gauge fields theory
Embed Size (px)
TRANSCRIPT

Classical gauge field theoryBertrand BERCHE
Groupe de Physique Statistique
UHP Nancy 1
References
D.J. Gross, Gauge TheoryPast, Present, and Future ?, Chinese Journal ofPhysics 30 955 (1992),
C. Quigg, Gauge theory of the strong, weak, and electromagnetic interactions, Westwiew Press, 1997,
L.H. Ryder, Quantum field theory, Cambridge University Press, Cambridge1985,
S. Weinberg, The quantum theory of fields, Vol. I and II, Cambridge University 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 KleinGordon equation 1 (+
m2) = 0 which follows from the conservation equation ppm2 = 0 with the
1. We forget about factors ~ and c in this chapter.
1

2correspondence 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 overarrowdenotes a vector in ordinary space, or = (1
ct,~) (
t,~). Most of
the time, ~ and c are fixed to unity.The Lagrangian density from which this equation follows must satisfy 2
d4xL(, , , ) = L
L
()= 0, (1)
L
= m2, (2)
L
()=
. (3)
The first condition is fulfilled if
L = m2+ terms in , (4)and the second if 3
L = + terms in . (5)Eventually the KleinGordon 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 momenta, 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 obtain L
()= 0
L(0)
+ iL
(i)=
= (00+ i
i) =
(00 ii), the solution L = 00 ii = 00 + ii = (upto terms in ) follows.
4. We have a space and a time part in = 1
c2
t2 ~2, and only the time
derivatives are reminiscent of a kinetic energy.

3classical field theory, which means that although the fields correspond to thewave functions of quantum objects in the firstquantized form of the theory,they are treated by classical field theory (e.g. EulerLagrange equations) andno quantum fluctuations are allowed.
The idea behind nonAbelian gauge theory
According to Salam and Ward, cited by Novaes in hepph/0001283 :
Our basic postulate is that it should be possible to generate strong, weak,and electromagnetic interaction terms ... by making local gauge transformations on the kineticenergy 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 gaugeinvariance 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 spacetime to an other at a distance dx,the scale is changed from 1 to (1 + Sdx
) in such a way that a spacetimedependent function (of dimension of a length) f(x) is changed according to
f(x) f(x+ dx) = (f +fdx))(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 of gauge invariance as aninvariance under a change of length scaled, or a change of the gauge.
5. See Cheng and Li pp235236, and Gross p956

4 Abelian U(1) gauge theory
The complex nature of the field as an internal structure
The prototype of gauge theory is the theory of electromagnetism, or Abelian 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 KleinGordon field) canbe modified by a global phase transformation (x) exp(iq)(x) (Abelian U(1) gauge transformation) which leaves the matter Lagrangian L =
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 twocomponent field (x) =
(1
2
). The gauge
transformation now appears as an Abelian rotation in a twodimensional (internal) 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/~, or i i~.

5in the expression of L0 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 KleinGordonLagrangian, the conserved current takes the form 7
j = iq( ). (13)
Local gauge symmetry
Extending the gauge symmetry to local transformations requires the introduction 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)) 6=G(x), and the derivative of the field does not transform like the field itselfdoes. Let us define the covariant derivative 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 currentdensity after being multiplied by q.
8. The term covariant refers to covariance with respect to the introduction of the localtransformation, and not to covariantcontravariant indices.

6Since 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 transformation 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 preserve local gauge invariance. In a sense, the interaction is created by theprinciple of local gauge symmetry, while the principle of global gauge symmetry 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 different approaches. In a first approach, we successively add terms to L0 inorder to get at the end a locally gauge invariant Lagrangian. Since the prescription of local gauge invariance induces Maxwell interactions, this shouldautomatically incorporate interaction terms in L. Starting with the observation that L0 is no longer gauge invariant through local transformations, sinceL0 = (x)j ((x))j now contains the second term, we have tokill this last term by the introduction of a L1 = jA term 9. This againgenerates one more contribution in (L0 + L1) which is canceled if we addL2 = q2AA. The combination L0 + L1 + L2 = L0 + Lint is now locallygauge invariant.
In a shorter approach, called minimal coupling, we simply replace the kinetic term
in L0 by (D)(D) which was especially constructed inorder to be gauge invariant (under local gauge transformations). One also hasto add the pure field contribution LA = 14FF to get the full Lagrangian
9. Note that here j is the current which was conserved in the absence of gauge interaction. The sign here is also coherent with the expression (~j ~A).

7density 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 KleinGordoncase) :
(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 with D instead of , the conservedNoether current in the presence of gauge interaction reads as 11
J = iqL0
(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 in A, as Lint = JA.
The equations of motion follow from EulerLagrange equations,
StotA
=LtotA
Ltot(A)
= 0, (21)
which simply yield
LintA
= LA
(A). (22)
10. Remember that we call L0 = F (, , , ).11. See e.g. V. Rubakov, Classical Theory of Gauge fields, Princeton University Press
2002, pp2127.12. 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.

8The 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 + F
F
(A)
). Both terms in
parenthesis are equal to F F = 2F and eventually we obtain aftermultiplication by 1
4, 13
F = F = J (23)
and, since F is antisymmetric, we recover the conservation equation for J,
J = 0. (24)
NonAbelian (YangMills) SU(2) gauge theory
Internal structure
The global phase transformation (x) exp(iq)(x) as mentioned aboveappears as an Abelian rotation in a twodimensional (internal) space. Thisgauge transformation, when extended to local phase transformations (x), generates the electromagnetic interaction. It is possible to generalize tononAbelian gauge transformations by extending the internal (isospin) structure. The field is for example a 3component 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 nonAbelian)of the rotations in three dimensions
1 =
0 0 00 0 i0 i 0
, 2 =
0 0 i0 0 0i 0 0
, 3 =
0 i 0i 0 00 0 0
. (27)
13. Note again that our choice of sign for the current density makes the expression coherent with the usual Maxwell equations.14. See Weinberg II p3.

9Use 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) transformations acting on complex twocomponent 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 (12ig
)(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 = 12 (
2 2), 2 = 12i(2 + 2), 3 = , see Ryder pp3238.
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 nonrelativistic spin 12 electron in an electromagnetic field reads as
H
((x)(x)
)=
(1
2m(~p q ~A)2 q
)1I
((x)(x)
) q~2m
B((x)(x)
)= E
((x)(x)
). (31)
Note that here the hat H notation stands for a 2 by 2 matrix.

10
with ijk the totally antisymmetric tensor, and is a 2 2 matrix
=
(3 1 i2
1 + i2 3)
(34)
Here 12i are the generators of SU(2) transformations.
The two representations can be written in a unified way in componentform 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 components (resp. 3) in the SU(2) representation (resp SO(3)) and t stands for thegenerator ( (resp. )).
Together with the internal degrees of freedom, the fields of course dependon spacetime position x.
Pure matter field and Noether current
From now on, we choose the SU(2) representation. Let L0 be a gaugeinvariant 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) = 12ig(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.

11
and using the equations of motion
L0
L0()
= 0,L0
L0()
= 0 (42)
we obtain the variation of the Lagrangian density
L0 =(1
2ig
)
L0()
+(1
2ig
) L0()
+ ( )= j (43)
where the Noether current
j = (12ig
) L0()
L0()
(12ig
)(44)
is conserved, j = 0, since the variation of the Lagrangian density vanishes
for a symmetry.
The current density is a vector in internal space (isospin current density) 19.With the Lagrangian density given in eqn. (??), the conserved current isexplicitly given by
j = 12ig 1
2ig (45)
= 12ig(,
)
(0 11 0
)(
)i () ()
+12ig(,
)
(0 ii 0
)(
)j () ()
+12ig(,
)
(1 00 1
)(
)k () ()
= 12ig(
+
)i () ()+1
2ig(i + i)j () ()
+12ig(
+
)k () () (46)
where i, j et k are the unit vectors in isospin space.
19. See Quigg p34

12
Introduction of a gauge potential 20
Now we extend the formalism to local gauge transformations
(x) = exp(12ig(x)
)(x) G(x)(x), (47)
(where G(x) = exp(12ig(x)
)is a 2 2 matrix) or locally to
(x) = 12ig(x)(x), (48)
but a problem occurs which will make the Lagrangian density L0 not gaugeinvariant. The fact that does not obey the same gauge transformationthan itself 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 =
12aBa (summation over a understood) is another 2 by 2 matrix (in
fact there is one such matrix for each of the 4 spacetime components, B is agauge potential (with three internal components which all are 4spacetimevectors)). We demand the following transformation
D(x) D(x) = G(x)(D(x)). (50)
We obtain D = ( + igB) = (G) + G() + igBG. From therequirement (??), this quantity should be equal to G(+igB) = G()+
igG(B). It follows an equation for the transformation of B, igBG =
igG(B) (G). Written in terms of operators, this equation is BG =GB +
igG. We multiply both sides by G
1 on the right to get
B = GBG1 +
i
g(G)G
1 = G
(B +
i
gG1(G)
)G1. (51)
In the case of electromagnetism, the local gauge transformation is performed by the operator (now an ordinary function) GEM(x) = exp(iq(x)) with
20. See Quigg pp555721. In analogy with electromagnetism where the covariant derivative i~D is given by
p qA with p = i~, which yields D = + i q~A.

13
(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 of B =12B, we can deduce the
gauge transformation of B as well. Consider an infinitesimal gauge transformation
G(x) = 1I+ 12ig(x). (53)
Eqn. (??) reads as (to linear order in i)
12B =
12B +
14ig(() (B) (B) ()) 12(). (54)
The term in the middle, written in components, has the form
12ijBk(
jkkj) = 12ijBk[
j , k] = jkl(jBk)l = (B) (55)
and it follows that
12B =
12B 12g(B) 12(). (56)
Another common expression uses the identity (B) = 2i[12, 1
2B
]such that
12B =
12B + ig
[12, 1
2B
] 12(). (57)
We can also write directly
B = B gB . (58)
The gauge transformation of B appears as a gradient term (like in electromagnetism) plus a rotation in internal space.
The fieldstrength tensor and field equations 22
Let us introduce a fieldstrength tensor by the 2 by 2 matrix
F =12F (59)
22. See Quigg pp5859

14
from which we construct the gaugeinvariant kinetic energy
Lfield = 14F F = 12 Tr(FF ), (60)where we used Tr(ab) = 2ab 23. The fieldstrength tensor is an observablequantity. It is thus supposed to be a gauge scalar, that is to be independentof the choice of gauge 24 :
F = GFG1. (61)
A simple transcription of the QED Faraday tensor F = A A is notsatisfactory, but if we note that the QED Faraday tensor can also be writtenas
F =1
iq[D,D] = A A + iq[A, A ] (62)
(the commutator vanishes in the Abelian case), we can define
F =1
ig[D,D ] = B B + ig[B, B ]. (63)
It is easy to check that this definition has the correct gauge invariance property.
In terms of isovectors, the same relation becomes
12F =
12B 12B + ig
[12B,
12B
]F = B B gB B , (64)
where we used[12B,
12B
]= 1
2i(BB) . The nonAbelian character
of the theory is obvious in the definition of the fieldstrength tensor. From thefield Lagrangian (??) and the EulerLagrange equations
LfieldBa
[Lfield(Ba)
]= 0, (65)
one can deduce the equations of motion (equivalent to the Maxwell equationsin the absence of matter charge current) :
F gB F = 0. (66)23. This kinetic energy has the same form as in QED, LU(1) field = 14FF .24. In the same sense than a Lorentz scalar (i.e. a contraction) does not depend on the
reference frame.

15
Note that the equations of motion are not as simple as the familiar Maxwellequations which are linear. The YangMills 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 (Procalike) equations of motionbecome
F gB F = m2B . (68)
Construction of a gaugeinvariant interaction 25
Since (x) depends on spacetime, the variations () (or ())
contain an extra term which contributes to L0
L0 =(1
2ig(x)
)
[L0
()
]
+(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 L0 is not gauge invariant under local gauge transformations. In order to compensate this new term, we must add another contribution to the Lagrangian,
L1 = jB (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 presented later. The approach used here follows the presentation of Ryder pp9698 in the case ofAbelian U(1) gauge symmetry.

16
where only the first term vanishes identically. Performing the variation of j
yields
j = 12ig 1
2ig() + ( )
= 12ig(1
2ig
)
12ig
(12ig + 1
2ig
)+ ( )
= 14g2 [,] + 1
4g2 [,] + 1
4g2 {,}
= 12ig2 1
2ig2 + 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
+12g2B
= +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
= 12g2B(g(B) + )
= 12g2B
(78)
and we obtain the expected vanishing variation
L0 + L1 + L2 = 0 (79)

17
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 freeparticle contribution of eqn (??).
Covariant derivative, minimal coupling
The introduction of the gauge covariant derivative facilitates the calculations. 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 on which 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 field D 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 Lagrangiandensity of eqn. (??) 26 :
L = (D)(D) V () (81)=
( 12 igB
)( + 1
2igB
) V ()
= 1
2igB
+ 12igB
+14g2(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.

18
with = 0 the adjoint spinor and the Dirac matrices 27 instead of thekinetic energy
, and the gauge invariant Lagrangian becomes
L = (iD m) 14F F . (84)
Conserved current in the presence of gauge fields
In the presence of gauge fields, the conserved Noether current can be written in terms of the covariant derivative,
J = (12ig
) L0((D))
L0(D)
(12ig
)(85)
In the case of the Lagrangian (??), it becomes
J = 12igD (D) 1
2ig
= 12ig
( + 1
2igB
) ( 12igB
)12ig
= j 14g2 [,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 + 12ig2 B. (87)
In component form we have
Ja = ja 12ig2abc
bBc. (88)
The current density J is conserved in the ordinary sense 28,
J = 0 (89)
while j satisfies a gaugecovariant conservation law
Dj = 0. (90)27. See e.g. Ryder pp434628. See Weinberg II pp1213

19
Now, the EulerLagrange equations of the matter field in the presence ofthe gauge field contain a new term,
LintBa
+LfieldBa
[Lfield(Ba)
]= 0, (91)
and lead to the equations of motion in the presence of charged matter :
F gB F = J . (92)
Spontaneous gauge symmetry breaking
Spontaneous breaking of global symmetries
Discrete symmetries
Let us first consider spontaneous breaking of a global discrete symmetry.It is illustrated by the case of the real scalar field,
L = 12
V ((x)), (93)
where V ((x)) is a potential which depends on the field configuration. Wewill consider two cases
V () =
44
2
22, , 2 > 0. (94)
Case 1 with the sign + corresponds to a scalar field theory with square mass2. Let us first build the Hamiltonian,
H = 12(0)
2 + 12()2 + V (). (95)
The field with the lowest energy (also called the ground state configuration,which would be denoted 0 =  in the quantized version of the theory)is a constant field which minimizes the potential. In case 1, it corresponds to avanishing field 0 = 0. The discrete Z2 symmetry of the Lagrangianis also a symmetry of the ground state. In case 2, with sign in the potential,the constant ground state field is given by
0 = = v. (96)

20
While the Lagrangian still possesses the symmetry, in any of thetwo degenerate ground states, this Z2 symmetry is broken. This situationoccurs in the low temperature phase of second order phase transitions, whenan ordered ground state emerges (for example a ferromagnetic ground state),which does not respect the full symmetry of the Hamiltonian (e.g. the updown symmetry in a Ising model, or rotational symmetry (this is a continuoussymmetry in this case) in the case of an Heisenberg model). It is instructiveto study the field fluctuations around the ground state. For this purpose, welet
= v + h, h v, (97)
(v is chosen positive without loss of generality) and we remind that V () =32, V () = 322, V () = 6, and V () = 6. The potentialis now
V () = V (v) + hV (v) + 12h2V (v) + 1
6h3V (v) + 1
24h4V (v)
= 14
4
+ 0 + 1
222h2 + 2
h3 + 1
4h4. (98)
We note that the new field v acquired a mass2 (the coefficient of the
quadratic term in h is now positive).
Continuous symmetries
Spontaneous breaking of a global continuous symmetry can be encounteredin the SO(2) model (rotations in the plane). We consider now a theory withtwo real scalar fields 1(x) and 2(x), and with the potential
V (1, 2) =14(21 +
22 v2)2 = 14(2 v2)2. (99)
The fields 1 and 2 are massless (the coefficients of the quadratic terms arenegative) and the theory is invariant under rotations in the plane,
(12
)=
(cos sin sin cos
)(12
). (100)
The minima of the potential lie on the circle
02 = 210 + 220 = v2. (101)

21
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 + h
22 + 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) symmetry. The massless field is called a Goldstone mode (or NambuGoldstonemode). 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 polar representation of the model, while 2 fluctuations correspond to phasefluctuations which do not cost any energy.
Spontaneous breaking of local symmetries
A new phenomenon occurs with local gauge theories, where the selectionof 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 of U(1) gauge theory,
L = 14FF
+ (D)(D) V (), (104)

22
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)/v12(v + h(x)). (108)
The local gauge transformation defined by q(x) = (x)/v eliminates (x),since
(x) = ei(x)/v(x) =12(v + h(x)), (109)
A(x) = A(x) +1
qv(x). (110)
The net effect in the Lagrangian density is the following,
L = 14F F
+(D)(D)+ 122(v+h2(x))2 14(v+h(x))4, (111)with D = + 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 redefinition of the gauge field.
This mechanism is known in condensed matter physics as the Andersonmechanism (see next section), and it occurs in superconductivity, where thenonzero 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.

23
The Anderson mechanism 29
We will first reproduce all the generic arguments given before in the relativistic U(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 mechanics, 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
willthus simply be denoted as i
i = ii, and 2i = ii stands for~2. Due to this noncovariant 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 EulerLagrange equation (variation with respect to ) indeed leads theSchrodinger equation,
L0
= 12i~ 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.

24
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 = 12i~( ) ~
2
2m(Di)(Di) V 14FijFij . (123)
The gauge field Ai is changed by a local gauge transformation,
(x) = (x)ei~e(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.

25
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
= 12i~ 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 +1
2m[i~i 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 amplitude fluctuations couple to the gauge field in such a way that the gaugefield becomes massive. The initial Lagrangian density
L = 12i~( ) ~
2
2m(Di)(Di) V () 14FijFij (134)
is gauge invariant. The potential energy has a minimum 0 (e.g. in the following 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)

26
with h(x) and (x) two real functions (and h(x) small w.r.t. 0).A convenient gauge transformation
(x) = (x)ei~e(x),
e
~(x) = (x) (136)
makes the analysis simpler, since it eliminates the phase fluctuations,
(x) = 0 + h(x) (137)
and also changes the gauge field
Ai(x) = Ai(x)~
ei(x). (138)
The Lagrangian density is left unchanged by the gauge transformation, butwritten in terms of the gauged variables it reads
L = 12i~(
) ~
2
2m(Di)(Di) V () 14F ijF ij. (139)
The first term is identically zero, the second term once expanded leads to
~2
2m
(ihih+
e2
~2AiA
i(0 + h)
2
), (140)
the third term to
2(0 + h)2 + (0 + h)4 (141)
(or any other form depending on the potential), and the last gauge invariantterm
14F ijF
ij (142)
has the usual form for the field kinetic energy, but here given in terms ofthe gauged vector potential. The essential novelty stands in the second termwhere an extra dependence of the gauge field occurs, 0A2i , coupled to theground state expectation value. This is a mass term which is also denoted as
12m2A =
e2
~220. (143)

27
Variation of L w.r.t. h(x) leads to the equation of motion known as theGinzburgLandau 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
m2i h(x).(145)
This equation contains information about the Cooper pair wave function0+ h(x) and it involves a characteristic length scale known as the coherencelength,
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 of L w.r.t. the gauge field leads toL
Aj(x)= i
L(iAj(x))
, (147)
e2
mAi(x)(0 + h(x))
2 = iFij . (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 expelthe 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).

28
SU(2)W U(1)YThe electroweak symmetry breaking scenario 31 discovered by Salam and
Weinberg describes the emergence of the present structure of electromagnetic and weak interactions as the broken gauge symmetry phase of a symmetric (unbroken) phase SU(2)W U(1)Y which existed in earlier times (higherenergy scales) of the Universe. With the spontaneous symmetry breaking scenario, some of the bosonic degrees of freedom (the gauge fields) acquire mass.In the symmetric phase, the relevant (non massive) fermionic particles (theelectron and the neutrino) consist in a righthanded 32 electron R = eR in an(weak) isospin singlet IW = 0 and an isospin doublet IW =
12made of the left
hande electron and the unique (lefthanded) neutrino L =
(eeL
). The bosons
are all non massive. The charges carried by the leptons follow from their weakisospin component I3W and 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 nonAbelian weak isospin gauge transformation SU(2)W ,the fields change according to
R SU(2)W
R, (151)
L SU(2)W
exp(12ig
)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 conventionally called g/2.
31. See Ryder pp30731232. In the Dirac Lagrangian i m, the righthanded and lefthanded spinors
are defined as R = 12 (1 + 5) and L =12 (1 5). Since 5 and commute, it follows
that i = iLL+ iR
R.

29
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+ 12 igW L 12igXL, (155)DR = R igXR, (156)
where W is a weak triplet gauge (nonmassive) boson IW = 1 with hypercharge zero and X is also a nonmassive 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( +
12igW 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 doublet). 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 =12with hypercharge unity Y = 1,
D =( +
12igW +
12igX
), (159)
and a Lagrangian of the form
LHiggs = DDm2 ()2 + interaction with leptons. (160)33. We consider non massive fermions, otherwise a term like m2LL would assign the same
mass to the electron and the neutrino.

30
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) =12
(0
v + h(x)
). (161)
Acting with the covariant derivative gives
D = 12
(12ig(W 1 iW 2)(v + h(x))
h 12i(gW 3 gX)(v + h(x)))
(162)
and reported in the Lagrangian density, this leads to (up to cubic terms)
LHiggs = 12[h
h 12m2(v + h(x))2 1
4(v + h(x))4
+14g2v2(W 1W
1 +W 2W2) + 1
4(gW 3 gX)(gW 3 gX)v2
]= 1
2
[h
h 12m2(v + h(x))2 1
4(v + h(x))4
]+M2WW
+ W
+ 12M2ZZZ
, (163)
where the charged massive vector bosons are
W = (W1 iW 2)/
2 (164)
with masses M2W =14g2v2 and the neutral massive boson is such that 34
12M2ZZZ
= 18v2(gW 3 gX)(gW 3 gX)
= 18v2(W 3
, X)
(g2 gggg g2
)(W 3
X
)
= 12(Z
, A)
(M2Z 00 0
)(Z3
A
). (165)
The last line is obtained by a diagonalization of the mass matrix by an orthogonal transformation
Z = cos WW3 sin WX (166)
A = sin WW3 + cos WX, (167)
34. See Cheng and Li p351

31
and the masses of the neutral fields are
M2Z =14v2(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