„¦ „¦χ = 0 „¦χ = 0. (237)

It follows immediately, that any BRST-closed vector „¦ψ = 0 is determined

uniquely by requiring it to be co-closed as well. Indeed, let — „¦ψ = 0; then

— —

” ”

„¦(ψ + „¦χ) = 0 „¦ „¦χ = 0 „¦χ = 0. (238)

Thus, if we regard the BRST transformations as gauge transformations on states

in the extended Hilbert space generated by „¦, then — „¦ represents a gauge-¬xing

operator determining a single particular state out of the complete BRST orbit.

States which are both closed and co-closed are called (BRST) harmonic.

Denoting the subspace of harmonic states by Hharm , we can now prove the

following theorem: the extended Hilbert space Hext can be decomposed exactly

into three subspaces (Fig. 1):

Hext = Hharm + Im „¦ + Im — „¦. (239)

140 J.W. van Holten

Ker „¦ Im * „¦

Hharm = Ker ∆

Hext

Im „¦ Ker * „¦

Fig. 1. Decomposition of the extended Hilbert space

Equivalently, any vector in Hext can be decomposed as

ψ = ω + „¦χ + — „¦φ, „¦ω = — „¦ω = 0.

where (240)

We sketch the proof. Denote the space of zero modes of the BRST operator (the

BRST-closed vectors) by Ker „¦, and the zero modes of the co-BRST operator

(co-closed vectors) by Ker — „¦. Then

(ψ, — „¦φ) = 0 ∀φ.

ψ ∈ Ker „¦ ” („¦ψ, φ) = 0 ∀φ ” (241)

With ψ being orthogonal to all vectors in Im — „¦, it follows that

⊥

Ker „¦ = (Im — „¦) , (242)

the orthoplement of Im — „¦. Similarly we prove

⊥

Ker — „¦ = (Im „¦) . (243)

Therefore, any vector which is not in Im „¦ and not in Im — „¦ must belong to the

orthoplement of both, i.e. to Ker — „¦ and Ker „¦ simultaneously; such a vector

is therefore harmonic.

Now as the BRST-operator and the co-BRST operator are both nilpotent,

⊥ ⊥

Im „¦ ‚ Ker „¦ = (Im — „¦) , Im — „¦ ‚ Ker — „¦ = (Im „¦) . (244)

Therefore Im „¦ and Im — „¦ have no elements in common (recall that the null-

vector is not in the space of states). Obviously, they also have no elements in

common with their own orthoplements (because of the non-degeneracy of the

inner product), and in particular with Hharm , which is the set of common states

in both orthoplements. This proves the theorem.

Aspects of BRST Quantization 141

We can de¬ne a BRST-laplacian ∆BRST as the semi positive de¬nite self-

adjoint operator

∆BRST = („¦ + — „¦)2 = — „¦ „¦ + „¦ — „¦, (245)

which commutes with both „¦ and — „¦. Consider its zero-modes ω:

—

„¦ „¦ ω + „¦ — „¦ ω = 0.

”

∆BRST ω = 0 (246)

The left-hand side of the last expression is a sum of a vector in Im „¦ and one

in Im — „¦; as these subspaces are orthogonal with respect to the non-degenerate

inner product, it follows that

—

„¦ — „¦ ω = 0,

§

„¦„¦ω =0 (247)

separately. This in turn implies „¦ω = 0 and — „¦ω = 0, and ω must be a harmonic

state:

∆BRST ω = 0 ” ω ∈ Hharm ; (248)

hence Ker ∆BRST = Hharm . The BRST-Hodge decomposition theorem can there-

fore be expressed as

Hext = Ker ∆BRST + Im „¦ + Im— „¦. (249)

The BRST-laplacian allows us to discuss the representation theory of BRST-

transformations. First of all, the BRST-laplacian commutes with the BRST-

and co-BRST operators „¦ and — „¦:

[∆BRST , — „¦] = 0.

[∆BRST , „¦] = 0, (250)

As a result, BRST-multiplets can be characterized by the eigenvalues of ∆BRST :

the action of „¦ or — „¦ does not change this eigenvalue. Basically we must then

distinguish between zero-modes and non-zero modes of the BRST-laplacian. The

zero-modes, the harmonic states, are BRST-singlets:

—

„¦|ω = 0, „¦|ω = 0.

In contrast, the non-zero modes occur in pairs of BRST- and co-BRST-exact

states:

—

∆BRST |φ± = »2 |φ± ’ „¦|φ+ = » |φ’ , „¦|φ’ = » |φ+ . (251)

Equation (232) guarantees that |φ± have zero (physical) norm; we can however

rescale these states such that

φ’ |φ+ = φ+ |φ’ = 1. (252)

It follows, that the linear combinations

1

|χ± = √ (|φ+ ± |φ’ ) (253)

2

142 J.W. van Holten

de¬ne a pair of positive- and negative-norm states:

χ± |χ± = ±1, χ“ |χ± = 0. (254)

They are eigenstates of the operator „¦ + — „¦ with eigenvalues (», ’»):

(„¦ + — „¦)|χ± = ±»|χ± . (255)

As physical states must have positive norm, all BRST-doublets must be unphys-

ical, and only BRST-singlets (harmonic) states can represent physical states.

Conversely, if all harmonic states are to be physical, only the components of

the BRST-doublets are allowed to have non-positive norm. Observe, however,

that this condition can be violated if the inner product ( , ) becomes degenerate

on the subspace Im „¦; in that case the harmonic gauge does not remove all

freedom to make BRST-transformations and zero-norm states can survive in the

subspace of harmonic states.

2.6 BRST Operator Cohomology

The BRST construction replaces a complete set of constraints, imposed by the

generators of gauge transformations, by a single condition: BRST invariance.

However, the normalizable solutions of the BRST condition (224):

Ψ |Ψ = 1,

„¦|Ψ = 0,

are not unique: from any solution one can construct an in¬nite set of other

solutions

|Ψ = |Ψ + „¦|χ , Ψ |Ψ = 1, (256)

provided the BRST operator is self-adjoint with respect to the physical inner

product. Under the conditions discussed in Sect. 2.5, the normalizable part of

the state vector is unique. Hence, the transformed state is not physically di¬erent

from the original one, and we actually identify a single physical state with the

complete class of solutions (256). As observed before, in this respect the quantum

theory in the extended Hilbert space behaves much like an abelian gauge theory,

with the BRST transformations acting as gauge transformations.

Keeping this in mind, it is clearly necessary that the action of dynamical

observables of the theory on physical states is invariant under BRST transfor-

mations: an observable O maps physical states to physical states; therefore if

|Ψ is a physical state, then

„¦O|Ψ = [„¦, O] |Ψ = 0. (257)

Again, the solution of this condition for any given observable is not unique: for

an observable with ghost number Ng = 0, and any operator ¦ with ghost number

Ng = ’1,

O = O + [„¦, ¦]+ (258)

Aspects of BRST Quantization 143

also satis¬es condition (257). The proof follows directly from the Jacobi identity:

„¦, [„¦, ¦]+ = „¦ 2 , ¦ = 0. (259)

This holds in particular for the hamiltonian; indeed, the time-evolution of states

in the unphysical sector (the gauge and ghost ¬elds) is not determined a priori,

and can be chosen by an appropriate BRST extension of the hamiltonian:

Hext = Hphys + [„¦, ¦]+ . (260)

Here Hphys is the hamiltonian of the physical degrees of freedom. The BRST-

exact extension [„¦, ¦]+ acts only on the unphysical sector, and can be used to

de¬ne the dynamics of the gauge- and ghost degrees of freedom.

2.7 Lie-Algebra Cohomology

We illustrate the BRST construction with a simple example: a system of con-

straints de¬ning an ordinary n-dimensional compact Lie algebra [25]. The Lie

algebra is taken to be a direct sum of semi-simple and abelian u(1) algebras, of

the form

[Ga , Gb ] = ifabc Gc , (a, b, c) = 1, ..., n, (261)

where the generators Ga are hermitean, and the fabc = ’fbac are real structure

constants. We assume the generators normalized such that the Killing metric is

unity:

1

’ facd fbdc = δab . (262)

2

Then fabc = fabd δdc is completely anti-symmetric. We introduce ghost operators

(ca , bb ) with canonical anti-commutation relations (213):

[ca , bb ]+ = δb ,

a

ca , cb = [ba , bb ]+ = 0.

+

This implies, that in the ˜co-ordinate representation™, in which the ghosts ca are

represented by Grassmann variables, the ba can be represented by a Grassmann

derivative:

‚

ba = a . (263)

‚c

The nilpotent BRST operator takes the simple form

i ab c

„¦ = ca Ga ’ „¦ 2 = 0.

c c fab bc , (264)

2

We de¬ne a ghost-extended state space with elements

n

1 a1 ak (k)

ψ[c] = c ...c ψa1 ...ak . (265)

k!

k=0

(k)

The coe¬cients ψa1 ..ak of ghost number k carry completely anti-symmetric prod-

uct representations of the Lie algebra.

144 J.W. van Holten

On the state space we introduce an inde¬nite inner product, with respect

to which the ghosts ca and ba are self-adjoint; this is realized by the Berezin

integral over the ghost variables

n

1 a1 ...an n

†

φ(n’k) — 1 ψan’k+1 ...an . (266)

n 1 (k)

φ, ψ = [dc ...dc ] φ ψ = µ an’k ...a

k

n!

k=0

In components, the action of the ghosts is given by

(ca ψ)(k) k = δa1 ψa2 a3 ...ak ’ δa2 ψa1 a3 ...ak + ... + (’1)k’1 δak ψa1 a2 ...ak’1 , (267)

a (k’1) a (k’1) a (k’1)

a1 ...a

and similarly

(k)

‚ψ

(ba ψ)(k) k (k+1)

= = ψaa1 ...ak . (268)

a1 ...a

‚ca a1 ...ak

It is now easy to check that the ghost operators are self-adjoint with respect to

the inner product (266):

φ, ca ψ = ca φ, ψ , φ, ba ψ = ba φ, ψ . (269)

It follows directly that the BRST operator (264) is self-adjoint as well:

φ, „¦ψ = „¦φ, ψ . (270)

Now we can introduce a second inner product, which is positive de¬nite and

therefore manifestly non-degenerate: