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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  [312898175Jump To The Next Citation Point272Jump To The Next Citation Point], 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 C is an arbitrary Cauchy surface, then a point in the phase space determines initial data on C, which can be evolved to give a unique solution, while, conversely, a classical solution restricted to C determines a point in the phase space. Moreover, the space of solutions has natural symplectic structure  [175272], 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  [81Jump To The Next Citation Point] .

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 R × S with S closed and /\ < 0, 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 /\ > 0 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 /\ > 0 .

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  [207Jump To The Next Citation Point] . 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  [50Jump To The Next Citation Point], of whether classical perturbation theory can be used to define a perturbative covariant canonical quantum theory.


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