1

φ(k) — (k)

(φ, ψ) = ψa1 ...ak . (271)

k!

k=0

It is related to the ¬rst inde¬nite inner product by Hodge duality: de¬ne the

Hodge —-operator by

1

—

ψ (k) a1 ...ak = µa1 ...ak ak+1 ...an ψak+1 ...an .

(n’k)

(272)

(n ’ k)!

Furthermore, de¬ne the ghost permutation operator P as the operator which

reverses the order of the ghosts in ψ[c]; equivalently:

(Pψ)(k) k = ψak ...a1 .

(k)

(273)

a1 ...a

Then the two inner products are related by

(φ, ψ) = P — φ, ψ . (274)

An important property of the non-degenerate inner product is, that the ghosts

ca and ba are adjoint to one another:

(φ, ca ψ) = (ba φ, ψ). (275)

Aspects of BRST Quantization 145

Then the adjoint of the BRST operator is given by the co-BRST operator

i c ab

—

„¦ = ba G a ’

c f c ba bb . (276)

2

Here raising and lowering indices on the generators and structure constants is

done with the help of the Killing metric (δab in our normalization). It is easy to

check that — „¦ 2 = 0, as expected.

The harmonic states are both BRST- and co-BRST-closed: „¦ψ = — „¦ψ = 0.

They are zero-modes of the BRST-laplacian:

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

2

(277)

as follows from the observation that

(ψ, ∆BRST ψ) = („¦ψ, „¦ψ) + (— „¦ψ, — „¦ψ) = 0 „¦ψ = — „¦ψ = 0.

” (278)

For the case at hand, these conditions become

Ga ψ = 0, Σa ψ = 0, (279)

where Σa is de¬ned as

†

Σa = Σa = ’ifabc c b bc . (280)

From the Jacobi identity, it is quite easy to verify that Σa de¬nes a representa-

tion of the Lie algebra:

[Σa , Σb ] = ifabc Σc , [Ga , Σb ] = 0. (281)

The conditions (279) are proven as follows. Substitute the explicit expressions

for „¦ and — „¦ into (277) for ∆BRST . After some algebra one then ¬nds

12 1 1

∆BRST = G2 + G · Σ + Σ = G2 + (G + Σ)2 . (282)

2 2 2

This being a sum of squares, any zero mode must satisfy (279). Q.E.D.

Looking for solutions, we observe that in components the second condition

reads

(k)

(Σa ψ)(k) k = ’ifa[ab ψa2 ...ak ]b = 0. (283)

a1 ...a 1

It acts trivially on states of ghost number k = 0; hence bona ¬de solutions are

the gauge-invariant states of zero ghost number:

ψ = ψ (0) , Ga ψ (0) = 0. (284)

However, other solutions with non-zero ghost number exist. A general solution

is for example

1

ψ = fabc ca cb cc χ, Ga χ = 0. (285)

3!

(3)

The 3-ghost state ψabc = fabc χ indeed satis¬es (283) as a result of the Jacobi

identity. The states χ are obviously in one-to-one correspondence with the states

ψ (0) . Hence, in general there exist several copies of the space of physical states in

the BRST cohomology, at di¬erent ghost number. We infer that in addition to

requiring physical states to belong to the BRST cohomology, it is also necessary

to ¬x the ghost number for the de¬nition of physical states to be unique.

146 J.W. van Holten

3 Action Formalism

The canonical construction of the BRST cohomology we have described, can be

given a basis in the action formulation, either in lagrangean or hamiltonian form.

The latter one relates most directly to the canonical bracket formulation. It is

then straightforward to switch to a gauge-¬xed lagrangean formulation. Once

we have the lagrangean formulation, a covariant approach to gauge ¬xing and

quantization can be developed. In this section these constructions are presented

and the relations between various formulations are discussed.

3.1 BRST Invariance from Hamilton™s Principle

We have observed in Sect. 2.6, that the e¬ective hamiltonian in the ghost-

extended phase space is de¬ned only modulo BRST-exact terms:

He¬ = Hc + i {„¦, Ψ } = Hc ’ iδ„¦ Ψ, (286)

where Ψ is a function of the phase space variables with ghost number Ng (Ψ ) =

’1. Moreover, the ghosts (c, b) are canonically conjugate:

{c± , bβ } = ’iδβ .

±

Thus, we are led to construct a pseudo-classical action of the form

dt pi q i + ib± c± ’ He¬ .

Se¬ = ™ ™ (287)

That this is indeed the correct action for our purposes follows from the ghost

equations of motion obtained from this action, reading

‚He¬ ‚He¬

™

c± = ’i b± = ’i

™ . . (288)

‚c±

‚b±

These equations are in full agreement with the de¬nition of the extended Poisson

brackets (182):

™

c± = ’ {He¬ , c± } , b± = ’ {He¬ , b± } .

™ (289)

As Hc is BRST invariant, He¬ is BRST invariant as well: the BRST variations

2

are nilpotent and therefore δ„¦ ¦ = 0. It is then easy to show that the action Se¬

is BRST-symmetric and that the conserved Noether charge is the BRST charge

as de¬ned previously:

™

δ„¦ pi q i ’ δ„¦ q i pi + iδ„¦ b± c± + iδ„¦ c± b± ’ δ„¦ He¬

δ„¦ Se¬ = dt ™ ™ ™

d

(pi δ„¦ q i ’ ib± δ„¦ c± )] (290)

+

dt

d

pi δ„¦ q i ’ ib± δ„¦ c± ’ „¦ .

= dt

dt

Aspects of BRST Quantization 147

To obtain the last equality, we have used (194) and (195), which can be summa-

rized

‚„¦ ‚„¦

δ „¦ pi = ’ i ,

δ„¦ q i = ,

‚pi ‚q

‚„¦ ‚„¦

δ„¦ c± = i , δ„¦ b± = i ± .

‚b± ‚c

The therefore action is invariant up to a total time derivative. By comparison

with (59), we conclude that „¦ is the conserved Noether charge.

3.2 Examples

The Relativistic Particle. A simple example of the procedure presented above

is the relativistic particle. The canonical hamiltonian H0 is constrained to vanish

itself. As a result, the e¬ective hamiltonian is a pure BRST term:

He¬ = i {„¦, Ψ } . (291)

A simple choice for the gauge fermion is Ψ = b, which has the correct ghost

number Ng = ’1. With this choice and the BRST generator „¦ of (203), the

e¬ective hamiltonian is

c 1

(p2 + m2 ), b = p 2 + m2 .

He¬ = i (292)

2m 2m

Then the e¬ective action becomes

1

p · x + ibc ’ (p2 + m2 ) .

Se¬ = d„ ™ ™ (293)

2m

This action is invariant under the BRST transformations (204) :

cpµ

δ„¦ x = {x , „¦} = , δ„¦ pµ = {pµ , „¦} = 0,

µ µ

m

i

δ„¦ c = ’ {c, „¦} = 0, δ„¦ b = ’ {b, „¦} = (p2 + m2 ),

2m

up to a total proper-time derivative:

p 2 ’ m2

d

δ„¦ Se¬ = d„ c . (294)

d„ 2m

Implementing the Noether construction, the conserved charge resulting from the

BRST transformations is

c c

„¦ = p · δ„¦ x + ib δ„¦ c ’ (p2 ’ m2 ) = (p2 + m2 ). (295)

2m 2m

Thus, we have reobtained the BRST charge from the action (293) and the trans-

formations (204) con¬rming that together with the BRST-cohomology principle,

they correctly describe the dynamics of the relativistic particle.

148 J.W. van Holten

From the hamiltonian formulation (293) it is straightforward to construct

a lagrangean one by using the hamilton equation pµ = mxµ to eliminate the

™

momenta as independent variables; the result is

m2

(x ’ 1) + ibc .

Se¬ d„ ™ ™ (296)

2

Maxwell“Yang“Mills Theory. The BRST generator of the Maxwell“Yang“

Mills theory in the temporal gauge has been given in (207):

ig

ca Ga ’

d3 x fabc ca cb bc ,

„¦=

2

with Ga = (D · E)a . The BRST-invariant e¬ective hamiltonian takes the form

1

E 2 + B 2 + i {„¦, Ψ } .

He¬ = (297)

a a

2

Then, a simple choice of the gauge fermion, Ψ = »a ba , with some constants »a

gives the e¬ective action

‚A 1

d4 x ’E · + iba ca ’ E 2 + B 2 ’ »a (D · E)a + ig»a fabc cb bc .

Se¬ = ™ a a

‚t 2

(298)

a

The choice » = 0 would in e¬ect turn the ghosts into free ¬elds. However, if

we eliminate the electric ¬elds E a as independent degrees of freedom by the

substitution Ei = Fi0 = ‚i Aa ’ ‚0 Aa ’ gfbc a Ab Ac and recalling the classical

a a

0 i0

i

hamiltonian (127), we observe that we might actually interpret »a as a constant

scalar potential, Aa = »a , in a BRST-extended relativistic action

0

1a

d4 x ’ (Fµν )2 + iba (D0 c)a

Se¬ = , (299)

4 Aa =»a

0

where (D0 c)a = ‚0 ca ’ gfbc a Ab cc . The action is invariant under the classical

0

BRST transformations (208):

δ„¦ Aa = (Dc)a , δ„¦ E a = gfabc cb E c ,

g

δ„¦ ca = fbc a cb cc , δ„¦ ba = i Ga + gfabc cb bc ,

2

with the above BRST generator (207) as the conserved Noether charge. All of

the above applies to Maxwell electrodynamics as well, except that in an abelian

theory there is only a single vector ¬eld, and all structure constants vanish:

fabc = 0.

3.3 Lagrangean BRST Formalism

From the hamiltonian formulation of BRST-invariant dynamical systems it is

straightforward to develop an equivalent lagrangean formalism, by eliminating

Aspects of BRST Quantization 149

the momenta pi as independent degrees of freedom. This proceeds as usual by