’ i c1 ‚σ c0 + c0 ‚σ c1 b0 ’ i c0 ‚σ c0 + c1 ‚σ c1 b1 .

The BRST transformations generated by the Poisson brackets of this charge read

δ„¦ X µ = {X µ , „¦} = c0 Π µ + c1 ‚σ X µ ≈ ca ‚a X µ ,

δ„¦ Πµ = {Πµ , „¦} = ‚σ c0 ‚σ Xµ + c1 Πµ ≈ ‚σ µab ca ‚b X µ ,

δ„¦ c0 = ’ c0 , „¦ = c1 ‚σ c0 + c0 ‚σ c1 ,

δ„¦ c1 = ’ c0 , „¦ = c0 ‚σ c0 + c1 ‚σ c1 ,

δ„¦ b0 = ’ {b0 , „¦} = i

Π 2 + [‚σ X]2 + c1 ‚σ b0 + c0 ‚σ b1 + 2‚σ c1 b0 + 2 ‚σ c0 b1 ,

2

δ„¦ b1 = ’ {b1 , „¦} = i Π · ‚σ X + c0 ‚σ b0 + c1 ‚σ b1 + 2‚σ c0 b0 + 2 ‚σ c1 b1 .

(210)

A tedious calculation shows that these transformations are indeed nilpotent:

2

δ„¦ = 0.

2.4 Quantum BRST Cohomology

The construction of a quantum theory for constrained systems poses the fol-

lowing problem: to have a local and/or covariant description of the quantum

system, it is advantageous to work in an extended Hilbert space of states, with

unphysical components, like gauge and ghost degrees of freedom. Therefore we

need ¬rst of all a way to characterize physical states within this extended Hilbert

space and then a way to construct a unitary evolution operator, which does not

mix physical and unphysical components. In this section we show that the BRST

construction can solve both of these problems.

We begin with a quantum system subject to constraints G± ; we impose these

constraints on the physical states:

G± |Ψ = 0, (211)

implying that physical states are gauge invariant. In the quantum theory the

generators of constraints are operators, which satisfy the commutation rela-

tions (80):

’i [G± , Gβ ] = P±β (G), (212)

where we omit the hat on operators for ease of notation.

136 J.W. van Holten

Next, we introduce corresponding ghost ¬eld operators (c± , bβ ) with equal-

time anti-commutation relations

[c± , bβ ]+ = c± bβ + ββ c± = δβ .

±

(213)

(For simplicity, the time-dependence in the notation has been suppressed). In

the ghost-extended Hilbert space we now construct a BRST operator

«

in ±1 ±n

„¦ = c± G± + c ...c M±±1 ...βn bβ1 ...bβn ,

β

(214)

1 ...±n

2n!

n≥1

which is required to satisfy the anti-commutation relation

[„¦, „¦]+ = 2„¦ 2 = 0. (215)

In words, the BRST operator is nilpotent. Working out the square of the BRST

operator, we get

i ±β γ

c c ’i [G± , Gβ ] + M±β Gγ

„¦2 =

2

(216)

1

’ c± cβ cγ ’i G± , Mβγ + M±β Mγµ + M±βγ Gµ bδ + ...

δ µ δ δµ

2

As a consequence, the coe¬cients M± are de¬ned as the solutions of the set of

equations

γ

i [G± , Gβ ] = ’P±β = M±β Gγ ,

δ µ δ δµ (217)

i G[± , Mβγ] + M[±β Mγ]µ = M±βγ Gµ

...

These are operator versions of the classical equations (187). As in the classical

case, their solution requires the absence of a central charge: c±β = 0.

Observe, that the Jacobi identity for the generators G± implies some restric-

tions on the higher terms in the expansion of „¦:

0 = [G± , [Gβ , Gγ ]] + (terms cyclic in [±βγ]) = ’3i G[± , Mβγ] Gδ

δ

(218)

3i

= ’3 i G[± , Mβγ] + M[±β M±]δµ Gδ = ’

δ µ δµ σ

M±βγ Mδµ Gσ .

2

The equality on the ¬rst line follows from the ¬rst equation (217), the last

equality from the second one.

To describe the states in the extended Hilbert space, we introduce a ghost-

state module, a basis for the ghost states consisting of monomials in the ghost

operators c± :

1

|[±1 ±2 ...±p ] gh = c±1 c±2 ...c±p |0 gh , (219)

p!

Aspects of BRST Quantization 137

with |0 gh the ghost vacuum state annihilated by all bβ . By construction these

states are completely anti-symmetric in the indices [±1 ±2 ...±p ], i.e. the ghosts

satisfy Fermi-Dirac statistics, even though they do not carry spin. This con¬rms

their unphysical nature. As a result of this choice of basis, we can decompose an

arbitrary state in components with di¬erent ghost number (= rank of the ghost

polynomial):

1 (2)

|Ψ = |Ψ (0) + c± |Ψ± + c± cβ |Ψ±β + ...

(1)

(220)

2

(n)

where the states |Ψ±1 ...±n corresponding to ghost number n are of the form

(n) (n)

|ψ±1 ...±n (q) — |0 gh , with |ψ±1 ...±n (q) states of zero-ghost number, depending

only on the degrees of freedom of the constrained (gauge) system; therefore we

have

bβ |Ψ±1 ...±n = 0.

(n)

(221)

To do the ghost-counting, it is convenient to introduce the ghost-number oper-

ator

[Ng , b± ] = ’b± ,

c± b± , [Ng , c± ] = c± ,

Ng = (222)

±

where as usual the summation over ± has to be interpreted in a generalized sense

(it includes integration over space when appropriate). It follows, that the BRST

operator has ghost number +1:

[Ng , „¦] = „¦. (223)

Now consider a BRST-invariant state:

„¦|Ψ = 0. (224)

Substitution of the ghost-expansions of „¦ and |Ψ gives

1 ±β (1) γ

„¦|Ψ = c± G± |Ψ (0) + c c G± |Ψβ ’ Gβ |Ψ± + iM±β |Ψγ

(1) (1)

2

(225)

1 1

(2) (2) (2)

+ c± cβ cγ G± |Ψβγ ’ iM±β |Ψγδ + M±βγ |Ψδµ

δ δµ

+ ...

2 2

Its vanishing then implies

G± |Ψ (0) = 0,

(1) (1) (1)

γ

G± |Ψβ ’ Gβ |Ψ± + iM±β |Ψγ = 0,

(226)

1

(2) (2) (2)

G[± |Ψβγ] ’ iM[±β |Ψγ]δ + M±βγ |Ψδµ = 0,

δ δµ

2

...

138 J.W. van Holten

These conditions admit solutions of the form

(1)

|Ψ± = G± |χ(0) ,

(2) (1) (1) (1)

γ

|Ψ±β = G± |χβ ’ Gβ |χ± + iM±β |χγ (227)

,

...

where the states |χ(n) have zero ghost number: b± |χ(n) = 0. Substitution of

these expressions into (220) gives

i ±β

(1) γ

|Ψ = |Ψ (0) + c± G± |χ(0) + c± cβ G± |χβ c c M±β |χ(1) + ...

+ γ

2

(228)

(1)

= |Ψ (0) + „¦ |χ(0) + c± |χ± + ...

= |Ψ (0) + „¦ |χ .

The second term is trivially BRST invariant because of the nilpotence of the

BRST operator: „¦ 2 = 0. Assuming that „¦ is hermitean, it follows, that |Ψ is

normalized if and only if |Ψ (0) is:

Ψ |Ψ = Ψ (0) |Ψ (0) + 2 Re χ|„¦|Ψ (0) + χ|„¦ 2 |χ = Ψ (0) |Ψ (0) . (229)

We conclude, that the class of normalizable BRST-invariant states includes the

set of states which can be decomposed into a normalizable gauge-invariant state

|Ψ (0) at ghost number zero, plus a trivially invariant zero-norm state „¦|χ .

These states are members of the BRST cohomology, the classes of states which

are BRST invariant (BRST closed) modulo states in the image of „¦ (BRST-exact

states):

Ker „¦

H(„¦) = . (230)

Im „¦

2.5 BRST-Hodge Decomposition of States

We have shown by explicit construction, that physical states can be identi¬ed

with the BRST-cohomology classes of which the lowest, non-trivial, component

has zero ghost-number. However, our analysis does not show to what extent these

solutions are unique. In this section we present a general discussion of BRST

cohomology to establish conditions for the existence of a direct correspondence

between physical states and BRST cohomology classes [24].

We assume that the BRST operator is self-adjoint with respect to the physical

inner product. As an immediate consequence, the ghost-extended Hilbert space

of states contains zero-norm states. Let

|Λ = „¦|χ . (231)

Aspects of BRST Quantization 139

These states are all orthogonal to each other, including themselves, and thus

they have zero-norm indeed:

Λ |Λ = χ |„¦ 2 |χ = 0 ’ Λ|Λ = 0. (232)

Moreover, these states are orthogonal to all normalizable BRST-invariant states:

’

„¦|Ψ = 0 Λ|Ψ = 0. (233)

Clearly, the BRST-exact states cannot be physical. On the other hand, BRST-

closed states are de¬ned only modulo BRST-exact states. We prove, that if on

the extended Hilbert space Hext there exists a non-degenerate inner product (not

the physical inner product), which is also non-degenerate when restricted to the

subspace Im „¦ of BRST-exact states, then all physical states must be members

of the BRST cohomology.

A non-degenerate inner product ( , ) on Hext is an inner product with the

following property:

(φ, χ) = 0 ∀φ ” χ = 0. (234)

If the restriction of this inner product to Im „¦ is non-degenerate as well, then

(„¦φ, „¦χ) = 0 ∀φ ” „¦χ = 0. (235)

As there are no non-trivial zero-norm states with respect to this inner product,

the BRST operator cannot be self-adjoint; its adjoint, denoted by — „¦, then

de¬nes a second nilpotent operator:

(„¦φ, χ) = (φ, — „¦χ) („¦ 2 φ, χ) = (φ, — „¦ 2 χ) = 0,

’ ∀φ. (236)

The non-degeneracy of the inner product implies that — „¦ 2 = 0. The adjoint — „¦

is called the co-BRST operator. Note, that from (235) one infers

(φ, — „¦ „¦χ) = 0, —