{G[Λ1 ], G[Λ2 ]} = G[Λ3 ], Λa = Λb ‚b Λa . (151)

3 [1 2]

It takes quite a long and di¬cult calculation to check this result.

Most practitioners of string theory prefer to work in the restricted phase

space, in which the metric gab is not a dynamical variable, and there is no

126 J.W. van Holten

need to introduce its conjugate momentum π ab . Instead, gab is chosen to have a

convenient value by exploiting the reparametrization invariance (140) or (148):

’1 0

gab = ρ ·ab = ρ . (152)

01

Because of the Weyl invariance (144), ρ never appears explicitly in any physical

quantity, so it does not have to be ¬xed itself. In particular, the hamiltonian

becomes

1 1

dσ [‚0 X]2 + [‚1 X]2 = dσ Π 2 + [‚σ X]2 ,

Hred = (153)

2 2

whilst the constrained gauge generators (149) become

10

Λ Π 2 + [‚σ X]2 + Λ1 Π · ‚σ X .

Gred [Λ] = dσ (154)

2

Remarkably, these generators still satisfy a closed bracket algebra:

{Gred [Λ1 ], Gred [Λ2 ]} = Gred [Λ3 ], (155)

but the structure constants have changed, as becomes evident from the expres-

sions for Λa :

3

Λ0 = Λ1 ‚σ Λ0 + Λ0 ‚σ Λ1 ,

3 [1 2] [1 2]

(156)

1 0 0 1 1

Λ3 = Λ[1 ‚σ Λ 2] + Λ[1 ‚σ Λ 2]

The condition for Gred [Λ] to generate a symmetry of the hamiltonian Hred (and

hence to be conserved), is again Gred [Λ] = 0. Observe, that these expressions

reduce to those of (151) when the Λa satisfy

‚σ Λ1 = ‚„ Λ0 , ‚σ Λ0 = ‚„ Λ1 . (157)

In terms of the light-cone co-ordinates u = „ ’σ or v = „ +σ this can be written:

‚v (Λ1 ’ Λ0 ) = 0.

‚u (Λ1 + Λ0 ) = 0, (158)

As a result, the algebras are identical for parameters living on only one branch

of the (two-dimensional) light-cone:

Λ0 (u, v) = Λ+ (v) ’ Λ’ (u), Λ1 (u, v) = Λ+ (v) + Λ’ (u), (159)

with Λ± = (Λ1 ± Λ0 )/2.

2 Canonical BRST Construction

Many interesting physical theories incorporate constraints arising from a local

gauge symmetry, which forces certain components of the momenta to vanish

Aspects of BRST Quantization 127

in the physical phase space. For reparametrization-invariant systems (like the

relativistic particle or the relativistic string) these constraints are quadratic in

the momenta, whereas in abelian or non-abelian gauge theories of Maxwell“

Yang“Mills type they are linear in the momenta (i.e., in the electric components

of the ¬eld strength).

There are several ways to deal with such constraints. The most obvious one

is to solve them and formulate the theory purely in terms of physical degrees of

freedom. However, this is possible only in the simplest cases, like the relativis-

tic particle or an unbroken abelian gauge theory (electrodynamics). And even

then, there can arise complications such as non-local interactions. Therefore, an

alternative strategy is more fruitful in most cases and for most applications; this

preferred strategy is to keep (some) unphysical degrees of freedom in the theory

in such a way that desirable properties of the description “ like locality, and

rotation or Lorentz-invariance “ can be preserved at intermediate stages of the

calculations. In this section we discuss methods for dealing with such a situa-

tion, when unphysical degrees of freedom are taken along in the analysis of the

dynamics.

The central idea of the BRST construction is to identify the solutions of

the constraints with the cohomology classes of a certain nilpotent operator, the

BRST operator „¦. To construct this operator we introduce a new class of vari-

ables, the ghost variables. For the theories we have discussed in Sect. 1, which

do not involve fermion ¬elds in an essential way (at least from the point of

view of constraints), the ghosts are anticommuting variables: odd elements of a

Grassmann algebra. However, theories with more general types of gauge sym-

metries involving fermionic degrees of freedom, like supersymmetry or Siegel™s

κ-invariance in the theory of superparticles and superstrings, or theories with

reducible gauge symmetries, require commuting ghost variables as well. Never-

theless, to bring out the central ideas of the BRST construction as clearly as

possible, here we discuss theories with bosonic symmetries only.

2.1 Grassmann Variables

The BRST construction involves anticommuting variables, which are odd ele-

ments of a Grassmann algebra. The theory of such variables plays an important

role in quantum ¬eld theory, most prominently in the description of fermion

¬elds as they naturally describe systems satisfying the Pauli exclusion principle.

For these reasons we brie¬‚y review the basic elements of the theory of anticom-

muting variables at this point. For more detailed expositions we refer to the

references [18,19].

A Grassmann algebra of rank n is the set of polynomials constructed from

elements {e, θ1 , ..., θn } with the properties

e2 = e, eθi = θi e = θi , θi θj + θj θi = 0. (160)

Thus, e is the identity element, which will often not be written out explicitly.

2

The elements θi are nilpotent, θi = 0, whilst for i = j the elements θi and θj

128 J.W. van Holten

anticommute. As a result, a general element of the algebra consists of 2n terms

and takes the form

n n

1 ij

i

g = ±e + ± θi + ± θi θj + ... + ± θ1 ...θn ,

˜ (161)

2!

i=1 (i,j)=1

where the coe¬cients ±i1 ..ip are completely antisymmetric in the indices. The

elements {θi } are called the generators of the algebra. An obvious example of

a Grassmann algebra is the algebra of di¬erential forms on an n-dimensional

manifold.

On the Grassmann algebra we can de¬ne a co-algebra of polynomials in

¯ ¯

elements θ1 , ..., θn , which together with the unit element e is a Grassmann

algebra by itself, but which in addition has the property

¯

¯ ¯

[θi , θj ]+ = θi θj + θj θi = δj e.

i

(162)

This algebra can be interpreted as the algebra of derivations on the Grassmann

algebra spanned by (e, θi ).

¯

By the property (162), the complete set of elements e; θi ; θi is actually

turned into a Cli¬ord algebra, which has a (basically unique) representation in

terms of Dirac matrices in 2n-dimensional space. The relation can be estab-

lished by considering the following complex linear combinations of Grassmann

generators:

¯ ¯

“i = γi+n = i θi ’ θi ,

˜

“i = γi = θi + θi , i = 1, ..., n. (163)

By construction, these elements satisfy the relation

[γa , γb ]+ = 2 δab e, (a, b) = 1, ..., 2n, (164)

but actually the subsets {“i } and ˜

“i de¬ne two mutually anti-commuting

Cli¬ord algebras of rank n:

˜˜ ˜

[“i , “j ]+ = [“i , “j ]+ = 2 δij , [“i , “j ]+ = 0. (165)

Of course, the construction can be turned around to construct a Grassmann

algebra of rank n and its co-algebra of derivations out of a Cli¬ord algebra of

rank 2n.

In ¬eld theory applications we are mostly interested in Grassmann algebras

of in¬nite rank, not only n ’ ∞, but particularly also the continuous case

¯

[θ(t), θ(s)]+ = δ(t ’ s), (166)

where (s, t) are real-valued arguments. Obviously, a Grassmann variable ξ is a

quantity taking values in a set of linear Grassmann forms i ±i θi or its con-

tinuous generalization t ±(t) θ(t). Similarly, one can de¬ne derivative operators

‚/‚ξ as linear operators mapping Grassmann forms of rank p into forms of rank

p ’ 1, by

‚ ‚

ξ =1’ξ , (167)

‚ξ ‚ξ

Aspects of BRST Quantization 129

and its generalization for systems of multi-Grassmann variables. These derivative

¯ ¯

operators can be constructed as linear forms in θi or θ(t).

In addition to di¬erentiation one can also de¬ne Grassmann integration. In

fact, Grassmann integration is de¬ned as identical with Grassmann di¬erentia-

tion. For a single Grassmann variable, let f (ξ) = f0 + ξf1 ; then one de¬nes

dξ f (ξ) = f1 . (168)

This de¬nition satis¬es all standard properties of inde¬nite integrals:

1. linearity:

dξ [±f (ξ) + βg(ξ)] = ± dξ f (ξ) + β dξ g(ξ); (169)

2. translation invariance:

dξ f (ξ + ·) = dξ f (ξ); (170)

3. fundamental theorem of calculus (Gauss“Stokes):

‚f

dξ = 0; (171)

‚ξ

4. reality: for real functions f (ξ) (i.e. f0,1 ∈ R)

dξf (ξ) = f1 ∈ R. (172)

A particularly useful result is the evaluation of Gaussian Grassmann integrals.

First we observe that

[dξ1 ...dξn ] ξ±1 ...ξ±n = µ±1 ...±n . (173)

From this it follows, that a general Gaussian Grassmann integral is

1

= ± | det A|.

[dξ1 ...dξn ] exp ξ± A±β ξβ (174)

2

This is quite obvious after bringing A into block-diagonal form:

«

0 ω1

0·

¬ ’ω1 0

¬ ·

¬ ·

0 ω2

A=¬ ·. (175)

¬ ·

’ω2 0

¬ ·

·

0

·

130 J.W. van Holten

There are then two possibilities:

(i) If the dimensionality of the matrix A is even [(±, β) = 1, ..., 2r] and none of

the characteristic values ωi vanishes, then every 2 — 2 block gives a contribution

2ωi to the exponential:

r r

1

exp ξ± A±β ξβ = exp ωi ξ2i’1 ξ2i = 1 + ... + (ωi ξ2i’1 ξ2i ). (176)

2 i=1 i=1

The ¬nal result is then established by performing the Grassmann integrations,

which leaves a non-zero contribution only from the last term, reading

r

ωi = ± | det A| (177)

i=1

with the sign depending on the number of negative characteristic values ωi .

(ii) If the dimensionality of A is odd, the last block is one-dimensional represent-

ing a zero-mode; then the integral vanishes, as does the determinant. Of course,

the same is true for even-dimensional A if one of the values ωi vanishes.

Another useful result is, that one can de¬ne a Grassmann-valued delta-

function:

δ(ξ ’ ξ ) = ’δ(ξ ’ ξ) = ξ ’ ξ , (178)

with the properties

dξ δ(ξ ’ ξ ) = 1, dξ δ(ξ ’ ξ )f (ξ) = f (ξ ). (179)

The proof follows simply by writing out the integrants and using the fundamental

rule of integration (168).

2.2 Classical BRST Transformations

Consider again a general dynamical system subject to a set of constraints G± = 0,

as de¬ned in (41) or (60). We take the algebra of constraints to be ¬rst-class, as

in (69):

{G± , Gβ } = P±β (G), {G± , H} = Z± (G). (180)

Here P (G) and Z(G) are polynomial expressions in the constraints, such that

P (0) = Z(0) = 0; in particular this implies that the constant terms vanish: