. 26
( 78 .)


It is straightforward to check that under these transformations the canoni-
cal lagrangean (the integrand of (89)) transforms into a total derivative, and
δcov Scan = [Bcov ]2 with

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

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

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

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
δ xµ = (δcov ’ δG )xµ = ’ pµ ≈ 0,
m e d»
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
Scan |e=1 = p·x’ p2 + m2 c2
d„ ™ . (101)

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
{xµ , pν } = δν ,
p2 + m2 c2 , µ
Hf = (102)
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
p2 = p 2 + m2 c2 . (103)

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:
Sem [φ, A, E] = dt Lem (φ, A, E),
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:

π A = ’E = (107)

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:

πφ = = 0. (108)

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
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
‚A ‚φ
d3 x ’E · ’ Hem (E, A, πφ , φ)
Sem = dt + πφ
‚t ‚t
d3 x ’E · ’ Hem (E, A, φ)|πφ =0
= dt
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
δA = ∇Λ, δφ = , δE = δB = 0. (113)
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] =
‚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),

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
E2 + B2 , {Ai (x, t), Ej (y, t)} = ’δij δ 3 (x ’ y).
d3 x
Hred = (119)
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:


. 26
( 78 .)