###
3.3
Covariant canonical quantization

The technique of Chern-Simons quantization relies on special
features of general relativity in 2+1 dimensions, and does not
readily generalize to higher dimensions. It is, however, closely
related to a much more general approach, covariant canonical
quantization
[31, 28, 98, 175, 272], or “quantization of the space of classical solutions”.
Our starting point is the observation that the
phase space of a well-behaved classical theory is isomorphic to
the space of classical solutions. Indeed, if
is an arbitrary Cauchy surface, then a point in the phase space
determines initial data on
, which can be evolved to give a unique solution, while,
conversely, a classical solution restricted to
determines a point in the phase space. Moreover, the space of
solutions has natural symplectic structure
[175, 272], which can be shown to be equivalent to the standard symplectic
structure on phase space. For the case of (2+1)-dimensional
gravity, this equivalence is demonstrated in Section 6.1
of
[81]
.

For (2+1)-dimensional gravity, the space of
classical solutions is the space of geometric strictures of
Section
2.2
. If we restrict our attention to spacetimes with the topology
with
closed and
, the holonomies of a geometric structure determine a unique
maximal domain of dependence
[200], exactly the right setting for covariant canonical quantization.
But as we saw in Section
2.3, the holonomies of a geometric structure are precisely the
holonomies of the Chern-Simons formalism, and the symplectic
structures are the same as well. Thus in this setting,
Chern-Simons quantization
is
covariant canonical quantization. If
or point particles are present, the holonomies do not quite
determine a unique geometric structure, and the Chern-Simons
theory is not quite equivalent to general relativity. In that
case, additional discrete variables might be necessary; see, for
example,
[117]
for the case of a torus universe with
.

As we shall see in Section
3.4, the construction of dynamical observables and time-dependent
states in covariant canonical quantum theory requires an explicit
isomorphism between the phase space and the space of classical
solutions. For the torus universe, such an isomorphism is known.
For higher genus spaces, however - and certainly for realistic
(3+1)-dimensional gravity - it is not
[207]
. Often, however, we can determine such an isomorphism
perturbatively in the neighborhood of a known classical solution.
This raises the interesting question, so far answered only in
simple models
[50], of whether classical perturbation theory can be used to define
a perturbative covariant canonical quantum theory.