The BRST construction starts with the introduction of canonical pairs of

Grassmann degrees of freedom (c± , bβ ), one for each constraint G± , with Poisson

brackets

{c± , bβ } = {bβ , c± } = ’iδβ ,

±

(181)

These anti-commuting variables are known as ghosts; the complete Poisson

brackets on the extended phase space are given by

‚A ‚B ‚A ‚B ‚A ‚B ‚A ‚B

{A, B} = ’ + i(’1)A + , (182)

i ‚p ‚pi ‚q i ± ‚b ‚b± ‚c±

‚q ‚c

i ±

Aspects of BRST Quantization 131

where (’1)A denotes the Grassmann parity of A: +1 if A is Grassmann-even

(commuting) and ’1 if A is Grassmann-odd (anti-commuting).

With the help of these ghost degrees of freedom, one de¬nes the BRST charge

„¦, which has Grassmann parity (’1)„¦ = ’1, as

„¦ = c± (G± + M± ) , (183)

where M± is Grassmann-even and of the form

in ±1 ±n

c ...c M±±1 ...βn bβ1 ...bβn

β

M± = 1 ...±n

2n!

n≥1

(184)

i ±1 1

c M±±1 bβ1 ’ c±1 c±2 M±±1 β2 bβ1 bβ2 + ...

β β

= 1 ±2

1

2 4

β ...β

The quantities M±±1 ...±p are functions of the classical phase-space variables via

1 p

the constraints G± , and are de¬ned such that

{„¦, „¦} = 0. (185)

As „¦ is Grassmann-odd, this is a non-trivial property, from which the BRST

charge can be constructed inductively:

γ

{„¦, „¦} = c± cβ P±β + M±β Gγ

(186)

G± , Mβγ ’ M±β Mγ δ + M±βγ Gµ bδ + ...

+ ic± cβ cγ δ µ δµ

This vanishes if and only if

γ

M±β Gγ = ’P±β ,

δµ δ µ δ (187)

M±βγ Gµ = M[±β , G γ] + M[±β M γ]µ ,

...

Observe, that the ¬rst relation can only be satis¬ed under the condition c±β = 0,

with the solution

1

M±β = f±βγ + g±βγδ Gδ + ...

γ

(188)

2

The same condition guarantees that the second relation can be solved: the

bracket on the right-hand side is

δ

‚M±β 1

δ

Pµγ = g±βδµ fµγ Gσ + ...

σ

M±β , Gγ = (189)

‚Gµ 2

whilst the Jacobi identity (67) implies that

µ δ

f[±β f γ]µ = 0, (190)

132 J.W. van Holten

and therefore M[±β M γ]µ = O[Gσ ]. This allows to determine M±βγ . Any higher-

µ δ δµ

order terms can be calculated similarly. In practice P±β and M± usually contain

only a small number of terms.

Next, we observe that we can extend the classical hamiltonian H = H0 with

ghost terms such that

in ±1 ±n (n) β1 ...βn

{„¦, Hc } = 0.

Hc = H 0 + c ...c h±1 ...±n (G) bβ1 ...bβn , (191)

n!

n≥1

Observe that on the physical hypersurface in the phase space this hamiltonian

coincides with the original classical hamiltonian modulo terms which do not

a¬ect the time-evolution of the classical phase-space variables (q, p). We illustrate

the procedure by constructing the ¬rst term:

i i ±1 ±2 β

{„¦, Hc } = {c± G± , H0 } + c± G± , cγ h(1) β bβ + c c M±1 ±2 bβ , H0 + ...

γ

2 2

= c± Z± ’ h(1) β Gβ + ...

±

(192)

Hence the bracket vanishes if the hamiltonian is extended by ghost terms such

that

h(1) β (G) Gβ = Z± (G), ... (193)

±

This equation is guaranteed to have a solution by the condition Z(0) = 0.

As the BRST charge commutes with the ghost-extended hamiltonian, we can

use it to generate ghost-dependent symmetry transformations of the classical

phase-space variables: the BRST transformations

‚„¦ ‚G±

δ„¦ q i = ’ „¦, q i = = c± + ghost extensions,

‚pi ‚pi

(194)

‚„¦ ‚G±

δ„¦ pi = ’ {„¦, pi } = ’ = c± + ghost extensions.

‚q i

‚qi

These BRST transformations are just the gauge transformations with the param-

eters ± replaced by the ghost variables c± , plus (possibly) some ghost-dependent

extension.

Similarly, one can de¬ne BRST transformations of the ghosts:

‚„¦ 1

δ„¦ c± = ’ {„¦, c± } = i = ’ cβ cγ Mβγ + ...,

±

‚b± 2

(195)

‚„¦ γ

δ„¦ b± = ’ {„¦, b± } = i = iG± ’ cβ M±β bγ + ...

±

‚c

An important property of these transformations is their nilpotence:

2

δ„¦ = 0. (196)

Aspects of BRST Quantization 133

This follows most directly from the Jacobi identity for the Poisson brackets of

the BRST charge with any phase-space function A:

1

δ„¦ A = {„¦, {„¦, A}} = ’ {A, {„¦, „¦}} = 0.

2

(197)

2

Thus, the BRST variation δ„¦ behaves like an exterior derivative. Next we observe

that gauge invariant physical quantities F have the properties

‚F ‚F

{F, c± } = i {F, b± } = i {F, G± } = δ± F = 0. (198)

= 0, = 0,

‚c±

‚b±

As a result, such physical quantities must be BRST invariant:

δ„¦ F = ’ {„¦, F } = 0. (199)

In the terminology of algebraic geometry, such a function F is called BRST

closed. Now because of the nilpotence, there are trivial solutions to this condition,

of the form

F0 = δ„¦ F1 = ’ {„¦, F1 } . (200)

These solutions are called BRST exact; they always depend on the ghosts (c± , b± ),

and cannot be physically relevant. We conclude, that true physical quantities

must be BRST closed, but not BRST exact. Such non-trivial solutions of the

BRST condition (199) de¬ne the BRST cohomology, which is the set

Ker(δ„¦ )

H(δ„¦ ) = . (201)

Im(δ„¦ )

We will make this more precise later on.

2.3 Examples

As an application of the above construction, we now present the classical BRST

charges and transformations for the gauge systems discussed in Sect. 1.

The Relativistic Particle. We consider the gauge-¬xed version of the rela-

tivistic particle. Taking c = 1, the only constraint is

12

(p + m2 ) = 0,

H0 = (202)

2m

and hence in this case P±β = 0. We only introduce one pair of ghost variables,

and de¬ne

c2

(p + m2 ).

„¦= (203)

2m

134 J.W. van Holten

It is trivially nilpotent, and the BRST transformations of the phase space vari-

ables read

cpµ

δ„¦ xµ = {xµ , „¦} = , δ„¦ pµ = {pµ , „¦} = 0,

m

(204)

i

δ„¦ c = ’ {c, „¦} = 0, δ„¦ b = ’ {b, „¦} = (p2 + m2 ) ≈ 0.

2m

The b-ghost transforms into the constraint, hence it vanishes on the physical

2

hypersurface in the phase space. It is straightforward to verify that δ„¦ = 0.

Electrodynamics. In the gauge ¬xed Maxwell™s electrodynamics there is again

only a single constraint, and a single pair of ghost ¬elds to be introduced. We

de¬ne the BRST charge

d3 x c∇ · E.

„¦= (205)

The classical BRST transformations are just ghost-dependent gauge transfor-

mations:

δ„¦ A = {A, „¦} = ∇c, δ„¦ E = {E, „¦} = 0,

(206)

δ„¦ c = ’ {c, „¦} = 0, δ„¦ b = ’ {b, „¦} = i∇ · E ≈ 0.

Yang“Mills Theory. One of the simplest non-trivial systems of constraints

is that of Yang“Mills theory, in which the constraints de¬ne a local Lie alge-

bra (137). The BRST charge becomes

ig a b c

ca Ga ’

d3 x

„¦= c c fab bc , (207)

2

with Ga = (D · E)a . It is now non-trivial that the bracket of „¦ with itself

vanishes; it is true because of the closure of the Lie algebra, and the Jacobi

identity for the structure constants.

The classical BRST transformations of the ¬elds become

δ„¦ Aa = {Aa , „¦} = (Dc)a , δ„¦ E a = {E a , „¦} = gfabc cb E c ,

(208)

g

δ„¦ c = ’ {c , „¦} = fbc a cb cc , δ„¦ ba = ’ {ba , „¦} = i Ga + gfabc cb bc .

a a

2

2

Again, it can be checked by explicit calculation that δ„¦ = 0 for all varia-

tions (208). It follows, that

δ„¦ Ga = gfabc cb Gc ,

2

and as a result δ„¦ ba = 0.

Aspects of BRST Quantization 135

The Relativistic String. Finally, we discuss the free relativistic string. We

take the reduced constraints (154), satisfying the algebra (155), (156). The BRST

charge takes the form

10

c Π 2 + [‚σ X]2 + c1 Π · ‚σ X

„¦= dσ

2