cal lagrangean (the integrand of (89)) transforms into a total derivative, and

δcov Scan = [Bcov ]2 with

1

dxµ 12

’ (p + m2 c2 ) .

Bcov [σ] = σ pµ (96)

ed» 2m

Using (60), we ¬nd that the generator of the local transformations (94) is given

by

σ dσ

Gcov [σ] = (δcov xµ )pµ + (δcov e)pe ’ Bcov = p2 + m2 c2 + pe . (97)

2m d»

It is easily veri¬ed that dGcov /d» = 0 on physical trajectories for arbitrary σ(»)

if and only if the two earlier constraints are satis¬ed at all times:

p2 + m2 c2 = 0, pe = 0. (98)

It is also clear that the Poisson brackets of these constraints among themselves

vanish. On the canonical variables, Gcov generates the transformations

σpµ

δG xµ = {xµ , Gcov [σ]} = , δG pµ = {pµ , Gcov [σ]} = 0,

m

(99)

dσ

δG e = {e, Gcov [σ]} = δG pe = {pe , Gcov [σ]} = 0.

,

d»

These transformation rules actually di¬er from the original ones, cf. (95). How-

ever, all di¬erences vanish when applying the equations of motion:

m dxµ

σ

δ xµ = (δcov ’ δG )xµ = ’ pµ ≈ 0,

m e d»

(100)

σ dpµ

δ pµ = (δcov ’ δG )pµ = ≈ 0.

e d»

The transformations δ are in fact themselves symmetry transformations of the

canonical action, but of a trivial kind: as they vanish on shell, they do not imply

any conservation laws or constraints [17]. Therefore, the new transformations δG

are physically equivalent to δcov .

118 J.W. van Holten

The upshot of this analysis is, that we can describe the relativistic particle

by the hamiltonian (88) and the Poisson brackets (90), provided we impose on

all physical quantities in phase space the constraints (98).

A few comments are in order. First, the hamiltonian is by construction the

generator of translations in the time coordinate (here: »); therefore, after the

general exposure in Sects. 1.2 and 1.3, it should not come as a surprise, that when

promoting such translations to a local symmetry, the hamiltonian is constrained

to vanish.

Secondly, we brie¬‚y discuss the other canonical procedure, which takes direct

advantage of the the local parametrization invariance (10) by using it to ¬x the

einbein; in particular, the choice e = 1 leads to the identi¬cation of » with

proper time: d„ = ed» ’ d„ = d». This procedure is called gauge ¬xing. Now

the canonical action becomes simply

2

1

Scan |e=1 = p·x’ p2 + m2 c2

d„ ™ . (101)

2m

1

This is a regular action for a hamiltonian system. It is completely Lorentz covari-

ant, only the local reparametrization invariance is lost. As a result, the constraint

p2 + m2 c2 = 0 can no longer be derived from the action; it must now be imposed

separately as an external condition. Because we have ¬xed e, we do not need to

introduce its conjugate momentum pe , and we can work in a restricted physical

phase space spanned by the canonical pairs (xµ , pµ ). Thus, a second consistent

way to formulate classical hamiltonian dynamics for the relativistic particle is

to use the gauge-¬xed hamiltonian and Poisson brackets

1

{xµ , pν } = δν ,

p2 + m2 c2 , µ

Hf = (102)

2m

whilst adding the constraint Hf = 0 to be satis¬ed at all (proper) times. Ob-

serve, that the remaining constraint implies that one of the momenta pµ is not

independent:

p2 = p 2 + m2 c2 . (103)

0

As this de¬nes a hypersurface in the restricted phase space, the dimensionality of

the physical phase space is reduced even further. To deal with this situation, we

can again follow two di¬erent routes; the ¬rst one is to solve the constraint and

work in a reduced phase space. The standard procedure for this is to introduce

√

light-cone coordinates x± = (x0 ± x3 )/ 2, with canonically conjugate momenta

√

p± = (p0 ± p3 )/ 2, such that

x± , p± = 1, x± , p“ = 0. (104)

The constraint (103) can then be written

2p+ p’ = p2 + p2 + m2 c2 , (105)

1 2

which allows us to eliminate the light-cone co-ordinate x’ and its conjugate

momentum p’ = (p2 +p2 +m2 c2 )/2p+ . Of course, by this procedure the manifest

1 2

Aspects of BRST Quantization 119

Lorentz-covariance of the model is lost. Therefore one often prefers an alternative

route: to work in the covariant phase space (102), and impose the constraint on

physical phase space functions only after solving the dynamical equations.

1.6 The Electro-magnetic Field

The second example to be considered here is the electro-magnetic ¬eld. As our

starting point we take the action of (18) modi¬ed by a total time-derivative,

and with the magnetic ¬eld written as usual in terms of the vector potential as

B(A) = ∇ — A:

2

Sem [φ, A, E] = dt Lem (φ, A, E),

1

(106)

1 ‚A

’ E 2 + [B(A)]2 ’ φ ∇ · E ’ E ·

d3 x

Lem =

2 ‚t

It is clear, that (A, ’E) are canonically conjugate; by adding the time derivative

we have chosen to let A play the role of co-ordinates, whilst the components of

’E represent the momenta:

δSem

π A = ’E = (107)

δ(‚A/‚t)

Also, like the einbein in the case of the relativistic particle, here the scalar

potential φ = A0 plays the role of Lagrange multiplier to impose the constraint

∇ · E = 0; therefore its canonical momentum vanishes:

δSem

πφ = = 0. (108)

δ(‚φ/‚t)

This is the generic type of constraint for Lagrange multipliers, which we en-

countered also in the case of the relativistic particle. Observe, that the la-

grangean (106) is already in the canonical form, with the hamiltonian given

by

1 ‚φ

E 2 + B 2 + φ ∇ · E + πφ

Hem = d3 x . (109)

2 ‚t

Again, as in the case of the relativistic particle, the last term can be taken to

vanish upon imposing the constraint (108), but in any case it cancels in the

canonical action

2

‚A ‚φ

d3 x ’E · ’ Hem (E, A, πφ , φ)

Sem = dt + πφ

‚t ‚t

1

(110)

2

‚A

d3 x ’E · ’ Hem (E, A, φ)|πφ =0

= dt

‚t

1

120 J.W. van Holten

To proceed with the canonical analysis, we have the same choice as in the case of

the particle: to keep the full hamiltonian, and include the canonical pair (φ, πφ )

in an extended phase space; or to use the local gauge invariance to remove φ by

¬xing it at some particular value.

In the ¬rst case we have to introduce Poisson brackets

{Ai (x, t), Ej (y, t)} = ’δij δ 3 (x ’ y), {φ(x, t), πφ (y, t)} = δ 3 (x ’ y). (111)

It is straightforward to check, that the Maxwell equations are reproduced by the

brackets with the hamiltonian:

¦ = {¦, H} ,

™ (112)

where ¦ stands for any of the ¬elds (A, E, φ, πφ ) above, although in the sector

of the scalar potential the equations are empty of dynamical content.

Among the quantities commuting with the hamiltonian (in the sense of Pois-

son brackets), the most interesting for our purpose is the generator of the gauge

transformations

‚Λ

δA = ∇Λ, δφ = , δE = δB = 0. (113)

‚t

Its construction proceeds according to (60). Actually, the action (106) is gauge

invariant provided the gauge parameter vanishes su¬ciently fast at spatial in-

¬nity, as δLem = ’ d3 x ∇ · (E‚Λ/‚t). Therefore the generator of the gauge

transformations is

d3 x (’δA · E + δφ πφ )

G[Λ] =

(114)

‚Λ ‚Λ

’E · ∇Λ + πφ Λ∇ · E + πφ

d3 x d3 x

= = .

‚t ‚t

The gauge transformations (113) are reproduced by the Poisson brackets

δ¦ = {¦, G[Λ]} . (115)

From the result (114) it follows, that conservation of G[Λ] for arbitrary Λ(x, t)

is due to the constraints

∇ · E = 0, πφ = 0, (116)

which are necessary and su¬cient. These in turn imply that G[Λ] = 0 itself.

One reason why this treatment might be preferred, is that in a relativistic

notation φ = A0 , πφ = π 0 , the brackets (111) take the quasi-covariant form

{Aµ (x, t), π ν (y, t)} = δµ δ 3 (x ’ y),

ν

(117)

and similarly for the generator of the gauge transformations :

G[Λ] = ’ d3 x π µ ‚µ Λ. (118)

Aspects of BRST Quantization 121

Of course, the three-dimensional δ-function and integral show, that the covari-

ance of these equations is not complete.

The other procedure one can follow, is to use the gauge invariance to set

φ = φ0 , a constant. Without loss of generality this constant can be chosen

equal to zero, which just amounts to ¬xing the zero of the electric potential. In

any case, the term φ ∇ · E vanishes from the action and for the dynamics it

su¬ces to work in the reduced phase space spanned by (A, E). In particular,

the hamiltonian and Poisson brackets reduce to

1

E2 + B2 , {Ai (x, t), Ej (y, t)} = ’δij δ 3 (x ’ y).

d3 x

Hred = (119)

2

The constraint ∇ · E = 0 is no longer a consequence of the dynamics, but

has to be imposed separately. Of course, its bracket with the hamiltonian still

vanishes: {Hred , ∇ · E} = 0. The constraint actually signi¬es that one of the

components of the canonical momenta (in fact an in¬nite set: the longitudinal

electric ¬eld at each point in space) is to vanish; therefore the dimensionality of

the physical phase space is again reduced by the constraint. As the constraint

is preserved in time (its Poisson bracket with H vanishes), this reduction is

consistent. Again, there are two options to proceed: solve the constraint and

obtain a phase space spanned by the physical degrees of freedom only, or keep

the constraint as a separate condition to be imposed on all solutions of the

dynamics. The explicit solution in this case consists of splitting the electric ¬eld

in transverse and longitudinal parts by projection operators:

1 1

E = ET + EL = 1’∇ ∇ · E + ∇ ∇ · E, (120)

∆ ∆

and similarly for the vector potential. One can now restrict the phase space to

the transverse parts of the ¬elds only; this is equivalent to requiring ∇ · E = 0

and ∇·A = 0 simultaneously. In practice it is much more convenient to use these

constraints as such in computing physical observables, instead of projecting out

the longitudinal components explicitly at all intermediate stages. Of course, one

then has to check that the ¬nal result does not depend on any arbitrary choice

of dynamics attributed to the longitudinal ¬elds.

1.7 Yang“Mills Theory

Yang“Mills theory is an important extension of Maxwell theory, with a very

similar canonical structure. The covariant action is a direct extension of the

covariant electro-magnetic action used before: