For spatial surfaces of genus
, the complexity of the constraint (32
) seems to make this approach to quantization impractical
[207
]
. A perturbative expression for
may still exist, though, as discussed in
[217, 218], and the Gauss map has been proposed as a useful tool
[226]
.
For genus one, on the other hand, a full
quantization is possible. The classical Hamiltonian (44) becomes, up to operator ordering ambiguities,
A related form of quantization comes from
reexpressing the moduli space for the torus as a quotient space
[193, 273
]
. Here, the symmetric space
describes the transverse traceless deformations of the spatial
metric, while
is the modular group. As Waldron has observed
[273], this makes it possible to reinterpret the quantum mechanical
problem as that of a fictitious free particle, with mass
proportional to
, moving in a quotient space of the (flat) three-dimensional
Milne universe. With a suitable choice of coordinates, though,
the problem again reduces to that of understanding the
Hamiltonian (53
) and the corresponding Maass forms.
While the choice (53) of operator ordering is not unique, the number of alternatives
is smaller than one might expect. The key restriction is
diffeomorphism invariance: The eigenfunctions of the Hamiltonian
should transform under a one-dimensional unitary representation
of the mapping class group (50
). The representation theory of this group is
well-understood
[120, 184]
; one finds that the possible Hamiltonians are all of the
form (53
), but with
replaced by
This ordering ambiguity may be viewed as arising from the structure of the classical phase space. The torus moduli space is not a manifold, but rather has orbifold singularities, and quantization on an orbifold is generally not unique. Since the space of solutions of the Einstein equations in 3+1 dimensions has a similar orbifold structure [163], we might expect a similar ambiguity in realistic (3+1)-dimensional quantum gravity.
The quantization described here is an example
of what KuchaĆ has called an “internal Schrödinger
interpretation”
[173]
. It appears to be self-consistent, and like ordinary quantum
mechanics, it is guaranteed to have the correct classical limit
on the reduced phase space of Section
2.4
. The principal drawback is that the method relies on a
classical
choice of time coordinate, which occurs as part of the
gauge-fixing needed to solve the constraints. In particular, the
analysis of Section
2.4
required that we choose the York time-slicing from the start; a
different choice might lead to a different quantum theory, as it
is known to do in quantum field theory
[258
]
. In other words, it is not clear that this approach to quantum
gravity preserves general covariance.
The problem may be rephrased as a statement
about the kinds of questions we can ask in this quantum theory.
The model naturally allows us to compute the transition amplitude
between the spatial geometry of a time slice of constant mean
curvature
and the geometry of a later slice of constant mean curvature
. Indeed, such amplitudes are given explicitly in
[118
], where it is shown that they are peaked around the classical
trajectory. But it is far less clear how to ask for transition
amplitudes between other spatial slices, on which
is not constant. Such questions would seem to require a
different classical time-slicing, and thus a different - and
perhaps inequivalent - quantum theory.
We will eventually find a possible way out of this difficulty in Section 3.4 . As a first step, we next turn to an alternative approach to quantization, one that starts from the first order formalism.
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