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(209)
’ 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

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

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

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

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

䄦 b0 = ’ {b0 , „¦} = i
Π 2 + [‚σ X]2 + c1 ‚σ b0 + c0 ‚σ b1 + 2‚σ c1 b0 + 2 ‚σ c0 b1 ,
2

䄦 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
䄦 = 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, —

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