A A

δF δG δF δG

(F, G) = (’1)F +G+F G (G, F ) = (’1)A(F +1) + (’1)F .

δ¦A δ¦— δ¦— δ¦A

A A

(320)

These brackets are symmetric in F and G if both are Grassmann-even (bosonic),

and anti-symmetric in all other cases. Sometimes one introduces the notion of

right derivative:

←

Fδ δF

≡ (’1)A(F +1) A . (321)

δ¦A δ¦

Then the anti-brackets take the simple form

←’ ←’

Fδ δG Fδ δG

’

(F, G) = , (322)

δ¦A δ¦— δ¦— δ¦A

A A

where the derivatives with a right arrow denote the standard left derivatives. In

terms of the anti-brackets, the BRST transformations (319) can be written in

the form

iδ„¦ ¦A = (S — , ¦A ). (323)

In analogy, we can de¬ne

δS —

iδ„¦ ¦— = (S — , ¦— ) = (’1)A . (324)

A A

δ¦A

Then the BRST transformation of any functional Y (¦A , ¦— ) is given by

A

iδ„¦ Y = (S — , Y ). (325)

154 J.W. van Holten

In particular, the BRST invariance of the action S — can be expressed as

(S — , S — ) = 0. (326)

This equation is known as the master equation. Next we observe, that on the

physical hypersurface Σ[Ψ ] the BRST transformations of the anti¬elds are given

by the classical ¬eld equations; indeed, introducing an anti-commuting parame-

ter µ for in¬nitesimal BRST transformations

δS — δSe¬

Σ[Ψ ]

iµ δ„¦ ¦— ’’

= µ µ 0, (327)

A

δ¦A δ¦A

where the last equality holds only for solutions of the classical ¬eld equations.

Because of this result, it is customary to rede¬ne the BRST transformations of

the anti¬elds such that they vanish:

δ„¦ ¦— = 0, (328)

A

instead of (324). As the BRST transformations are nilpotent, this is consistent

with the identi¬cation ¦— = ‚Ψ/‚¦A in the action; indeed, it now follows that

A

δ„¦ δ„¦ ¦A ¦— = 0, (329)

A

which holds before the identi¬cation as a result of (328), and after the iden-

2

ti¬cation because it reduces to δ„¦ Ψ = 0. Note, that the condition for BRST

invariance of the action now becomes

1——

iδ„¦ S — = (S , S ) = 0, (330)

2

which still implies the master equation (326).

3.5 Path-Integral Quantization

The construction of BRST-invariant actions Se¬ = S — [¦— = ‚Ψ/‚¦A ] and the

A

anti-bracket formalism is especially useful in the context of path-integral quan-

tization. The path integral provides a representation of the matrix elements of

the evolution operator in the con¬guration space:

qf T /2

’iT H i L(q,q)dt

™

|qi , ’T /2 = ’T /2

qf , T /2|e Dq(t) e . (331)

qi

In ¬eld theory one usually considers the vacuum-to-vacuum amplitude in the

presence of sources, which is a generating functional for time-ordered vacuum

Green™s functions:

Z[J] = D¦ eiS[¦]+i J¦ , (332)

such that

δ k Z[J]

0|T (¦1 ...¦k )|0 = . (333)

δJ1 ...δJk J=0

Aspects of BRST Quantization 155

The corresponding generating functional W [J] for the connected Green™s func-

tions is related to Z[J] by

Z[J] = ei W [J] . (334)

For theories with gauge invariances, the evolution operator is constructed from

the BRST-invariant hamiltonian; then the action to be used is the in the path

integral (332) is the BRST invariant action:

—

[¦A ,¦— ]+i JA ¦A

Z[J] = ei W [J] = D¦A ei S , (335)

A

¦— =‚Ψ/‚¦A

A

where the sources JA for the ¬elds are supposed to be BRST invariant them-

selves. For the complete generating functional to be BRST invariant, it is not

su¬cient that only the action S — is BRST invariant, as guaranteed by the master

equation (326): the functional integration measure must be BRST invariant as

well. Under an in¬nitesimal BRST transformation µδ„¦ ¦A the measure changes

by a graded jacobian (superdeterminant) [18,19]

δ(δ„¦ ¦A ) δ(δ„¦ ¦A )

J = SDet δB + µ(’1)B ≈ 1 + µ Tr

A

. (336)

δ¦B δ¦B

We now de¬ne

δ2 S —

δ(iδ„¦ ¦A )

≡ ∆S — .

¯

A

= (’1) (337)

δ¦A δ¦—

δ¦A A

¯

The operator ∆ is a laplacian on the ¬eld/anti-¬eld con¬guration space, which

for an arbitrary functional Y (¦A , ¦— ) is de¬ned by

A

δ2 Y

¯

∆Y = (’1)A(1+Y ) . (338)

δ¦A δ¦—A

The condition of invariance of the measure requires the BRST jacobian (336) to

be unity:

J = 1 ’ iµ ∆S — = 1,

¯ (339)

which reduces to the vanishing of the laplacian of S — :

∆S — = 0.

¯ (340)

The two conditions (326) and (340) imply the BRST invariance of the path in-

tegral (335). Actually, a somewhat more general situation is possible, in which

neither the action nor the functional measure are invariant independently, only

the combined functional integral. Let the action generating the BRST transfor-

mations be denoted by W — [¦A , ¦— ]:

A

iδ„¦ ¦A = (W — , ¦A ), iδ„¦ ¦— = 0. (341)

A

As a result the graded jacobian for a transformation with parameter µ is

δ(δ„¦ ¦A )

≈ 1 ’ iµ ∆W — .

¯

A B

SDet δB + µ(’1) (342)

δ¦B

156 J.W. van Holten

Then the functional W — itself needs to satisfy the generalized master equation

1

(W — , W — ) = i∆W — ,

¯ (343)

2

for the path-integral to be BRST invariant. This equation can be neatly sum-

marized in the form —

¯

∆ eiW = 0. (344)

Solutions of this equation restricted to the hypersurface ¦— = ‚Ψ/‚¦A are

A

acceptable actions for the construction of BRST-invariant path integrals.

4 Applications of BRST Methods

In the ¬nal section of these lecture notes, we turn to some applications of BRST-

methods other than the perturbative quantization of gauge theories. We deal

with two topics; the ¬rst is the construction of BRST ¬eld theories, presented in

the context of the scalar point particle. This is the simplest case [34]; for more

complicated ones, like the superparticle [35,36] or the string [35,37,32], we refer

to the literature.

The second application concerns the classi¬cation of anomalies in gauge the-

ories of the Yang“Mills type. Much progress has been made in this ¬eld in recent

years [40], of which a summary is presented here.

4.1 BRST Field Theory

The examples of the relativistic particle and string show that in theories with

local reparametrization invariance the hamiltonian is one of the generators of

gauge symmetries, and as such is constrained to vanish. The same phenomenon

also occurs in general relativity, leading to the well-known Wheeler-deWitt equa-

tion. In such cases, the full dynamics of the system is actually contained in the

BRST cohomology. This opens up the possibility for constructing quantum ¬eld

theories for particles [32“34], or strings [32,35,37], in a BRST formulation, in

which the usual BRST operator becomes the kinetic operator for the ¬elds. This

formulation has some formal similarities with the Dirac equation for spin-1/2

¬elds.

As our starting point we consider the BRST-operator for the relativistic

quantum scalar particle, which for free particles, after some rescaling, reads

„¦ = c(p2 + m2 ), „¦ 2 = 0. (345)

It acts on ¬elds Ψ (x, c) = ψ0 (x) + cψ1 (x), with the result

„¦Ψ (x, c) = c(p2 + m2 ) ψ0 (x). (346)

As in the case of Lie-algebra cohomology (271), we introduce the non-degenerate

(positive de¬nite) inner product

dd x (φ— ψ0 + φ— ψ1 ) .

(¦, Ψ ) = (347)

0 1

Aspects of BRST Quantization 157

With respect to this inner product the ghosts (b, c) are mutually adjoint:

b = c† .

”

(¦, cΨ ) = (b¦, Ψ ) (348)

Then, the BRST operator „¦ is not self-adjoint but rather

„¦ † = b(p2 + m2 ), „¦ † 2 = 0. (349)

Quite generally, we can construct actions for quantum scalar ¬elds coupled to

external sources J of the form

1

(Ψ, G „¦Ψ ) ’ (Ψ, J) ,

SG [J] = (350)

2

where the operator G is chosen such that

G„¦ = (G„¦)† = „¦ † G† . (351)

This guarantees that the action is real. From the action we then derive the ¬eld

equation

G„¦ Ψ = „¦ † G† Ψ = J. (352)

Its consistency requires the co-BRST invariance of the source:

„¦ † J = 0. (353)

This re¬‚ects the invariance of the action and the ¬eld equation under BRST

transformations

Ψ ’ Ψ = Ψ + „¦χ. (354)

In order to solve the ¬eld equation we therefore have to impose a gauge condition,

selecting a particular element of the equivalence class of solutions (354).

A particularly convenient condition is

„¦G† Ψ = 0. (355)

In this gauge, the ¬eld equation can be rewritten in the form

∆G† Ψ = „¦ † „¦ + „¦„¦ † G† Ψ = „¦ J. (356)

Here ∆ is the BRST laplacean, which can be inverted using a standard analytic

continuation in the complex plane, to give

1

G† Ψ = „¦ J. (357)

∆

We interpret the operator ∆’1 „¦ on the right-hand side as the (tree-level) prop-

agator of the ¬eld.

We now implement the general scheme (350)“(357) by choosing the inner

product (347), and G = b. Then