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the same lines of argument as presented for classical theories in Sect. 1.3. Con-
sider a theory of canonical pairs of operators (ˆ, p) with commutation relations


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:

= {¦, H} , (91)

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(») = .

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)

δcov e = .

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