Kiefer C. Quantum gravity

THE PROBLEM OF TIME 139
One can derive from H true Hamilton’s equations of motion for φ r and p s .The
variables Xt A (x) can only be interpreted as describing embeddings in a space–
time after these equations (together with the choice for lapse and shift) have
been solved.
The constraint (5.10) can be quantized in a straightforward manner by introducing
wave functionals Ψ[φ r (x)], with the result
i δΨ[φr (x)]
δX A (x)
= h A
(x; X B , ˆφ r , ˆp s] Ψ[φ r (x)] , (5.14)
in which the X A have not been turned into an operator. In this respect, the
quantization is of a hybrid nature: momenta occurring linearly in the constraints
are formally turned into derivatives, although the corresponding configuration
variables stay classical—like the t in the Schrödinger equation. Equation (5.14)
has the form of a local Schrödinger equation. Such an equation is usually called a
‘Tomonaga–Schwinger equation’; strictly speaking, it consists of infinitely many
equations with respect to the local ‘bubble time’ X A (x). We shall say more
about such equations in Section 5.4. The main advantages of this approach to
quantization are:
1. One has isolated already at the classical level a time variable (here: ‘embedding
variables’) that is external to the quantum system described by
ˆφ r , ˆp s . The formalism thus looks similar to ordinary quantum field theory.
2. Together with such a distinguished notion of time comes a natural Hilbertspace
structure and its ensuing probability interpretation.
3. One would consider observables to be any function of the ‘genuine’ operators
ˆφ r and ˆp s . As in the linearized approximation (Chapter 2), the
gravitational field would have two degrees of freedom.
On the other hand, one faces many problems:
1. ‘Multiple-choice problem’: The canonical transformation (5.9) is certainly
non-unique and the question arises which choice should be made. One
would expect that different choices of ‘time’ lead to non-unitarily connected
quantum theories.
2. ‘Global-time problem’: One cannot find a canonical transformation (5.9)
to find a global time variable (Torre 1993).
3. ‘h A -problem’: The ‘true’ Hamiltonian (5.13) depends on ‘time’, that is, on
the embedding variables X A . This dependence is expected to be very complicated
(leading to square roots of operators, etc.), prohibiting in general
a rigorous definition.
4. ‘X A -problem’: In the classical theory, the ‘bubble time’ X A describes a hypersurface
in space–time only after the classical equations have been solved.
Since no classical equations and therefore no space–time are available in
the quantum theory, (5.14) has no obvious space–time interpretation. In
particular, an operational treatment of time is unknown.