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two functionals F (¦A , ¦— ) and G(¦A , ¦— ) on the large con¬guration space by
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
i䄦 ¦A = (S — , ¦A ). (323)
In analogy, we can de¬ne
δS —
i䄦 ¦— = (S — , ¦— ) = (’1)A . (324)
A A
δ¦A
Then the BRST transformation of any functional Y (¦A , ¦— ) is given by
A

i䄦 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 䄦 ¦— ’’
= 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:

䄦 ¦— = 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

䄦 䄦 ¦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 䄦 Ψ = 0. Note, that the condition for BRST
invariance of the action now becomes
1——
i䄦 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(䄦 ¦A ) δ(䄦 ¦A )
J = SDet δB + µ(’1)B ≈ 1 + µ Tr
A
. (336)
δ¦B δ¦B
We now de¬ne
δ2 S —
δ(i䄦 ¦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

i䄦 ¦A = (W — , ¦A ), i䄦 ¦— = 0. (341)
A

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

δ(䄦 ¦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

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