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Classical gauge field theory Bertrand BERCHE Groupe de Physique Statistique UHP Nancy 1 References – D.J. Gross, Gauge Theory-Past, Present, and Future?, Chinese Journal of Physics 30 955 (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 Field Theory, (ABP) 1995. Introduction Classical field theories Consider as an example the free particle Klein-Gordon equation 1 (μ μ + m 2 )φ = 0 which follows from the conservation equation p μ p μ m 2 = 0 with the 1. We forget about factors and c in this chapter. 1

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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 30 955 (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, Cambridge1985,

    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 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 over-arrowdenotes 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 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 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 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 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 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 +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 pp235-236, 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 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 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 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/~, 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 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)) 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 covariant-contravariant 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 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 in L. Starting with the observa-tion 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 kine-tic 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 inter-action. 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 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 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 Euler-Lagrange 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, 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.

  • 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)

    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 tonon-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 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 co-herent 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) 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 (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 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- 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 compo-nents (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 space-time 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 space-time components, B is agauge 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 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 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 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 transfor-mation

    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 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

  • 14

    from which we construct the gauge-invariant kinetic energy

    Lfield = 14F F = 12 Tr(FF ), (60)where we used Tr(ab) = 2ab 23. The field-strength 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 pro-perty.

    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 non-Abelian character

    of the theory is obvious in the definition of the field-strength tensor. From thefield Lagrangian (??) and the Euler-Lagrange 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 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

    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 transfor-mations. In order to compensate this new term, we must add another contri-bution 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 presen-ted later. The approach used here follows the presentation of Ryder pp96-98 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 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 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 writ-ten 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 gauge-covariant conservation law

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

  • 19

    Now, the Euler-Lagrange 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 up-down 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

    |0|2 = 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) 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 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 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.

  • 23

    The Anderson mechanism 29

    We will first reproduce all the generic arguments given before in the rela-tivistic 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 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

    willthus simply be denoted as i

    i = ii, and 2i = ii stands for~2. 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 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 am-plitude 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 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)

  • 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 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

    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 electromagne-tic and weak interactions as the broken gauge symmetry phase of a symme-tric (unbroken) phase SU(2)W U(1)Y which 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 (theelectron and the neutrino) consist in a right-handed 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 (left-handed) 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 non-Abelian 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 conven-tionally called g/2.

    31. See Ryder pp307-31232. In the Dirac Lagrangian i 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 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 (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( +

    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 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 =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 ortho-gonal 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