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• Find a representation of a subset of classical variables on the physical state space, such that the operators commute with all of the quantum constraints. (One must also find an inner product making these self-adjoint.)

The classical observables argument filters through into this quantum setup since, by analogy with the classical observables, the quantum observables Oi are defined as follows:

Note that the weak equality is now defined on the solution space of the quantum constraints; i.e. F0 = (&: H& = 0}. Clearly, if Eq. (7.7) did not hold, then there could be possible observables whose measurement would knock a state & out of F0. The state version of the problem then follows simply from the fact that the quantum Hamiltonian annihilates physical states: H& = 0. What motivates this view is the idea common to gauge theories that if a pair of classical configurations q and q' are gauge related then O(q) = O(q'), so we should impose gauge invariance at the level of quantum states too: f (q) = f (q'). The diffeomorphism constraint, Eq. (7.5), is particularly easy to comprehend along such lines; it simply says that for any diffeomorphism d: £ ^ £, and state & [q], &[q] = &[d*q]. Were this not the case, one could use the quantum theory to distinguish between classically indistinguishable states. The Hamiltonian constraint is more problematic, for it generates changes in data 'flowing off' £, and is seen as generating evolution. If

207 Where £2(Riem(£, are the square integrable functions on the space of Riemannian metrics on a 3-surface, with suitable measure

208 Gabcd is the DeWitt supermetric defined by |det q|1/2 [(qab^cd —19ac9bd)], and3R(q) is the scalar curvature of q. Eq. (7.6) is known as the 'Wheeler-DeWitt equation'.

we forbid quantum states to distinguish between states related by the Hamiltonian constraint, then there is no evolution.

According to the alternative method—reduced phase space quantization—the constraints are solved for prior to quantization (i.e. at the classical level). To solve the constraints, one divides r by its gauge orbits [x]i. This yields a space rred equipped with a symplectic form <y. The resulting symplectic geometry (rred, <w) is the reduced phase space, and in the case of general relativity corresponds to the space of non-isometric (vacuum) spacetimes. Thus, the symmetries generated by the constraints are factored out and one is left with an intrinsic geometrical structure of standard Hamiltonian form. In this form the canonical quantization is carried out as usual, and the observables are automatically gauge invariant when considered as functions on the extended (unreduced) space. However, since one of the constraints (the Hamiltonian constraint) was associated with time evolution, in factoring its action out the dynamics is eliminated, since time evolution unfolded along a gauge orbit (i.e. instants of time correspond to points in a gauge orbit). Thus, on this approach, states of general relativity are given by points in the reduced phase space, as opposed to the extended phase space used in constrained quantization approach.209

Of course, one can completely remove the ambiguity associated with gauge freedom by imposing gauge conditions, thus allowing for an unproblematic direct interpretation. However, in the case of general relativity (and other non-Abelian gauge theories) the geometrical structure of the constraint surface and the gauge orbits can prohibit the implementation of gauge conditions, so that some gauge slices will intersect some gauge orbits more than once, or not at all. One frequently finds that the reduced phase space method is mixed with gauge fixation methods, so that one has a partially reduced space, with the remaining gauge freedom frozen by imposing gauge conditions. Such an approach is used by a number of internal time responses to the problem of time. The idea is that one first solves the diffeomorphism constraint and then imposes gauge conditions on the gauge freedom generated by the Hamiltonian constraint. This is essentially the position of Kuchar (see below), and constant mean curvature approaches (see Carlip [1998] for a clear and thorough review).

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