‚H

qi =

™ ,

‚pi

for the momenta in terms of the velocities, and performing the inverse Legendre

transformation. We have already seen how this works for the examples of the

relativistic particle and the Maxwell“Yang“Mills theory. As the lagrangean is a

scalar function under space-time transformations, it is better suited for the de-

velopment of a manifestly covariant formulation of gauge-¬xed BRST-extended

dynamics of theories with local symmetries, including Maxwell“Yang“Mills the-

ory and the relativistic particle as well as string theory and general relativity.

The procedure follows quite naturally the steps outlined in the previous

Sects. 3.1 and 3.2:

a. Start from a gauge-invariant lagrangean L0 (q, q). ™

b. For each gauge degree of freedom (each gauge parameter), introduce a ghost

variable ca ; by de¬nition these ghost variables carry ghost number Ng [ca ] = +1.

Construct BRST transformations δ„¦ X for the extended con¬guration-space vari-

ables X = (q i , ca ), satisfying the requirement that they leave L0 invariant (pos-

2

sibly modulo a total derivative), and are nilpotent: δ„¦ X = 0.

c. Add a trivially BRST-invariant set of terms to the action, of the form δ„¦ Ψ

for some anti-commuting function Ψ (the gauge fermion).

The last step is to result in an e¬ective lagrangean Le¬ with net ghost num-

ber Ng [Le¬ ] = 0. To achieve this, the gauge fermion must have ghost number

Ng [Ψ ] = ’1. However, so far we only have introduced dynamical variables with

non-negative ghost number: Ng [q i , ca ] = (0, +1). To solve this problem we in-

troduce anti-commuting anti-ghosts ba , with ghost number Ng [ba ] = ’1. The

BRST-transforms of these variables must then be commuting objects ±a , with

ghost number Ng [±] = 0. In order for the BRST-transformations to be nilpotent,

we require

δ„¦ ba = i±a , δ„¦ ±a = 0, (300)

2

which indeed satisfy δ„¦ = 0 trivially. The examples of the previous section

illustrate this procedure.

The Relativistic Particle. The starting point for the description of the rela-

tivistic particle was the reparametrization-invariant action (8). We identify the

integrand as the lagrangean L0 . Next we introduce the Grassmann-odd ghost

variable c(»), and de¬ne the BRST transformations

dxµ d(ce) dc

δ„¦ xµ = c , δ„¦ e = , δ„¦ c = c . (301)

d» d» d»

As c2 = 0, these transformations are nilpotent indeed. In addition, introduce

the anti-ghost representation (b, ±) with the transformation rules (300). We can

now construct a gauge fermion. We make the choice

d(ce)

Ψ (b, e) = b(e ’ 1) ’ δ„¦ Ψ = i±(e ’ 1) ’ b . (302)

d»

150 J.W. van Holten

As a result, the e¬ective lagrangean (in natural units) becomes

m dxµ dxµ em d(ce)

Le¬ = L0 ’ iδ„¦ Ψ = ’ + ±(e ’ 1) + ib . (303)

2e d» d» 2 d»

Observing that the variable ± plays the role of a Lagrange multiplier, ¬xing the

einbein to its canonical value e = 1 such that d» = d„ , this lagrangean is seen

to reproduce the action (296):

m2

(x ’ 1) + ib c .

Se¬ = d„ Le¬ d„ ™ ™

2

Maxwell“Yang“Mills Theory. The covariant classical action of the Maxwell“

Yang“Mills theory was presented in (121):

1 2

S0 = ’ d4 x Fµν

a

.

4

Introducing the ghost ¬elds ca , we can de¬ne nilpotent BRST transformations

g

a

δ„¦ Aa = (Dµ c) , δ„¦ ca = fbc a cb cc . (304)

µ

2

Next we add the anti-ghost BRST multiplets (ba , ±a ), with the transformation

rules (300). Choose the gauge fermion

Ψ (Aa , ba ) = ba (Aa ’ »a ) ’ δ„¦ Ψ = i±a (Aa ’ »a ) ’ ba (D0 c)a , (305)

0 0 0

where »a are some constants (possibly zero). Adding this to the classical action

gives

1a

Se¬ = d4 x ’ (Fµν )2 + ±a (Aa ’ »a ) + iba (D0 c)a . (306)

0

4

Again, the ¬elds ±a act as Lagrange multipliers, ¬xing the electric potentials to

the constant values »a . After substitution of these values, the action reduces to

the form (299).

We have thus demonstrated that the lagrangean and canonical procedures lead

to equivalent results; however, we stress that in both cases the procedure involves

the choice of a gauge fermion Ψ , restricted by the requirement that it has ghost

number Ng [Ψ ] = ’1.

The advantage of the lagrangean formalism is, that it is easier to formulate

the theory with di¬erent choices of the gauge fermion. In particular, it is possible

to make choices of gauge which manifestly respect the Lorentz-invariance of

Minkowski space. This is not an issue for the study of the relativistic particle,

but it is an issue in the case of Maxwell“Yang“Mills theory, which we have

constructed so far only in the temporal gauge Aa = constant.

0

We now show how to construct a covariant gauge-¬xed and BRST-invariant

e¬ective lagrangean for Maxwell“Yang“Mills theory, using the same procedure.

In stead of (305), we choose the gauge fermion

»a i» 2

Ψ = b a ‚ · Aa ’ ’ δ„¦ Ψ = i±a ‚ · Aa ’ ± ’ ba ‚ · (Dc)a . (307)

±

2a

2

Aspects of BRST Quantization 151

Here the parameter » is a arbitrary real number, which can be used to obtain a

convenient form of the propagator in perturbation theory. The e¬ective action

obtained with this choice of gauge-¬xing fermion is, after a partial integration:

1a »2

d4 x ’ (Fµν )2 + ±a ‚ · Aa ’ ±a ’ i‚ba · (Dc)a .

Se¬ = (308)

4 2

As we have introduced quadratic terms in the bosonic variables ±a , they now

behave more like auxiliary ¬elds, rather than Lagrange multipliers. Their varia-

tional equations lead to the result

1

‚ · Aa .

±a = (309)

»

Eliminating the auxiliary ¬elds by this equation, the e¬ective action becomes

1a 1

d4 x ’ (Fµν )2 + (‚ · Aa )2 ’ i‚ba · (Dc)a .

Se¬ = (310)

4 2»

This is the standard form of the Yang“Mills action used in covariant perturbation

theory. Observe, that the elimination of the auxiliary ¬eld ±a also changes the

BRST-transformation of the anti-ghost ba to:

i i

‚ · Aa ’ ‚ · (Dc)a

δ „¦ ba = δ „¦ ba =

2

0. (311)

» »

The transformation is now nilpotent only after using the ghost ¬eld equation.

The BRST-Noether charge can be computed from the action (310) by the

standard procedure, and leads to the expression

ig

πa (Dµ c)a ’

d3 x µ

fabc ca cb γc ,

„¦= (312)

2

where πa is the canonical momentum of the vector potential Aa , and (β a , γa )

µ

µ

denote the canonical momenta of the ghost ¬elds (ba , ca ):

‚Le¬ ‚Le¬ 1

= ’Fa = ’Ea , πa = = ’ ‚ · Aa ,

i 0i i 0

πa =

™ ™

a a »

‚ Ai ‚ A0

(313)

‚Le¬ ‚Le¬

= ’(D0 c)a ,

βa = i γa = i = ‚0 ba .

™a ‚ ca

™

‚b

Each ghost ¬eld (ba , ca ) now has its own conjugate momentum, because the

ghost terms in the action (310) are quadratic in derivatives, rather than linear

as before. Note also, that a factor i has been absorbed in the ghost momenta to

make them real; this leads to the standard Poisson brackets

{ca (x; t), γb (y; t)} = ’iδb δ 3 (x ’ y), ba (x; t), β b (y; t) = ’iδa δ 3 (x ’ y).

a b

(314)

152 J.W. van Holten

As our calculation shows, all explicit dependence on (ba , β a ) has dropped out of

the expression (312) for the BRST charge.

The parameter » is still a free parameter, and in actual calculations it is

often useful to check partial gauge-independence of physical results, like cross

sections, by establishing that they do not depend on this parameter. What needs

to be shown more generally is, that physical results do not depend on the choice

of gauge fermion. This follows formally from the BRST cohomology being inde-

pendent of the choice of gauge fermion. Indeed, from the expression (312) for

„¦ we observe that it is of the same form as the one we have used previously

0

in the temporal gauge, even though now πa no longer vanishes identically. In

the quantum theory this implies, that the BRST-cohomology classes at ghost

number zero correspond to gauge-invariant states, in which

a

(D · E) = 0, ‚ · Aa = 0. (315)

The second equation implies, that the time-evolution of the 0-component of the

vector potential is ¬xed completely by the initial conditions and the evolution

of the spatial components Aa . In particular, Aa = »a = constant is a consistent

0

solution if by a gauge transformation we take the spatial components to satisfy

∇ · Aa = 0.

In actual computations, especially in perturbation theory, the matter is more

subtle however: the theory needs to be renormalized, and this implies that the

action and BRST-transformation rules have to be adjusted to the introduction

of counter terms. To prove the gauge independence of the renormalized theory it

must be shown, that the renormalized action still possesses a BRST-invariance,

and the cohomology classes at ghost-number zero satisfy the renormalized condi-

tions (315). In four-dimensional space-time this can indeed be done for the pure

Maxwell“Yang“Mills theory, as there exists a manifestly BRST-invariant reg-

ularization scheme (dimensional regularization) in which the theory de¬ned by

the action (310) is renormalizable by power counting. The result can be extended

to gauge theories interacting with scalars and spin-1/2 fermions, except for the

case in which the Yang“Mills ¬elds interact with chiral fermions in anomalous

representations of the gauge group.

3.4 The Master Equation

Consider a BRST-invariant action Se¬ [¦A ] = S0 + dt (iδ„¦ Ψ ), where the vari-

ables ¦A = (q i , ca , ba , ±a ) parametrize the extended con¬guration space of the

system, and Ψ is the gauge fermion, which is Grassmann-odd and has ghost

number Ng [Ψ ] = ’1. Now by construction,

‚Ψ

δ„¦ Ψ = δ„¦ ¦A , (316)

‚¦A

and therefore we can write the e¬ective action also as

dt δ„¦ ¦A ¦—

Se¬ [¦A ] = S0 + i . (317)

‚Ψ

¦— =

A

‚¦A

A

Aspects of BRST Quantization 153

In this way of writing, one considers the action as a functional on a doubled

con¬guration space, parametrized by variables (¦A , ¦— ) of which the ¬rst set

A

¦A is called the ¬elds, and the second set ¦— is called the anti-¬elds. In the

A

generalized action

S — [¦A , ¦— ] = S0 + i dt δ„¦ ¦A ¦— , (318)

A A

the anti-¬elds play the role of sources for the BRST-variations of the ¬elds ¦A ;

the e¬ective action Se¬ is the restriction to the hypersurface Σ[Ψ ] : ¦— = A

‚Ψ/‚¦A . We observe, that by construction the anti¬elds have Grassmann par-

ity opposite to that of the corresponding ¬elds, and ghost number Ng [¦— ] =

A

’(Ng [¦ ] + 1).

A

In the doubled con¬guration space the BRST variations of the ¬elds can be

written as —

A δS

A

iδ„¦ ¦ = (’1) , (319)

δ¦—

A

where (’1)A is the Grassmann parity of the ¬eld ¦A , whilst ’(’1)A = (’1)A+1

is the Grassmann parity of the anti-¬eld ¦— . We now de¬ne the anti-bracket of

A