™™ ‚q ‚ q

™ ‚q

™ dt

Aspects of BRST Quantization 109

By antisymmetrization this immediately gives

‚L ‚L d

(δ2 B1 ’ δ1 B2 ) .

[δ2 , δ1 ] L = [δ1 , δ2 ] q i + [δ2 , δ1 ] q i

™ = (46)

‚q i ‚ qi

™ dt

By assumption of the completeness of the set of symmetry transformations it

follows, that there must exist a symmetry transformation

δ3 q i = [δ2 , δ1 ] q i , δ3 q i = [δ2 , δ1 ] q i ,

™ ™ (47)

with the property that the associated B3 = δ2 B1 ’ δ1 B2 . Implementing these

conditions gives

j j

i i i

j ‚R1 k ‚R2 ‚R1 k ‚R2 ‚R1

’ [1 ” 2] = R3 ,

i i

[δ2 , δ1 ] q = R2 j + q

™ +q

¨ (48)

‚q k j k ‚ qj

‚q ‚q

™ ‚q

™ ™

where we use a condensed notation Ra ≡ Ri [ a ], a = 1, 2, 3. In all standard

i

cases, the symmetry transformations δq i = Ri involve only the coordinates and

velocities: Ri = Ri (q, q). Then R3 cannot contain terms proportional to q , and

™ ¨

the conditions (48) reduce to two separate conditions

j j

i i i i

j ‚R1 j ‚R2 ‚R2 ‚R1 ‚R1 ‚R2

’ ’

k i

R2 j R1 j +q

™ = R3 ,

‚q k ‚ q j ‚q k ‚ q j

‚q ‚q ™ ™

(49)

j j

i i

‚R2 ‚R1 ‚R1 ‚R2

’ = 0.

‚ qk ‚ qj ‚ qk ‚ qj

™ ™ ™ ™

Clearly, the parameter 3 of the transformation on the right-hand side must be

an antisymmetric bilinear combination of the other two parameters:

= ’f ± ( 2 ,

±

= f ±( 1, 2) 1 ). (50)

3

1.3 Canonical Formalism

The canonical formalism describes dynamics in terms of phase-space coordinates

(q i , pi ) and a hamiltonian H(q, p), starting from an action

2

pi q i ’ H(q, p) dt.

Scan [q, p] = ™ (51)

1

Variations of the phase-space coordinates change the action to ¬rst order by

2

‚H ‚H d

dt δpi q i ’ ’ δq i pi + pi δq i

δScan = ™ ™ + . (52)

‚q i

‚pi dt

1

The action is stationary under variations vanishing at times (t1 , t2 ) if Hamilton™s

equations of motion are satis¬ed:

‚H ‚H

qi = ’

pi =

™ , ™ . (53)

‚q i ‚pi

110 J.W. van Holten

This motivates the introduction of the Poisson brackets

‚F ‚G ‚F ‚G

{F, G} = ’ , (54)

‚q i ‚pi ‚pi ‚q i

which allow us to write the time derivative of any phase-space function G(q, p)

as

‚G ‚G

= {G, H} .

™

G = q i i + pi

™ ™ (55)

‚q ‚pi

It follows immediately that G is a constant of motion if and only if

{G, H} = 0 (56)

everywhere along the trajectory of the physical system in phase space. This is

guaranteed to be the case if (56) holds everywhere in phase space, but as we

discuss below, more subtle situations can arise.

Suppose (56) is satis¬ed; then we can construct variations of (q, p) de¬ned

by

‚G ‚G

δpi = {pi , G} = ’ i ,

δq i = q i , G = , (57)

‚pi ‚q

which leave the hamiltonian invariant:

‚H ‚H ‚G ‚H ‚G ‚H

’i = {H, G} = 0.

δH = δq i + δpi = (58)

‚q i ‚pi ‚q i

‚pi ‚q ‚pi

They represent in¬nitesimal symmetries of the theory provided (56), and hence

(58), is satis¬ed as an identity, irrespective of whether or not the phase-space co-

ordinates (q, p) satisfy the equations of motion. To see this, consider the variation

of the action (52) with (δq, δp) given by (57) and δH = 0 by (58):

2 2

‚G ‚G d ‚G d

‚G

dt ’ i q i ’ pi ’ G .

δScan = ™ pi +

™ pi = dt

‚q ‚pi dt ‚pi dt

‚pi

1 1

(59)

If we call the quantity inside the parentheses B(q, p), then we have rederived (37)

and (38); indeed, we then have

‚G

pi ’ B = δq i pi ’ B,

G= (60)

‚pi

where we know from (55), that G is a constant of motion on classical trajectories

(on which Hamilton™s equations of motion are satis¬ed). Observe that “ whereas

in the lagrangean approach we showed that symmetries imply constants of mo-

tion “ here we have derived the inverse Noether theorem: constants of motion

generate symmetries. An advantage of this derivation over the lagrangean one

is, that we have also found explicit expressions for the variations (δq, δp).

A further advantage is, that the in¬nitesimal group structure of the tran-

formations (the commutator algebra) can be checked directly. Indeed, if two

Aspects of BRST Quantization 111

symmetry generators G± and Gβ both satisfy (56), then the Jacobi identity for

Poisson brackets implies

{{G± , Gβ } , H} = {G± , {Gβ , H}} ’ {Gβ , {G± , H}} = 0. (61)

Hence if the set of generators {G± } is complete, we must have an identity of the

form

{G± , Gβ } = P±β (G) = ’Pβ± (G) , (62)

where the P±β (G) are polynomials in the constants of motion G± :

1

P±β (G) = c±β + f±βγ Gγ + g±βγδ Gγ Gδ + .... (63)

2

The coe¬cients c±β , f±βγ , g±βγδ , ... are constants, having zero Poisson brackets

with any phase-space function. As such the ¬rst term c±β may be called a central

charge.

It now follows that the transformation of any phase-space function F (q, p),

given by

δ± F = {F, G± } , (64)

satis¬es the commutation relation

[δ± , δβ ] F = {{F, Gβ } , G± } ’ {{F, G± } , Gβ } = {F, {Gβ , G± }}

(65)

γ

= Cβ± (G) δγ F,

where we have introduced the notation

‚Pβ± (G)

γ

= f±βγ + g±βγδ Gδ + ....

Cβ± (G) = (66)

‚Gγ

In particular this holds for the coordinates and momenta (q, p) themselves; taking

F to be another constraint Gγ , we ¬nd from the Jacobi identity for Poisson

brackets the consistency condition

C[±β P γ]δ = f[±βδ c γ]δ + f[±βδ f γ]δµ + g[±β c γ]δ Gµ + .... = 0.

δ δ

(67)

By the same arguments as in Sect. 1.2 (cf. (41 and following), it is estab-

lished, that whenever the theory generated by G± is a local symmetry with

time-dependent parameters, the generator G± turns into a constraint:

G± (q, p) = 0. (68)

However, compared to the case of rigid symmetries, a subtlety now arises: the

constraints G± = 0 de¬ne a hypersurface in the phase space to which all physical

trajectories of the system are con¬ned. This implies that it is su¬cient for the

constraints to commute with the hamiltonian (in the sense of Poisson brackets)

on the physical hypersurface (i.e. on shell). O¬ the hypersurface (i.e. o¬ shell),

the bracket of the hamiltonian with the constraints can be anything, as the

112 J.W. van Holten

physical trajectories never enter this part of phase space. Thus, the most general

allowed algebraic structure de¬ned by the hamiltonian and the constraints is

{G± , Gβ } = P±β (G), {H, G± } = Z± (G), (69)

where both P±β (G) and Z± (G) are polynomials in the constraints with the prop-

erty that P±β (0) = Z± (0) = 0. This is su¬cient to guarantee that in the physical

sector of the phase space {H, G± }|G=0 = 0. Note, that in the case of local symme-

tries with generators G± de¬ning constraints, the central charge in the bracket

of the constraints must vanish: c±β = 0. This is a genuine restriction on the

existence of local symmetries. A dynamical system with constraints and hamil-

tonian satisfying (69) is said to be ¬rst class. Actually, it is quite easy to see

that the general ¬rst-class algebra of Poisson brackets is more appropriate for

systems with local symmetries. Namely, even if the brackets of the constraints

and the hamiltonian genuinely vanish on and o¬ shell, one can always change

the hamiltonian of the system by adding a polynomial in the constraints:

1 ±β

R(G) = ρ0 + ρ± G± +

H = H + R(G), ρ G± Gβ + ... (70)

1

22

This leaves the hamiltonian on the physical shell in phase space invariant (up

to a constant ρ0 ), and therefore the physical trajectories remain the same. Fur-

thermore, even if {H, G± } = 0, the new hamiltonian satis¬es

{H , G± } = {R(G), G± } = Z± (G) ≡ ρβ Pβ± (G) + ...,

(R)

(71)

1

which is of the form (69). In addition the equations of motion for the variables

(q, p) are changed by a local symmetry transformation only, as

‚R

(q™i ) = q i , H = q i , H + q i , G± = q i + µ± δ ± q i ,

™ (72)

‚G±

where µ± are some “ possibly complicated “ local functions which may depend

on the phase-space coordinates (q, p) themselves. A similar observation holds

of course for the momenta pi . We can actually allow the coe¬cients ρ± , ρ±β , ...

1 2

to be space-time dependent variables themselves, as this does not change the

general form of the equations of motion (72), whilst variation of the action with

respect to these new variables will only impose the constraints as equations of

motion:

δS

= G± (q, p) = 0, (73)

δρ±1

in agreement with the dynamics already established.

The same argument shows however, that the part of the hamiltonian depend-

ing on the constraints in not unique, and may be changed by terms like R(G).

In many cases this allows one to get rid of all or part of Z± (G).

Aspects of BRST Quantization 113

1.4 Quantum Dynamics

In quantum dynamics in the canonical operator formalism, one can follow largely