sider a theory of canonical pairs of operators (ˆ, p) with commutation relations

qˆ

q i , pj = iδj ,

i

ˆˆ (74)

ˆqˆ

and a hamiltonian H(ˆ, p) such that

dˆ i

q dˆi

p

ˆˆ ˆˆ

= q i, H ,

i i = pi , H . (75)

dt dt

The δ-symbol on the right-hand side of (74) is to be interpreted in a generalized

sense: for continuous parameters (i, j) it represents a Dirac delta-function rather

than a Kronecker delta.

ˆ

In the context of quantum theory, constants of motion become operators G

which commute with the hamiltonian:

ˆ

ˆ H = i dG = 0,

ˆ

G, (76)

dt

and therefore can be diagonalized on stationary eigenstates. We henceforth as-

sume we have at our disposal a complete set {G± } of such constants of motion,

ˆ

in the sense that any operator satisfying (76) can be expanded as a polynomial

ˆ

in the operators G± .

In analogy to the classical theory, we de¬ne in¬nitesimal symmetry transfor-

mations by

δ± q i = ’i q i , G± , δ± pi = ’i pi , G± .

ˆˆ ˆˆ

ˆ ˆ (77)

By construction they have the property of leaving the hamiltonian invariant:

δ± H = ’i H, G± = 0.

ˆ ˆˆ (78)

ˆ

Therefore, the operators G± are also called symmetry generators. It follows by

the Jacobi identity, analogous to (61), that the commutator of two such gener-

ators commutes again with the hamiltonian, and therefore

’i G± , Gβ = P±β (G) = c±β + f±βγ Gγ + ....

ˆˆ ˆ ˆ (79)

ˆqˆ

A calculation along the lines of (65) then shows, that for any operator F (ˆ, p)

one has

[δ± , δβ ] F = if±βγ δγ F + ...

δ± F = ’i F , G± ,

ˆ ˆˆ ˆ ˆ (80)

Observe, that compared to the classical theory, in the quantum theory there

is an additional potential source for the appearance of central charges in (79),

to wit the operator ordering on the right-hand side. As a result, even when no

central charge is present in the classical theory, such central charges can arise in

114 J.W. van Holten

the quantum theory. This is a source of anomalous behaviour of symmetries in

quantum theory.

As in the classical theory, local symmetries impose additional restrictions; if

ˆ

a symmetry generator G[ ] involves time-dependent parameters a (t), then its

evolution equation (76) is modi¬ed to:

ˆ ˆ

dG[ ] ‚ G[ ]

ˆ ˆ

i = G[ ], H + i , (81)

dt ‚t

where

ˆ ˆ

‚ a δ G[ ]

‚ G[ ]

= . (82)

‚t δ a

‚t

ˆ

It follows, that G[ ] can generate symmetries of the hamiltonian and be conserved

at the same time for arbitrary a (t) only if the functional derivative vanishes:

ˆ

δ G[ ]

= 0, (83)

δ a (t)

which de¬nes a set of operator constraints, the quantum equivalent of (44). The

important step in this argument is to realize, that the transformation properties

of the evolution operator should be consistent with the Schr¨dinger equation,

o

which can be true only if both conditions (symmetry and conservation law) hold.

To see this, recall that the evolution operator

U (t, t ) = e’i(t’t )H ,

ˆ

ˆ (84)

is the formal solution of the Schr¨dinger equation

o

‚

’ H U = 0,

ˆˆ

i (85)

‚t

ˆ

satisfying the initial condition U (t, t) = ˆ Now under a symmetry transforma-

1.

tion (77) and (80), this equation transforms into

‚ ‚

’ H U = ’i ’ H U , G[ ]

ˆˆ ˆ ˆˆ

δ i i

‚t ‚t

(86)

‚ ‚

= ’i i ’H U , G[ ] ’ i ’ H , G[ ] U

ˆ ˆˆ ˆ ˆ ˆ

i

‚t ‚t

For the transformations to respect the Schr¨dinger equation, the left-hand side of

o

this identity must vanish, hence so must the right-hand side. But the right-hand

side vanishes for arbitrary (t) if and only if both conditions are met:

ˆ

‚ G[ ]

ˆˆ

H, G[ ] = 0, and = 0.

‚t

This is what we set out to prove. Of course, like in the classical hamiltonian

formulation, we realize that for generators of local symmetries a more general

Aspects of BRST Quantization 115

¬rst-class algebra of commutation relations is allowed, along the lines of (69).

Also here, the hamiltonian may then be modi¬ed by terms involving only the

constraints and, possibly, corresponding Lagrange multipliers. The discussion

parallels that for the classical case.

1.5 The Relativistic Particle

In this section and the next we revisit the two examples of constrained systems

discussed in Sect. 1.1 to illustrate the general principles of symmetries, conser-

vation laws, and constraints. First we consider the relativistic particle.

The starting point of the analysis is the action (8):

2

1 dxµ dxµ

m

’ ec2 d».

µ

S[x ; e] =

2 e d» d»

1

Here » plays the role of system time, and the hamiltonian we construct is the

one generating time-evolution in this sense. The canonical momenta are given

by

δS m dxµ δS

pµ = = , pe = = 0. (87)

δ(dxµ /d») e d» δ(de/d»)

The second equation is a constraint on the extended phase space spanned by

the canonical pairs (xµ , pµ ; e, pe ). Next we perform a Legendre transformation

to obtain the hamiltonian

e de

p2 + m2 c2 + pe

H= . (88)

2m d»

The last term obviously vanishes upon application of the constraint pe = 0. The

canonical (hamiltonian) action now reads

2

dxµ e

’ p2 + m2 c2

Scan = d» pµ . (89)

d» 2m

1

Observe, that the dependence on pe has dropped out, irrespective of whether we

constrain it to vanish or not. The role of the einbein is now clear: it is a Lagrange

multiplier imposing the dynamical constraint (7):

p2 + m2 c2 = 0.

Note, that in combination with pe = 0, this constraint implies H = 0, i.e. the

hamiltonian consists only of a polynomial in the constraints. This is a general

feature of systems with reparametrization invariance, including for example the

theory of relativistic strings and general relativity.

In the example of the relativistic particle, we immediately encounter a generic

phenomenon: any time we have a constraint on the dynamical variables imposed

by a Lagrange multiplier (here: e), its associated momentum (here: pe ) is con-

strained to vanish. It has been shown in a quite general context, that one may al-

ways reformulate hamiltonian theories with constraints such that all constraints

116 J.W. van Holten

appear with Lagrange multipliers [16]; therefore this pairing of constraints is a

generic feature in hamiltonian dynamics. However, as we have already discussed

in Sect. 1.3, Lagrange multiplier terms do not a¬ect the dynamics, and the mul-

tipliers as well as their associated momenta can be eliminated from the physical

hamiltonian.

The non-vanishing Poisson brackets of the theory, including the Lagrange

multipliers, are

{xµ , pν } = δν , {e, pe } = 1.

µ

(90)

As follows from the hamiltonian treatment, all equations of motion for any quan-

tity ¦(x, p; e, pe ) can then be obtained from a Poisson bracket with the hamilto-

nian:

d¦

= {¦, H} , (91)

d»

although this equation does not imply any non-trivial information on the dynam-

ics of the Lagrange multipliers. Nevertheless, in this formulation of the theory it

must be assumed a priori that (e, pe ) are allowed to vary; the dynamics can be

projected to the hypersurface pe = 0 only after computing the Poisson brackets.

The alternative is to work with a restricted phase space spanned only by the

physical co-ordinates and momenta (xµ , pµ ). This is achieved by performing a

Legendre transformation only with respect to the physical velocities3 . We ¬rst

explore the formulation of the theory in the extended phase space.

All possible symmetries of the theory can be determined by solving (56):

{G, H} = 0.

Among the solutions we ¬nd the generators of the Poincar´ group: translations

e

pµ and Lorentz transformations Mµν = xν pµ ’ xµ pν . Indeed, the combination

of generators

1 µν

G[ ] = µ pµ + Mµν . (92)

2

with constant ( µ , µν ) produces the expected in¬nitesimal transformations

δxµ = {xµ , G[ ]} = δpµ = {pµ , G[ ]} =

µ µ

xν , ν

+ pν . (93)

ν µ

The commutator algebra of these transformations is well-known to be closed: it

is the Lie algebra of the Poincar´ group.

e

For the generation of constraints the local reparametrization invariance of the

theory is the one of interest. The in¬nitesimal form of the transformations (10)

is obtained by taking » = » ’ (»), with the result

dxµ dpµ

δx = x (») ’ x (») =

µ µ µ

, δpµ = ,

d» d»

(94)

d(e )

δe = e (») ’ e(») = .

d»

3

This is basically a variant of Routh™s procedure; see e.g. Goldstein [15], Chap. 7.

Aspects of BRST Quantization 117

Now recall that ed» = d„ is a reparametrization-invariant form. Furthermore,

(») is an arbitrary local function of ». It follows, that without loss of generality

we can consider an equivalent set of covariant transformations with parameter

σ=e :

σ dxµ σ dpµ

δcov xµ = , δcov pµ = ,

e d» e d»

(95)

dσ

δcov e = .

d»