3.9
The Wheeler-DeWitt equation
The approaches to quantization of Sections
3.1,
3.2,
3.3,
3.4, and
3.5
share an important feature: All are “reduced phase space”
quantizations, quantum theories based on the true physical
degrees of freedom of the classical theory. That is, the
classical constraints have been solved before quantizing,
eliminating classically redundant “gauge” degrees of freedom. In
Dirac’s approach to quantization
[112, 113, 114], in contrast, one quantizes the entire space of degrees of
freedom of classical theory, and only then imposes the
constraints. States are initially determined from the full
classical phase space; in the ADM formulation of quantum gravity,
for instance, they are functionals
of the full spatial metric. The constraints then act as
operators on this auxiliary Hilbert space; the physical Hilbert
space consists of those states that are annihilated by the
constraints, with a suitable new inner product, acted on by
physical operators that commute with the constraints. For
gravity, in particular, the Hamiltonian constraint acting on
states leads to a functional differential equation, the
Wheeler-DeWitt equation
[110, 276]
.
In the first order formalism, it is
straightforward to show that Dirac quantization is equivalent to
the Chern-Simons quantum theory we have already seen. Details can
be found in Chapter 8 of
[81
], but the basic argument is fairly clear: At least for
, the first order constraints coming from Equations (15,
16) are at most linear in the momenta, and are thus uncomplicated
to solve.
In the second order formalism, matters become
considerably more complicated
[73]
. We begin with a wave function
, upon which we wish to impose the constraints (30), with momenta acting as functional derivatives,
The first difficulty is that we are no longer allowed to choose a
nice time-slicing such as York time; that would be a form of
gauge-fixing, and is not permitted in Dirac quantization. We can
still decompose the spatial metric and momentum as in
Equation (31), but only up to a spatial diffeomorphism, which depends on an
undetermined vector field
appearing in the momentum
[206]
. The momentum constraint fixes
in terms of the scale factor
, but it does so nonlocally. As a consequence, the Hamiltonian
constraint becomes a nonlocal functional differential equation,
and very little is understood about its solutions, even for the
simplest case of the torus universe. Further complications come
from the fact that the inner product on the space of solutions of
the Wheeler-DeWitt equation must be gauge-fixed
[282, 144]
; again, little is understood about the resulting Hilbert space.
In view of the difficulty in finding exact
solutions to the Wheeler-DeWitt equation, it is natural to look
for perturbative methods, for example an expansion in powers of
Newton’s constant
. One can solve the momentum constraints order by order by
insisting that each term depend only on (nonlocal) spatially
diffeomorphism-invariant quantities. Such an expansion has been
studied by Banks, Fischler, and Susskind for the physically
trivial topology
[42], following much earlier work by Leutwyler
[176]
. Even in this simple case, computations quickly become extremely
difficult. Other attempts have been made
[47
, 186]
to write a discrete version of the Wheeler-DeWitt equation in
the Ponzano-Regge formalism of Section
3.6
. This approach has the advantage that the spatial
diffeomorphisms have already been largely eliminated, removing
the main source of nonlocality discussed above. The
Wheeler-DeWitt-like equation in
[47]
has been shown to agree with the the Ponzano-Regge model.