G„¦ = bc(p2 + m2 ) = „¦ † G† , (358)

158 J.W. van Holten

and therefore

1 1 —

dd x ψ0 (p2 + m2 )ψ0 ,

(Ψ, G „¦Ψ ) = (359)

2 2

which is the standard action for a free scalar ¬eld4 .

The laplacean for the BRST operators (346) and (349) is

∆ = „¦ „¦ † + „¦ † „¦ = (p2 + m2 )2 , (360)

which is manifestly non-negative, but might give rise to propagators with double

poles, or negative residues, indicating the appearance of ghost states. However,

in the expression (357) for the propagator, one of the poles is canceled by the

zero of the BRST operator; in the present context the equation reads

1

c(p2 + m2 ) J0 .

cψ0 = (361)

2 + m2 )2

(p

This leads to the desired result

1

ψ0 = J0 , (362)

p 2 + m2

and we recover the standard scalar ¬eld theory indeed. It is not very di¬cult

to extend the theory to particles in external gravitational or electromagnetic

¬elds5 , or to spinning particles [38].

However, a di¬erent and more di¬cult problem is the inclusion of self interac-

tions. This question has been addressed mostly in the case of string theory [32].

As it is expected to depend on spin, no unique prescription has been constructed

for the point particle so far.

4.2 Anomalies and BRST Cohomology

In the preceding sections we have seen how local gauge symmetries are encoded

in the BRST-transformations. First, the BRST-transformations of the classical

variables correspond to ghost-dependent gauge transformations. Second, the clo-

sure of the algebra of the gauge transformations (and the Poisson brackets or

commutators of the constraints), as well as the corresponding Jacobi-identities,

are part of the condition that the BRST transformations are nilpotent.

It is important to stress, as we observed earlier, that the closure of the classi-

cal gauge algebra does not necessarily guarantee the closure of the gauge algebra

in the quantum theory, because it may be spoiled by anomalies. Equivalently,

in the presence of anomalies there is no nilpotent quantum BRST operator, and

no local action satisfying the master equation (344). A particular case in point

is that of a Yang“Mills ¬eld coupled to chiral fermions, as in the electro-weak

standard model. In the following we consider chiral gauge theories in some detail.

4

Of course, there is no loss of generality here if we restrict the coe¬cients ψa to be

real.

5

See the discussion in [34], which uses however a less elegant implementation of the

action.

Aspects of BRST Quantization 159

The action of chiral fermions coupled to an abelian or non-abelian gauge ¬eld

reads

¯ /ψ

SF [A] = d4 x ψL D L . (363)

Here Dµ ψL = ‚µ ψL ’ gAa Ta ψL with Ta being the generators of the gauge group

µ

in the representation according to which the spinors ψL transform. In the path-

integral formulation of quantum ¬eld theory the fermions make the following

contribution to the e¬ective action for the gauge ¬elds:

¯

eiW [A] = DψL DψL eiSF [A] . (364)

An in¬nitesimal local gauge transformation with parameter Λa changes the ef-

fective action W [A] by

δW [A] δ δ

’ gfab Ab

d4 x (Dµ Λ)a d4 x Λa ‚µ c

δ(Λ)W [A] = = W [A],

µ

δAa δAa δAc

µ µ µ

(365)

assuming boundary terms to vanish. By construction, the fermion action SF [A]

itself is gauge invariant, but this is generally not true for the fermionic functional

integration measure. If the measure is not invariant:

d4 x Λa “a [A] = 0,

δ(Λ)W [A] =

(366)

δ δ

“a [A] = Da W [A] ≡ ’ gfab Ab

c

‚µ W [A].

µ

a δAc

δAµ µ

Even though the action W [A] may not be invariant, its variation should still be

covariant and satisfy the condition

Da “b [A] ’ Db “a [A] = [Da , Db ] W [A] = gfab Dc W [A] = gfab “c [A].

c c

(367)

This consistency condition was ¬rst derived by Wess and Zumino [41], and its

solutions determine the functional form of the anomalous variation “a [A] of the

e¬ective action W [A]. It can be derived from the BRST cohomology of the gauge

theory [39,44,40].

To make the connection, observe that the Wess“Zumino consistency condi-

tion (367) can be rewritten after contraction with ghosts as follows:

d4 x ca cb (Da “b [A] ’ Db “a [A] ’ gfab “c [A])

c

0=

(368)

gc

Da “b ’ fab “c = ’2 δ„¦

d4 x ca cb d4 xca “a ,

=2

2

provided we can ignore boundary terms. The integrand is a 4-form of ghost

number +1:

1

µµνκ» dxµ § dxν § dxκ § dx» ca “a [A].

I4 = d4 x ca “a [A] =

1

(369)

4!

160 J.W. van Holten

The Wess“Zumino consistency condition (368) then implies that non-trivial so-

lutions of this condition must be of the form

1 2

δ„¦ I4 = dI3 , (370)

2

where I3 is a 3-form of ghost number +2, vanishing on any boundary of the

space-time M.

Now we make a very interesting and useful observation: the BRST construc-

tion can be mapped to a standard cohomology problem on a principle ¬bre bun-

dle with local structure M — G, where M is the space-time and G is the gauge

group viewed as a manifold [42]. First note that the gauge ¬eld is a function of

both the co-ordinates xµ on the space-time manifold M and the parameters Λa

on the group manifold G. We denote the combined set of these co-ordinates by

ξ = (x, Λ). To make the dependence on space-time and gauge group explicit, we

introduce the Lie-algebra valued 1-form

A(x) = dxµ Aa (x)Ta , (371)

µ

with a generator Ta of the gauge group, and the gauge ¬eld Aa (x) at the point x

µ

in the space-time manifold M. Starting from A, all gauge-equivalent con¬gura-

tions are obtained by local gauge transformations, generated by group elements

a(ξ) according to

1

A(ξ) = ’ a’1 (ξ) da(ξ) + a’1 (ξ) A(x) a(ξ), A(x) = A(x, 0) (372)

g

where d is the ordinary di¬erential operator on the space-time manifold M:

‚a

da(x, Λ) = dxµ (x, Λ). (373)

‚xµ

Furthermore, the parametrization of the group is chosen such that a(x, 0) = 1,

the identity element. Then, if a(ξ) is close to the identity:

a(ξ) = e gΛ(x)·T ≈ 1 + g Λa (x)Ta + O(g 2 Λ2 ), (374)

and (372) represents the in¬nitesimally transformed gauge ¬eld 1-form (124). In

the following we interpret A(ξ) as a particular 1-form living on the ¬bre bundle

with local structure M — G.

A general one-form N on the bundle can be decomposed as

N(ξ) = dξ i Ni = dxµ Nµ + dΛa Na . (375)

Correspondingly, we introduce the di¬erential operators

‚ ‚

d = dxµ s = dΛa

, , d = d + s, (376)

‚xµ ‚Λa

with the properties

d2 = 0, s2 = 0, d2 = ds + sd = 0. (377)

Aspects of BRST Quantization 161

Next de¬ne the left-invariant 1-forms on the group C(ξ) by

C = a’1 sa, c(x) = C(x, 0). (378)

By construction, using sa’1 = ’a’1 sa a’1 , these forms satisfy

sC = ’C 2 . (379)

The action of the group di¬erential s on the one-form A is

1 1

DC = (dC ’ g[A, C]+ ) .

sA = (380)

g g

Finally, the ¬eld strength F(ξ) for the gauge ¬eld A is de¬ned as the 2-form

F = dA ’ gA2 = a’1 F a, F (x) = F(x, 0). (381)

The action of s on F is given by

sF = [F, C]. (382)

Clearly, the above equations are in one-to-one correspondence with the BRST

transformations of the Yang“Mills ¬elds, described by the Lie-algebra valued

one-form A = dxµ Aa Ta , and the ghosts described by the Lie-algebra valued

µ

Grassmann variable c = ca Ta , upon the identi¬cation ’gs|Λ=0 ’ δ„¦ :

’gsA|Λ=0 ’ δ„¦ A = ’dxµ (Dµ c)a Ta = ’Dc,

g cab g

’gsC|Λ=0 ’ δ„¦ c = fab c c Tc = ca cb [Ta , Tb ] = gc2 . (383)

2 2

g

’gsF|Λ=0 ’ δ„¦ F = ’ dxµ § dxν fab Fµν cb Tc = ’g[F, c],

ca

2

provided we take the BRST variational derivative δ„¦ and the ghosts c to anti-

commute with the di¬erential operator d:

dc + cd = dxµ (‚µ c).

dδ„¦ + δ„¦ d = 0, (384)

Returning to the Wess“Zumino consistency condition (370), we now see that it

can be restated as a cohomology problem on the principle ¬bre bundle on which

the 1-form A lives. This is achieved by mapping the 4-form of ghost number +1

to a particular 5-form on the bundle, which is a local 4-form on M and a 1-form

on G; similarly one maps the 3-form of ghost number +2 to another 5-form

which is a local 3-form on M and a 2-form on G:

I4 ’ ω4 , I3 ’ ω3 ,

1 1 2 2

(385)

where the two 5-forms must be related by

’gsω4 = dω3 .

1 2

(386)

162 J.W. van Holten

We now show how to solve this equation as part of a whole chain of equations

known as the descent equations. The starting point is a set of invariant polyno-

mials known as the Chern characters of order n. They are constructed in terms

of the ¬eld-strength 2-form:

1µ

F = dA ’ gA2 = dx § dxν Fµν Ta ,

a

(387)

2

which satis¬es the Bianchi identity

DF = dF ’ g [A, F ] = 0. (388)

The two-form F transforms covariantly under gauge transformations (372):

F ’ a’1 F a = F. (389)

It follows that the Chern character of order n, de¬ned by

Chn [A] = Tr F n = Tr F n , (390)

is an invariant 2n-form: Chn [A] = Chn [A]. It is also closed, as a result of the

Bianchi identity:

d Chn [A] = nTr [(DF )F n’1 ] = 0. (391)

The solution of this equation is given by the exact (2n ’ 1)-forms:

0

Chn [A] = dω2n’1 [A]. (392)

Note that the exact 2n-form on the right-hand side lies entirely in the local

space-time part M of the bundle because this is manifestly true for the left-

hand side.