wave vectors At fixed wave vec-tor the system is purely quantum mechanical and consists ofa pair of harmonic oscillators at each position in space-timeto cover transversely polarized photons A 16-dimensionalHilbert space describes the different states of an electron-positron field

The mediation of Coulomb forces by transverse photonscan be understood by analogy with the polaron problem ofSolid State PhysicsThe polaron [13] is a state binding an elec-tronwith quantized lattice vibrations In a polarizedmediumthe electric field of the electron is completely screened byredistribution of charges in its vicinity The remaining inter-actions between electron and medium result in an attractiveforce betweenpairs of polarons In some situations it is strongenough to form bipolarons

The present work shows that a free electron field can forma bound state with transversely polarized photons to form

HindawiAdvances in High Energy PhysicsVolume 2017 Article ID 7232798 4 pageshttpsdoiorg10115520177232798

2 Advances in High Energy Physics

a dressed electron field By analogy with the polaron caseone can then expect that dressed electrons interact with eachother and that Coulomb forces can be explained in this way

The next section introduces a transformation of fieldvariables which adds Coulomb forces to transverse photonsinteracting with a charge field The new field variables formwhat is called here the emergent picture of QED Theysatisfy the full Maxwell equations Section 3 highlights someaspects of reducible QED Section 4 discusses the proofof the existence of bound states of transverse photons ininteraction with the electron field The final section gives ashort discussion of what actually has been achieved

2 The Emergent Picture

In [3] the temporal gauge also called Weyl gauge or Hamil-tonian gauge is used A unitary transformation is definedby a generator (x) through

119881 = exp(119894 int d3x (x)) (1)

The generator is of the form

(x) = 1199024120587 int d3yA (119909) sdot x minus y

1003816100381610038161003816x minus y100381610038161003816100381631198950 (y) (2)

Here 119902 is the elementary unit of charge Bold characters areused to indicate 3-vectors The result of [3] in the context ofstandard QED is that

119881nabla sdot Eminus1 = nabla sdot E minus 1199021198950 (3)

where E(119909) are the electric field operators If they satisfyGaussrsquos law in the presence of a charge distribution 1198950(x) thenEminus1 satisfies Gaussrsquos law in absence of charges

In the present work the trick of [3] is directly applied todefine new electric field operators

119864耠耠훼 (119909)= 119864耠훼 (119909)

+ 12058301198884120587

120597120597119909훼 int dy 11003816100381610038161003816x minus y1003816100381610038161003816

(minus1199090) 119895 0 (y 0) (1199090) (4)

Here (1199090) = exp(minus1198941199090ℏ119888) is the time evolution of theinteracting system The new operators are marked with adouble prime to distinguish them from the operators of thenoninteracting system and those of the interacting systemThe latter are denoted with a single prime One verifiesimmediately that Gaussrsquo law is satisfied

sum훼

120597120597119909훼119864耠耠훼 (119909) = minus1205830119888119895 0耠 (119909) (5)

The second term of (4) is the Coulomb contribution tothe electric field The curl of this term vanishes Hence it isobvious to take

119861耠耠훼 (119909) equiv 119861耠훼 (119909) (6)

This implies the second of the four equations of Maxwellstating that the divergence of 119861耠耠훼 (119909) vanishes Also the fourthequation absence of magnetic charges follows immediatelybecause 119864耠耠(119909) and 119864耠(119909) have the same curl Faradayrsquos lawremains to be written as

(nabla times 119861耠耠 (119909))훼minus 1119888

1205971205971199090119864耠耠훼 (119909) = minus1205830119895 耠耠훼 (119909) (7)

with

119895 耠耠훼 (119909) = minus 11205830119888

1205971205971199090 (119864

耠耠훼 (119909) minus 119864耠훼 (119909)) (8)

Finally take 119895 耠耠0 (119909) = 119895 耠0 (119909) A short calculation shows that thenewly defined current operators 119895 耠耠휇 (119909) satisfy the continuityequation

One concludes that a formalism of QED is possiblewhich does not postulate the existence of longitudinal orscalar photons Two pictures coexist the original Heisenbergpicture and what is called here the emergent picture In bothpictures the time evolution of all operators is the same butthe definition of the electromagnetic field operators differs Inthe original description only transversely polarized photonsexist On the other hand the field operators of the emergentpicture satisfy the fullMaxwell equations includingCoulombforces

3 Reducible QED

A dominant characteristic of reducible QED in the versionused here is that many expressions look familiar fromstandard QED except that integrations over the wave vectoraremissingThey are postponed to the evaluation of quantumexpectation values As a consequence field operators dependon both position 119909 in spacetime and wave vector k inR3 Forinstance the electromagnetic potential operators are definedby

119860훼 (119909) = 12120582120576(H)훼 (kph) [119890minus푖푘ph120583 푥120583119886H + 119890푖푘ph120583 푥120583119886daggerH]

+ 12120582120576(V)훼 (kph) [119890minus푖푘ph120583 푥120583119886V + 119890푖푘

ph120583푥120583119886daggerV]

(9)

Here 119886H 119886daggerH 119886V and 119886daggerV are the creation and annihilationoperators of horizontally respectively vertically polarizedphotons The dispersion relation of the photon is 119896ph0 = |kph|as usual 120576(H)훼 (kph) and 120576(V)훼 (kph) are polarization vectors Theparameter 120582 is introduced for dimensional reasons

A consequence of the missing integration over the wavevector is that equal-time fields become noncommutative Forinstance a calculation starting from (9) shows that

[119860훼 (x 0) 119860훽 (y 0)]minus= 1198941205822

2 119865훼훽 (kph) sin (kph sdot (x minus y)) (10)

where 119865(kph) projects onto the plane orthogonal to kph If theintegration over kph is executed then the standard result ofvanishing equal-time commutators follows

Advances in High Energy Physics 3

Another feature of the theory is that wave functions areproperly normalized for each wave vector separately Forinstance if120595 describes an electronpositron field then120595k is awave function in the kth Fock space and satisfies ⟨120595k | 120595k⟩ =1 for each value of k Superpositions of wave functions withdifferent wave vector are allowed The general wave functionis therefore of the form

120595k = sum푋sub1234

radic120588푋 (k) |119883⟩ for any k (11)

The set 119883 selects one of the 16 possible states of an elec-tronpositron fieldThe empty set 0 refers to the vacuum state|0⟩ Normalization requires that

sum푋sub1234

120588푋 (k) = 1 for any k (12)

Ultraviolet divergences are avoided by requiring that 120588푋(k)with 119883 = 0 vanishes for large values of |k| Similarly thegeneral wave function of the free electromagnetic field is ofthe form

120595kph =infin

sum푚푛=0

radic120588푚푛 (kph)119890푖휙(푚푛) |119898 119899⟩ (13)

Normalization requires that

infin

sum푚푛=0

120588푚푛 (kph) = 1 for any kph (14)

The Dirac currents 119895휇(119909) are defined in terms of Diracspinors 푟(119909) which satisfy a free Dirac equation (see [11] fordetails) The Dirac equation is only used to define currents119895휇(119909) in absence of interaction with the electromagneticfields No interacting Dirac equation is considered Insteadthe interactions between the free field operators are describedby the usual interactionHamiltonian in aHeisenberg pictureSee (17) below

4 Bound States

Let daggeruarr denote the creation operator for an electron with spinup An example of a realistic electron field is described by

120595el = 119890푖휒(k)radic1 minus 120588el (k) |0⟩ + radic120588el (k)daggeruarr |0⟩ (15)

Similarly a realisticwave function for a horizontally polarizedphoton is

120595ph = radic120588ph (kph)119886daggerH |0⟩ + radic1 minus 120588ph (kph) |0⟩ (16)

The Hamiltonian is the usual one with interaction part

I = int dx119895 휇 (x 0) 119860휇 (x 0) (17)

Assume that 120588el(minuskel) = 120588el(kel)Then the average interactionenergy of a product state 120595 = 120595ph120595el vanishes for symmetry

reasonsHowever there exist entangledwave functionswhichlower the total energy

Choose for instance an entangled wave function of theform

120595kphk = [120591 (kph k) 119886daggerH + 1 minus 120591 (kph k)]

sdot radic120588 (kph) 120588el (k)daggeruarr |0⟩

+ radic1 minus 120588 (kph) 120588el (k) |0⟩

(18)

where 120591(kph k) equals either 1 or 0 Assume in addition thatthe electron density 120588el(k) has a Gaussian shape Then anexplicit calculation shows that the binding energy peaks forlong-wavelength horizontally polarized photons with wavevector kph in principle direction 1 (see [12] for detailedcalculations) Moreover the total energy is lower than that ofthe free electronOne concludes that there exist states bindingan electron field and a transversely polarized photon field

5 Discussion

In a theory with only transverse photons and no Coulombforces a simple transformation of the field variables given by(4) introduces new variables which satisfy the full Maxwellequations The new variables form what is called here theemergent picture of QED The time evolution of operators isthe same in the emergent picture as in the originalHeisenbergpicture Hence one can avoid introducing Coulomb forces ifone does not want to have them

Is this transformation more than a mathematical trickA plausible explanation of the physics behind this trans-formation is that long-wavelength transverse photons pro-duce effective forces between different parts of the elec-tronpositron field The expectation is that these effectiveforces coincide with what is known as Coulomb forces

Investigation of the scenario sketched above starts witha mathematical proof that transverse photons do interactwith the electronpositron field and even can form boundstates This proof is given in the context of reducible QEDbecause this formalism allows for a mathematically rigoroustreatment Details of the proof are found elsewhere [12] Thenext thing to do is an analysis of the time evolution of thesebound states This analysis is still missing

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this paper

References

[1] S N Gupta ldquoTheory of longitudinal photons in quantum elec-trodynamicsrdquo Proceedings of the Physical Society Section A vol63 no 7 article no 301 pp 681ndash691 1950

[2] K Bleuler ldquoEine neue methode zur Behandlung der longitudi-nalen und skalaren photonenrdquo Helvetica Physica Acta vol 23pp 567ndash586 1950


[3] M Creutz ldquoQuantum electrodynamics in the temporal gaugerdquoAnnals of Physics vol 117 no 2 pp 471ndash483 1979

[4] S J Brodsky H-C Pauli and S S Pinsky ldquoQuantum chromo-dynamics and other field theories on the light conerdquo PhysicsReports A Review Section of Physics Letters vol 301 no 4-6pp 299ndash486 1998

[5] E Verlinde ldquoOn the origin of gravity and the laws of NewtonrdquoJournal of High Energy Physics vol 4 p 29 2011

[6] E P Verlinde ldquoEmergent Gravity and the Dark Universerdquohttpsarxivorgabs161102269

[7] M Czachor ldquoNon-canonical quantum opticsrdquo Journal ofPhysics A Mathematical and General vol 33 no 45 pp 8081ndash8103 2000

[8] M Czachor and J Naudts ldquoRegularization as quantizationin reducible representations of CCRrdquo International Journal ofTheoretical Physics vol 46 no 1 pp 73ndash104 2007

[9] M Czachor and K Wrzask ldquoAutomatic regularization byquantization in reducible representations of CCR Point-formquantum optics with classical sourcesrdquo International Journal ofTheoretical Physics vol 48 no 9 pp 2511ndash2549 2009

[10] J Naudts ldquoReducible Quantum Electrodynamics I TheQuantum Dimension of the Electromagnetic Fieldrdquo httpsarxivorgabs150600098

[11] J Naudts ldquoReducible Quantum Electrodynamics II Thecharged states of the vacuumrdquo httpsarxivorgabs151002640

[12] J Naudts ldquoReducibleQuantumElectrodynamics IIIThe emer-gence of the Coulomb forcesrdquo httpsarxivorgabs170304952

[13] J T Devreese and A S Alexandrov ldquoFrohlich polaron andbipolaron recent developmentsrdquo Reports on Progress in Physicsvol 72 no 6 p 066501 2009

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a dressed electron field By analogy with the polaron caseone can then expect that dressed electrons interact with eachother and that Coulomb forces can be explained in this way

The next section introduces a transformation of fieldvariables which adds Coulomb forces to transverse photonsinteracting with a charge field The new field variables formwhat is called here the emergent picture of QED Theysatisfy the full Maxwell equations Section 3 highlights someaspects of reducible QED Section 4 discusses the proofof the existence of bound states of transverse photons ininteraction with the electron field The final section gives ashort discussion of what actually has been achieved

2 The Emergent Picture

In [3] the temporal gauge also called Weyl gauge or Hamil-tonian gauge is used A unitary transformation is definedby a generator (x) through

119881 = exp(119894 int d3x (x)) (1)

The generator is of the form

(x) = 1199024120587 int d3yA (119909) sdot x minus y

1003816100381610038161003816x minus y100381610038161003816100381631198950 (y) (2)

Here 119902 is the elementary unit of charge Bold characters areused to indicate 3-vectors The result of [3] in the context ofstandard QED is that

119881nabla sdot Eminus1 = nabla sdot E minus 1199021198950 (3)

where E(119909) are the electric field operators If they satisfyGaussrsquos law in the presence of a charge distribution 1198950(x) thenEminus1 satisfies Gaussrsquos law in absence of charges

In the present work the trick of [3] is directly applied todefine new electric field operators

119864耠耠훼 (119909)= 119864耠훼 (119909)

+ 12058301198884120587

120597120597119909훼 int dy 11003816100381610038161003816x minus y1003816100381610038161003816

(minus1199090) 119895 0 (y 0) (1199090) (4)

Here (1199090) = exp(minus1198941199090ℏ119888) is the time evolution of theinteracting system The new operators are marked with adouble prime to distinguish them from the operators of thenoninteracting system and those of the interacting systemThe latter are denoted with a single prime One verifiesimmediately that Gaussrsquo law is satisfied

sum훼

120597120597119909훼119864耠耠훼 (119909) = minus1205830119888119895 0耠 (119909) (5)

The second term of (4) is the Coulomb contribution tothe electric field The curl of this term vanishes Hence it isobvious to take

119861耠耠훼 (119909) equiv 119861耠훼 (119909) (6)

This implies the second of the four equations of Maxwellstating that the divergence of 119861耠耠훼 (119909) vanishes Also the fourthequation absence of magnetic charges follows immediatelybecause 119864耠耠(119909) and 119864耠(119909) have the same curl Faradayrsquos lawremains to be written as

(nabla times 119861耠耠 (119909))훼minus 1119888

1205971205971199090119864耠耠훼 (119909) = minus1205830119895 耠耠훼 (119909) (7)

with

119895 耠耠훼 (119909) = minus 11205830119888

1205971205971199090 (119864

耠耠훼 (119909) minus 119864耠훼 (119909)) (8)

Finally take 119895 耠耠0 (119909) = 119895 耠0 (119909) A short calculation shows that thenewly defined current operators 119895 耠耠휇 (119909) satisfy the continuityequation

One concludes that a formalism of QED is possiblewhich does not postulate the existence of longitudinal orscalar photons Two pictures coexist the original Heisenbergpicture and what is called here the emergent picture In bothpictures the time evolution of all operators is the same butthe definition of the electromagnetic field operators differs Inthe original description only transversely polarized photonsexist On the other hand the field operators of the emergentpicture satisfy the fullMaxwell equations includingCoulombforces

3 Reducible QED

A dominant characteristic of reducible QED in the versionused here is that many expressions look familiar fromstandard QED except that integrations over the wave vectoraremissingThey are postponed to the evaluation of quantumexpectation values As a consequence field operators dependon both position 119909 in spacetime and wave vector k inR3 Forinstance the electromagnetic potential operators are definedby

119860훼 (119909) = 12120582120576(H)훼 (kph) [119890minus푖푘ph120583 푥120583119886H + 119890푖푘ph120583 푥120583119886daggerH]

+ 12120582120576(V)훼 (kph) [119890minus푖푘ph120583 푥120583119886V + 119890푖푘

ph120583푥120583119886daggerV]

(9)

Here 119886H 119886daggerH 119886V and 119886daggerV are the creation and annihilationoperators of horizontally respectively vertically polarizedphotons The dispersion relation of the photon is 119896ph0 = |kph|as usual 120576(H)훼 (kph) and 120576(V)훼 (kph) are polarization vectors Theparameter 120582 is introduced for dimensional reasons

A consequence of the missing integration over the wavevector is that equal-time fields become noncommutative Forinstance a calculation starting from (9) shows that

[119860훼 (x 0) 119860훽 (y 0)]minus= 1198941205822

2 119865훼훽 (kph) sin (kph sdot (x minus y)) (10)

where 119865(kph) projects onto the plane orthogonal to kph If theintegration over kph is executed then the standard result ofvanishing equal-time commutators follows



120595k = sum푋sub1234

radic120588푋 (k) |119883⟩ for any k (11)


sum푋sub1234

120588푋 (k) = 1 for any k (12)


120595kph =infin

sum푚푛=0

radic120588푚푛 (kph)119890푖휙(푚푛) |119898 119899⟩ (13)


infin

sum푚푛=0



4 Bound States






I = int dx119895 휇 (x 0) 119860휇 (x 0) (17)







(18)


5 Discussion






References




















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





120595k = sum푋sub1234

radic120588푋 (k) |119883⟩ for any k (11)


sum푋sub1234

120588푋 (k) = 1 for any k (12)


120595kph =infin

sum푚푛=0

radic120588푚푛 (kph)119890푖휙(푚푛) |119898 119899⟩ (13)


infin

sum푚푛=0



4 Bound States






I = int dx119895 휇 (x 0) 119860휇 (x 0) (17)







(18)


5 Discussion






References




















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



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On the Emergence of the Coulomb Forces in Quantum