Teoria de Campo de Gauge

8/13/2019 Teoria de Campo de Gauge

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Classical gauge field theoryBertrand BERCHE

Groupe de Physique Statistique

UHP Nancy 1

References

D.J. Gross, Gauge Theory-Past, Present, and Future ?, Chinese Journal ofPhysics 30955 (1992),

C. Quigg, Gauge theory of the strong, weak, and electromagnetic interac-

tions, Westwiew Press, 1997, L.H. Ryder, Quantum field theory, Cambridge University Press, Cambridge

1985,

S. Weinberg, The quantum theory of fields, Vol. I and II, Cambridge Uni-versity Press, Cambridge 1996,

T.-P. Cheng and L.-F. Li, Gauge theory of elementary particle physics,Oxford University Press, Oxford 1984,

M. E. Peskin and D. V. Schroeder, An Introduction to Quantum FieldTheory, (ABP) 1995.

Introduction

Classical field theories

Consider as an example the free particle Klein-Gordon equation 1 ( +

m2)= 0 which follows from the conservation equation ppm2 = 0 with the

1. We forget about factorsandc in this chapter.

1


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correspondence p i. We use the convention g= diag (+1, 1, 1, 1)for the metric tensor. For later use, we choose the MKSA conventions, e.g.p = (E/c,p) (E, p) and A = (/c,A) (,A), where an over-arrowdenotes a vector in ordinary space, or = (1

ct

, ) ( t

, ). Most ofthe time, and c are fixed to unity.

The Lagrangian density from which this equation follows must satisfy 2

d4x L(, , , ) = L

L

()= 0, (1)

L

= m2, (2)

L

()=

. (3)

The first condition is fulfilled if

L = m2+ terms in , (4)and the second if3

L =

+ terms in

. (5)Eventually the Klein-Gordon Lagrangian is given by

L = m2. (6)The first term is usually referred to as kinetic energy, although the space part 4

is reminiscent from local interactions in the context of classical field theory,and the second term to the mass term, since it corresponds to the mass of theparticles after quantization. The quantization procedure, not discussed here,consists in the promotion of the classical fields into creation (or annihilation)field operators which obey, together with the corresponding conjugate mo-menta, to canonical commutation relations. We will stay here at the level of

2. We consider complex scalar fields. For real fields, factors of 12 would appear here andthere.

3. We develop to obtainL

() =0

L(0)

+ iL

(i) == (00 + i

i)=

(00 ii), the solutionL =00 ii= 00+ii= (upto terms in ) follows.

4. We have a space and a time part in = 1c2

t2 ||2, and only the time

derivatives are reminiscent of a kinetic energy.


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classical field theory, which means that although the fields correspond to thewave functions of quantum objects in the first-quantized form of the theory,they are treated by classical field theory (e.g. Euler-Lagrange equations) andno quantum fluctuations are allowed.

The idea behind non-Abelian gauge theory

According to Salam and Ward, cited by Novaes in hep-ph/0001283 :

Our basic postulate is that it should be possible to generate strong, weak,and electromagnetic interaction terms ... by making local gauge transforma-tions on the kinetic-energy terms in the free Lagrangian for all particles.

or Yang and Mills cited in A.C.T. Wu and C.N. Yang, Int. J. Mod. Phys. Vol.21, No. 16 (2006) 3235 :

The conservation of isotopic spin points to the existence of a fundamentalinvariance law similar to the conservation of electric charge. In the latter case,the electric charge serves as a source of electromagnetic field. An importantconcept in this case is gauge invariance which is closely connected with (1)the equation of motion of the electromagnetic field, (2) the existence of acurrent density, and (3) the possible interactions between a charged field andthe electromagnetic field. We have tried to generalize this concept of gauge

invariance to apply to isotopic spin conservation.

The origin of gauge invariance 5

The idea of gauging a theory, i.e. making local the symmetries, is dueto E. Noether, but gauge invariance was introduced by Weyl when he triedto incorporate electromagnetism into geometry through the idea of local scaletransformations. From one point of space-time to an other at a distance dx,the scale is changed from 1 to (1 +Sdx

) in such a way that a space-timedependent function (of dimension of a length) f(x) is changed according to

f(x) f(x + dx) = (f+ f dx))(1+ Sdx) f+ [(+ S)f]dx. (7)The original idea of Weyl was to identify S to the 4potential A, butwith the advent of quantum mechanics and the correspondence between pand i, it was later realized that the correct identification is S iqA.Weyl nevertheless retained his original terminology ofgauge invarianceas aninvariance under a change of length scaled, or a change of the gauge.

5. See Cheng and Li pp235-236, and Gross p956


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AbelianU(1)gauge theory

The complex nature of the field as an internal structure

The prototype of gauge theory is the theory of electromagnetism, or Abe-lian U(1) theory, where charge conservation is deeply connected to globalphase invariance in quantum mechanics (a connection probably made first byHermann Weyl).

The complex scalar field (x) (Schrodinger or Klein-Gordon field) canbe modified by a global phase transformation (x) exp(iq)(x) (Abe-lian U(1) gauge transformation) which leaves the matter LagrangianL =

V(||2) unchanged. We anticipate and introduce already the chargeq which couples the particle to the electromagnetic field 6. Let us write =

1 +i2 and introduce a real two-component field (x) =

1

2

. The gauge

transformation now appears as an Abelian rotation in a two-dimensional (in-ternal) space.

Noether theorem and matter current density

Consider the Langrangian density

L0= V(). (8)

For later use, we will call this Lagrangian the function

L0=F(, , , ). (9)

For any symmetry transformation,

L0=L0

+() L0

()+ = 0. (10)

The notation means that we add a similar term with all complexnumbers replaced by their conjugate. The global phase transformation = eiq being such a symmetry, we put = iq and () = iq

6. Here we set = 1. In most of the relations, is restored through the substitutionq q/, g g/, ori i.


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in the expression ofL0 to have

L0 = iq

L0

+ L0

()

+

= iq

L0

()

+

= j. (11)It follows that the Noether current

j = iq L0()

L0()

(12)

is conserved, j = 0. The electric charge conservation thus appears as the

consequence of the invariance of the theory under global phase changes, thisis called a global gauge symmetry. Note that in the case of the Klein-GordonLagrangian, the conserved current takes the form 7

j = iq( ). (13)

Local gauge symmetry

Extending the gauge symmetry to local transformations requires the intro-duction of a (vector) gauge field which will be seen later as the vector potentialof electromagnetism. In other words, making local the gauge symmetry buildsthe electromagnetic interaction.

Let us assume that the gauge transformation is local, i.e. =eiq(x) G(x). We note that now =G(x)+ (G(x))=G(x), and the derivative of the field does not transform like the field itselfdoes. Let us define the covariantderivative 8

D +iqA (14)

where A is still to be defined by its transformation properties. Like

=G(x), we demand that

D (D) G(x)D. (15)7. Note that at this point, the sign in front of the current density was arbitrary, but if one

wants to recover the usual expression of probability density current in quantum mechanics,e.g. in the Schrodinger case, j =

2mi+ , it leads to the present charge current

density after being multiplied by q.8. The term covariant refers to covariance with respect to the introduction of the local

transformation, and not to covariant-contravariant indices.


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Since we have (D) = ()+iqA =G(x)+(G(x))+iqAG(x)and G(x)D = G(x)+ iqAG(x), we must require that iqAG(x) =iqAG(x) G(x) or, multiplying by G1(x),

A=A+ i

qG1(x)G(x) =A (x). (16)

Demanding the invariance property of the kinetic term (according to Salamand Ward cited above) in the Lagrangian density under a local gauge trans-formation requires the introduction of a vector field which obeys the usualtransformation law of the vector potential of electromagnetism through gaugetransformations. These transformations which appeared before as a kind ofmathematical curiosity of Maxwell theory are now necessary in order to pre-serve local gauge invariance. In a sense, the interaction is created by theprinciple of local gauge symmetry, while the principle of global gauge symme-try implies the conservation of the electric charge.

The interaction of matter (as described by the Lagrangian density

L0)

with the electromagnetic field can be built in through essentially two dif-ferent approaches. In a first approach, we successively add terms to L0 inorder to get at the end a locally gauge invariant Lagrangian. Since the pres-cription of local gauge invariance induces Maxwell interactions, this shouldautomatically incorporate interaction terms inL. Starting with the observa-tion thatL0 is no longer gauge invariant throughlocal transformations, sinceL0 =(x)j ((x))j now contains the second term, we have tokill this last term by the introduction of aL1 =jA term 9. This againgenerates one more contribution in (L0+ L1) which is canceled if we addL2 =q2AA. The combinationL0+ L1+ L2 =L0+ Lint is now locallygauge invariant.

In a shorter approach, calledminimal coupling, we simply replace the kine-tic term

inL0 by (D)(D) which was especially constructed inorder to be gauge invariant (under localgauge transformations). One also hasto add the pure field contributionLA= 14FF to get the full Lagrangian

9. Note that here j is the current which was conserved in the absence of gauge inter-action. The sign here is also coherent with the expression(jA).


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density 10

Ltot = L0+ Lint+ LA F(, , (D), (D)) + LA= (D)(D) V() + LA. (17)

The interaction terms are recovered from the r.h.s. (here in the Klein-Gordoncase) :

(

D)

(

D) = (+iqA)

(+iqA)

= iqA+iqA +q2AA=

+iq( )A +q2AA,(18)

and

L = V() jA +q2AA 14FF. (19)Now, since the functional form of the interacting matter Lagrangian is thesame as the form of the free Lagrangian withD instead of, the conservedNoether current in the presence of gauge interaction reads as 11

J

= iq

L0

(D) +

= iqD+ .= j 2q2A, (20)

the last two lines being valid for the KG case. The interaction term can nowbe written, up to second order terms 12 inA, asLint= JA.

The equations of motion follow from Euler-Lagrange equations,

StotA

=Ltot

A Ltot

(A)= 0, (21)

which simply yieldLintA

=LA

(A). (22)

10. Remember that we callL0= F(,, ,).11. See e.g. V. Rubakov, Classical Theory of Gauge fields, Princeton University Press

2002, pp21-27.12. These terms are particularly important, since they restore gauge invariance of the

interaction Lagrangian which otherwise would not exhibit this gauge invariance propertyin the present form. Indeed, J is gauge invariant, but A is not.


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The l.h.s. is LintA

=j + 2q2A =J while at the r.h.s. we haveLA

(A) =1

4

(A)FF

= 14

F(A)

F +FF

(A)

. Both terms in

parenthesis are equal to F F =2F and eventually we obtain aftermultiplication by1

4, 13

F = F = J (23)

and, since F is antisymmetric, we recover the conservation equation forJ,

J

= 0. (24)

Non-Abelian (Yang-Mills) SU(2)gauge theory

Internal structure

The global phase transformation(x) exp(iq)(x) as mentioned aboveappears as an Abelian rotation in a two-dimensional (internal) space. Thisgauge transformation, when extended to local phase transformations (x), generates the electromagnetic interaction. It is possible to generalize to

non-Abelian gauge transformations by extending the internal (isospin) struc-ture. The field is for example a 3-component real scalar field

(x) =

1

2

3

(25)

and the transformation corresponds to a rotation in the internal space 14 (thecorresponding charge is now written g)

(x) exp(ig)(x), (26)with the generators (which do not commute, hence the name of non-Abelian)of the rotations in three dimensions

1 =

0 0 00 0 i

0 i 0

, 2 =

0 0 i0 0 0

i 0 0

, 3 =

0 i 0i 0 0

0 0 0

. (27)

13. Note again that our choice of sign for the current density makes the expression co-herent with the usual Maxwell equations.

14. See Weinberg II p3.


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Use of bold font stands for vectors in the internal space and the scalar product(with omitted) is a 3 3 matrix,

=

0 i

3 i2

i3 0 i1i2 i1 0

. (28)

The transformation (being a rotation in 3 dimensions) is non Abelian and itcan be rewritten as 15 (x) (x) (x). Here, is a vector in theinternal space whose length is the angle of rotation and whose direction isthe rotation axis.

Rotations in three space dimensions are equivalent to SU(2) transforma-tions acting on complex two-component spinors 16. The field is now representedby such a spinor,

(x) =

(x)(x)

. (29)

Each component is complex, but due to a normalization constraint we stillhave three independent real scalar fields. Under a rotation in the internalspace, (x) changes into

(x) exp 12

ig

(x) (30)

where are the three Pauli matrices 17,

1 =

0 11 0

, 2 =

0 ii 0

, 3 =

1 00 1

, (32)

which obey the Lie algebra

[i, j] = 2iijkk (33)

15. See Ryder p108.

16. An O(3) transformation on corresponds to an SU(2) transformation on =(x)(x)

with 1 = 1

2(2 2), 2 = 12i(2+2), 3 =, see Ryder pp32-38.

17. There is nothing very mysterious to introduce two component spinors and the Paulimatrices in the context of quantum mechanics. Indeed, the Pauli equation, which describesthe non-relativistic spin- 1

2electron in an electromagnetic field reads as

H

(x)(x)

=

1

2m(p qA)2 q

1I

(x)(x)

q

2m B

(x)(x)

= E

(x)(x)

. (31)

Note that here the hat Hnotation stands for a 2 by 2 matrix.


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with ijk the totally antisymmetric tensor, and is a 2 2 matrix

=

3 1 i2

1 +i2 3

(34)

Here 12

i are the generators ofSU(2) transformations.The two representations can be written in a unified way in component

form 18. The (infinitesimal) transformation of the field (let say (x), for or) is written

l(x) =iga

(ta

)ml m(x). (35)

The as are now the parameters of the infinitesimal transformation. Thesuperscript a is used as internal space index, l and m denote the 2 compo-nents (resp. 3) in the SU(2) representation (respSO(3)) and t stands for thegenerator ( (resp. )).

Together with the internal degrees of freedom, the fields of course dependon space-time position x.

Pure matter field and Noether current

From now on, we choose the SU(2) representation. Let

L0 be a gauge

invariant matter Lagrangian density

L0= V() (36)where (x) = ((x),

(x)) and V(

) is a potential to be defined later.The variation of the Lagrangian density yields

L0=L0

+()

L0()

+L0

+

L0()

(). (37)

With the infinitesimal transformation

(x) = 12

ig(x), (38)

(x) = 12ig(x), (39)which can be written in matrix form,

(x)(x)

= 1

2ig

3 1 i2

1 +i2 3

(x)(x)

(40)

((x), (x)) = 12ig((x), (x))

3 1 +i2

1 i2 3

, (41)

18. See Weinberg II p2.


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Introduction of a gauge potential 20

Now we extend the formalism to local gauge transformations

(x) = exp12

ig(x)

(x) G(x)(x), (47)

(where G(x) = exp12

ig(x)

is a 2 2 matrix) or locally to(x) = 1

2ig(x)(x), (48)

but a problem occurs which will make the Lagrangian densityL0 not gaugeinvariant. The fact that does not obey the same gauge transformationthanitself corrupts the transformation of the kinetic energy. We have

=

(G)+ G(). Let us introduce a covariant derivative21

D +igB. (49)

Here as before we use the short notation for 1I with 1I the 2 by 2 iden-

tity matrix. The context suffices to distinguish between and 1I. B =12B=

12

aBa (summation over a understood) is another 2 by 2 matrix (infact there is one such matrix for each of the 4 space-time components, Bis a

gauge potential (with three internal components which all are 4space-timevectors)). We demand the following transformation

D(x) D(x) = G(x)(D(x)). (50)

We obtainD = (+ igB) = (G)+ G() + ig BG. From therequirement (??), this quantity should be equal to G(+igB)= G()+

igG(B). It follows an equation for the transformation ofB, igB

G =

igG(B) (G). Written in terms of operators, this equation is BG =GB+

ig

G. We multiply both sides by G1 on the right to get

B=GBG

1 + i

g(G)G

1 = G

B+

i

gG1(G)

G1. (51)

In the case of electromagnetism, the local gauge transformation is perfor-med by the operator (now an ordinary function) GEM(x) = exp(iq(x)) with

20. See Quigg pp55-5721. In analogy with electromagnetism where the covariant derivative iD is given by

p qA with p= i, which yieldsD = +i qA.


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(x) some function, and eqn. (??) leads to the known transformation of thegauge potential of electromagnetism,

A=GEMAG1EM+

i

q(GEM)G

1EM=A . (52)

From eqn. (??) and the transformation ofB = 12B, we can deduce the

gauge transformation ofB as well. Consider an infinitesimal gauge transfor-mation

G(x) =1I+ 12

ig(x). (53)

Eqn. (??) reads as (to linear order in i)

12B =

12B+

14

ig(() (B) (B) ()) 12(). (54)The term in the middle, written in components, has the form

12

ijBk(jkkj) = 1

2ijBk[

j, k] = jkl(jBk)l = (B) (55)and it follows that

12B =

12B 12g(B) 12(). (56)

Another common expression uses the identity (B)= 2i12, 1

2B

such that

12B =

12B+ig

12, 1

2B

12

(). (57)

We can also write directly

B = B gB . (58)The gauge transformation ofB appears as a gradient term (like in electro-magnetism) plus a rotation in internal space.

The field-strength tensor and field equations 22

Let us introduce a field-strength tensor by the 2 by 2 matrix

F= 12F (59)

22. See Quigg pp58-59


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Note that the equations of motion are not as simple as the familiar Maxwellequations which are linear. The Yang-Mills equations of motion on the otherhand are not linear, and this is due to the fact that the gauge field carries thecharge associated to the interaction. Thus, even in the absence of matter, thederivative of the field tensor does not vanish.

In the massive case (which is not gauge invariant), a term

m2BB (67)

is added to the Lagrangian density and the (Proca-like) equations of motionbecome

F gB F=m2B. (68)

Construction of a gauge-invariant interaction 25

Since (x) depends on space-time, the variations () (or ())

contain an extra term which contributes to L0

L0 = 1

2 ig

(x)

L

0

()

+1

2ig

(x) 1

2ig[(x)]

L0

()+ ( )

= (x)(j) ((x))j (69)The first term vanishes thanks to Noether theorem, but the second term,((x))j persists, so L0is not gauge invariant under local gauge transfor-mations. In order to compensate this new term, we must add another contri-bution to the Lagrangian,

L1 = j

B (70)

and demand that B obeys the gauge transformation (??). Now,

L0+L1= (j) ()j jB jB, (71)25. This section may be omitted. It presents step by step the construction of a gauge

invariant Lagrangian which is obtained faster by the minimal coupling requirement presen-ted later. The approach used here follows the presentation of Ryder pp96-98 in the case ofAbelian U(1) gauge symmetry.


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where only the first term vanishes identically. Performing the variation ofj

yields

j = 12

ig 12

ig() + ( )= 1

2ig1

2ig

12

ig12

ig+ 12

ig

+ ( )= 1

4g2 [,] + 1

4g2 [,] + 1

4g2 {,}

=

12

ig2

12

ig2

+ 1

2g2 (72)

We have used the identities [,] =2i and{,} = 2which are proven by the use of Pauli matrices properties [i, j ] = 2iijk

k

and{i, j} = 2ij1I. The three remaining terms of eqn. (??) are equal to()j = 12ig+ ( ), (73)jB = gj (B) j, (74)jB = 12ig2( ) B 12 ig2( ) B

+12

g2B

= +gj (B) + 12g2B, (75)where we have used the cyclic property () B= (B) . The sumeventually gives only

L0+L1= 12g2B (76)We still have to add another term which should eventually make the whole

Lagrangian gauge invariant,

L2= 14g2BB= 14g2BB. (77)The variation of

L2 reads as

L2 = 14g2(BB+BB+ BB())= 1

2g2BB

= 12

g2B(g(B) +)= 1

2g2B

(78)

and we obtain the expected vanishing variation

L0+L1+L2= 0 (79)


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which proves that the gauge invariant interaction in the presence of gaugefields contains two terms,

L1+ L2= jB+ 14g2BB. (80)The kinetic energy of the gauge field itself was not included, like the free

particle contribution of eqn (??).

Covariant derivative, minimal coupling

The introduction of the gauge covariant derivative facilitates the calcula-tions. The action should not depend on the gauge choice, since the equations ofmotion are independent of the gauge. The Lagrangian density should thus bea gauge scalar. The potential term is already a gauge scalar, since it dependsonly onwhich transforms covariantly according to = (G1)(G).In order to become manifestly gauge covariant, the kinetic term should bewritten as (D)(D), since the covariant derivative of the fieldD wasconstructed for the purpose of obeying the same gauge transformation thanthe field itself, (D)(D) = (D)G1G(D).

The minimal coupling is the prescription that the interaction with thegauge field is obtained by the replacement

D is the Lagrangian

density of eqn. (??) 26 :

L = (D)(D) V() (81)=

12 igB

+ 12

igB

V()=

12

igB+ 1

2igB

+14

g2(B)(B) V()

= jB+ 14g2BB V() (82)

where in the last term use has been made of the identity

(B)(B) =

B1B

1 +B2B2 +B3B

3 00 B1B

1 +B2B2 +B3B

3

= BB

1I.

(83)

This operation is called gauging the Lagrangian. Note that in the case offermionic particles, we have to use the Dirac Lagrangian density i,

26. In an expression such that 12 igB

, it is understood that B acts on

on the left.


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with =0 the adjoint spinor and the Dirac matrices 27 instead of thekinetic energy

, and the gauge invariant Lagrangian becomes

L = (iD m) 14FF. (84)

Conserved current in the presence of gauge fields

In the presence of gauge fields, the conserved Noether current can be writ-ten in terms of the covariant derivative,

J = 12

ig L0

((D)) L0(D)

12

ig

(85)

In the case of the Lagrangian (??), it becomes

J = 12

igD (D) 12

ig

= 12

ig

+ 12

igB 1

2igB

12

ig

= j 14

g2 [,B] . (86)

We see that the conserved current in the presence of gauge fields has twocontributions, one coming from the ordinary matter current and the otherfrom the gauge field itself. Using the identity [,B] = 2i B, we get

J =j + 12

ig2 B. (87)

In component form we have

Ja =ja 12

ig2abcbBc. (88)

The current density J

is conserved in the ordinary sense

28

,

J = 0 (89)

while j satisfies a gauge-covariant conservation law

Dj = 0. (90)27. See e.g. Ryder pp43-4628. See Weinberg II pp12-13


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Figure 1 The Mexican hat potential (from E.A. Paschos, Electroweaktheory, Cambridge University Press, Cambridge 2007.

As a result of the continuous symmetry, they are infinitely degenerate. In orderto analyze the field fluctuations around the minimum, we choose a particularvacuum state 10 = v, 20 = 0 and denote the fluctuations by

1 = v +h1, 2=h2 (102)

in terms of which the potential becomes

V(h1, h2) = 14(h21+h22+ 2vh1)2. (103)

Expansion of this potential shows that h1 becomes massive while h2 remainsmassless, and the appearance of cubic terms breaks the original SO(2) sym-metry. The massless field is called a Goldstone mode (or Nambu-Goldstonemode). It is easy to understand why h2 remains massless while h1 acquireda mass : close to the minimum which we have selected, 1 fluctuations haveto survive to the potential growth, these are amplitude fluctuations in a po-lar representation of the model, while 2 fluctuations correspond to phasefluctuations which do not cost any energy.

Spontaneous breaking of local symmetriesA new phenomenon occurs with local gauge theories, where the selection

of a particular minimum and the fluctuations around this minimum lead tomassive gauge fields which would otherwise be forbidden, since mass terms forthe gauge field would break gauge invariance. At the same time, the Goldstonemode disappears.

We consider the Lagrangian density ofU(1) gauge theory,

L = 14

FF + (D)(D) V(), (104)


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with the potential

V() = 2+()2. (105)As we have seen before, the theory is invariant under the gauge transformationcorresponding to a local rotation of the scalar field in the complex plane

(x) eiq(x)(x) (106)A(x) A(x) (x). (107)

Let us define two real fields (x) and h(x), associated to the phase andthe amplitude fluctuations around a particular (chosen real) minimum v =(2/2)1/2,

(x) =ei(x)/v 1

2(v+h(x)). (108)

The local gauge transformation defined by q(x) = (x)/v eliminates (x),since

(x) = ei(x)/v(x) = 1

2(v+h(x)), (109)

A(x) = A(x) + 1qv(x). (110)

The net effect in the Lagrangian density is the following,

L = 14

FF+ (D)(D) + 122(v + h2(x))2 14(v + h(x))4, (111)

withD = +iqA. The kinetic energy term generates the mass for thegauge field A :

(D)(D) = 12hh+ 12q2AA(v2 + 2hv+h2), (112)

and, as we announced, the Goldstone mode (x) was absorbed in the re-definition of the gauge field.

This mechanism is known in condensed matter physics as the Andersonmechanism (see next section), and it occurs in superconductivity, where thenon-zero mass (which also defines a characteristic length scale) of the gaugefield is responsible for the Meissner effect (the fact that the magnetic field isexpelled from the bulk of the material). In particle physics, this mechanismenables to give a mass to the gauge bosons, as we discuss below. This is knownin this context as the Higgs mechanism.


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The Anderson mechanism 29

We will first reproduce all the generic arguments given before in the rela-tivisticU(1) case before considering the application to superconductivity.

Gauge invariance in non relativistic quantum mechanics

Since we are interested in this section by non relativistic quantum mecha-nics, we will not distinguish between contravariant and covariant indices, i.e.xi = x,y,z and i =

xi

. Summation is understood as soon as an index isrepeated in an expression. The space part of an expression like

will

thus simply be denoted as ii =ii, and 2i =ii stands for2. Due to this non-covariant notation, there are other minus signs here andthere, for instance in the definition of the field tensor Fij = (iAj jAi).

The Lagrangian density for non relativistic quantum mechanics is given byan expression due to Jordan and Wigner (here written in a symmetric form)

L0 = 12i( ) 2

2mi

i V . (113)

The Euler-Lagrange equation (variation with respect to ) indeed leads the

Schrodinger equation,

L0

= 12

i V , (114)

t

L0

= 1

2i, (115)

i

L0

(i)

=

2

2m2i. (116)

and we have

i= 2

2m2i+V . (117)

Once we have noticed that the Lagrangian density is a function of, , andi (as well as their complex conjugates), its variation under an infinitesimal

29. Caution : in all this section we forget about the covariant notation, all indices arespace indices written as subscripts and summed over when repeated. See the beginning ofthe next paragraph for more detailed explanations.


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global gauge transformation = i

e leads to

L0 = i

e

i

L0

(i)

+t

L0

(i)

+i

L0(i)

+ L0

+ ( )

= [iji+t], (118)(use has been made of the equations of motion) where

ji = e

2mi((i) (i)) , (119)

= e. (120)

Notice that this continuity equation is usually written in the standard form

j+ t

= 0. (121)

In the presence of an electromagnetic field (relativistic in essence), we usethe minimal coupling

Di = i i

eAi (122)

and we add the field Lagrangian contribution (for further purpose, we willonly consider the magnetic contribution and only in a static situation, i.e.1

4FijFij) (Fij = (iAj jAi)),

L = 12

i( ) 2

2m(Di)(Di) V 14FijFij. (123)

The gauge field Ai is changed by a local gauge transformation,

(x) = (x)eie(x), (124)

Ai(x) = Ai(x) i(x), (125)

but the field tensor Fij is unaffected. The equations of motion in the presenceof the gauge field are modified 30,

LAj

= e

2mi[(j) (j)] e

2

mAj

, (126)

i

L

(iAj)

= 1

4

(iAj)(FklFkl)

= iFij, (127)

30. Note here a modification in the usual signs.


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leading to Maxwell equations

iFij =Jj, (128)

where the current density is also recovered from the expression

Ji = e

2mi((Di) (Di))

= e

2mi((i) (i)) e

2

mAi

. (129)

The Schrodinger equation follows from the variation w.r.t. ,

L

= 12

i V 2

2m

i

eAii+

e2

2AiAi

, (130)

t

L0

= 1

2i, (131)

i

L0

(i)

=

2

2m

2i

i

eAi

. (132)

Collecting the different terms, we get

i= V + 12m

[ii eAi]2 (133)

provided that the Coulomb gauge iAi= 0 is chosen.

Gauge symmetry breaking

We will now suppose that the gauge symmetry is spontaneously broken,i.e. the uniform ground state wave function which minimizes the potentialenergy is allowed to amplitude and phase fluctuation (for convenience, thephase fluctuations are removed by a local gauge transformation) and the am-plitude fluctuations couple to the gauge field in such a way that the gaugefield becomes massive. The initial Lagrangian density

L = 12

i( ) 2

2m(Di)(Di) V() 14FijFij (134)

is gauge invariant. The potential energy has a minimum|0| (e.g. in the fol-lowing calculations V() =2+ ()2 has a minimum at|0| =

2/2) which can be chosen real positive 0. We now allow for local ampli-tude and phase fluctuations around this minimum,

(x) = (0+h(x))ei(x), (135)


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Variation ofL w.r.t. h(x) leads to the equation of motion known as theGinzburg-Landau equation in the context of superconductivity,

Lh(x)

=iL

(ih(x)), (144)

e2

mAi

2(x)(0+h(x)) + 2

2(0+h(x)) 4(0+h(x))2 = 2

m2ih(x).(145)

This equation contains information about the Cooper pair wave function0+ h(x) and it involves a characteristic length scale known as the coherence

length,

2 =2m2

2 . (146)

The coherence length is for example a measure of the length scale neededto attain the condensate wave function in the bulk of a superconductor froma free surface.

Variation ofLw.r.t. the gauge field leads toL

Aj(x)=i

L(iAj(x))

, (147)

e2m

Ai(x)(0+h(x))2 =iF

ij. (148)

The l.h.s. corresponds to the London current density Jj (proportional to thegauge field instead of the usual Ohm law Jj =Ej in a normal metal). Thisequation also involves a typical length scale inversely proportional to theCooper pair wave function,

2 =e2

m20 . (149)

This parameter gives an information about the length scale needed to expel

the gauge field from the bulk of a superconductor, a phenomenon knownas the Meissner effect. It is completely governed by the gauge field mass,2 =

2

2mm2A.

The phenomenology of type I and type II superconductors is essentiallydescribed in terms of the two length scales and . If , the magneticfield does not penetrate at all in the bulk of a superconductor (type I), whilein the other limit, there exist regions of normal phase with non zero magneticfield (Abrikosov vortices) inside superconducting regions (mixed phase of typeII superconductors).


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SU(2)W U(1)YThe electroweak symmetry breaking scenario 31 discovered by Salam and

Weinberg describes the emergence of the present structure of electromagne-tic and weak interactions as the broken gauge symmetry phase of a symme-tric (unbroken) phaseSU(2)W U(1)Ywhich existed in earlier times (higherenergy scales) of the Universe. With the spontaneous symmetry breaking sce-nario, some of the bosonic degrees of freedom (the gauge fields) acquire mass.In the symmetric phase, the relevant (non massive) fermionic particles (the

electron and the neutrino) consist in a right-handed

32

electronR

=eR in an(weak) isospin singletIW= 0 and an isospin doublet IW = 12

made of the left-

hande electron and the unique (left-handed) neutrino L=

eeL

. The bosons

are all non massive. The charges carried by the leptons follow from their weakisospin component I3Wand their hypercharge Y,

Q= I3W+Y

2. (150)

The hypercharge of the doublet is thus YL =1 and that of the singlet isYR= 2. Under the non-Abelian weak isospin gauge transformationS U(2)W,the fields change according to

R SU(2)W

R, (151)

L SU(2)W

exp12

ig

L, (152)

and under the Abelian U(1)Y symmetry, they become

R U(1)Y

exp(ig)R, (153)L

U(1)Yexp(ig/2)L. (154)

Note that the isospin coupling is g while the hypercharge coupling is conven-tionally called g/2.

31. See Ryder pp307-31232. In the Dirac Lagrangiani m, the right-handed and left-handed spinors

are defined as R = 12

(1 + 5) and L = 12

(1 5). Since 5 and commute, it followsthat i= iL

L+iRR.


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SU(2)WU(1)Y is made a local gauge symmetry through the introductionof gauge fields W and X with the covariant derivative

DL = L+ 12igWL 12igXL, (155)DR = R igXR, (156)

where W is a weak triplet gauge (non-massive) boson IW= 1 with hyper-charge zero and X is also a non-massive boson which has zero hypercharge,but is in an isospin singlet IW= 0.

If we forget about the pure gauge field contributions, the kinetic part ofthe Lagrangian 33 in the minimal coupling is given by Dirac Lagrangian (theleptonic particles are fermions with spin 1

2) i.e.

L =iR ( igX) R+iL

+ 12

igW 12 igX

L (157)

The weakness of the gauge invariant formulation is obviously that it contains4 massless gauge fields, while Nature (at the present energy scales) has onlyone, and that the fermions are similarly all non massive (if the electron wouldhave a non zero mass in this theory, the corresponding neutrino would sharethe same mass, since it appears as the second component of an isospin dou-blet). The spontaneous symmetry breaking scenario leads to 3 massive gaugefields and at the same time, the electron acquires mass as well (but not theneutrino !).

The Higgs mechanism

The symmetry is broken by introduction of a complex Higgs field

=

+

0

=

12

1+i23+i4

. (158)

This is an isospin doublet IW = 12 with hypercharge unity Y= 1,

D=

+ 12

igW+ 12

igX

, (159)

and a Lagrangian of the form

LHiggs= DD m2 ()2 + interaction with leptons. (160)33. We consider non massive fermions, otherwise a term likem2LLwould assign the same

mass to the electron and the neutrino.


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The potential V(||) = m2+()2 is chosen such that it gives riseto spontaneous symmetry breaking with||2 =m2/2 = v/2. For theclassical field, the choice 3 = v is made and a local gauge transformationeliminates the other is. Fluctuations around v are introduced through

(x) = 1

2

0

v+h(x)

. (161)

Acting with the covariant derivative gives

D= 12

1

2ig(W1 iW2)(v+h(x))

h 12i(gW3 gX)(v+h(x))

(162)

and reported in the Lagrangian density, this leads to (up to cubic terms)

LHiggs = 12

hh 1

2m2(v+h(x))2 1

4(v+h(x))4

+14

g2v2(W1W1 +W2W

2) + 14

(gW3 gX)(gW3 gX)v2

= 12

h

h 12

m2(v+h(x))2 14

(v+h(x))4

+M2WW+W

+ 12

M2ZZZ, (163)

where the charged massive vector bosons are

W = (W1 iW2)/

2 (164)

with masses M2W = 14

g2v2 and the neutral massive boson is such that 34

12

M2ZZZ = 1

8v2(gW3 gX)(gW3 gX)

= 18

v2(W3, X)

g2 gg gg g2

W3

X

= 12(Z

, A)M2Z 0

0 0Z3

A

. (165)

The last line is obtained by a diagonalization of the mass matrix by an ortho-gonal transformation

Z = cos WW3 sin WX (166)

A = sin WW3 + cos WX, (167)

34. See Cheng and Li p351


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and the masses of the neutral fields are

M2Z = 14

v2(g2 +g 2) (168)

M2A = 0. (169)

The coupling constant of the (charged) leptons and the electromagnetic gaugefield gets the value

e= g sin W. (170)

With the symmetry breaking scenario, the coupling between the Higgsfields and the leptons of the theory (Yukawa term which forms a Lorentzscalar by the coupling between a Dirac spinor with a scalar field) in

Ge(RL+LR) (171)

similarly leads to massive electrons 35

me=Gev/

2. (172)

35. See Quigg p110

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Teoria de Campo de Gauge