3.1 Reduced phase-space quantization

The reduced phase-space quantization is simply the quantization of the reduced space (&tidle;Γ ,Ω&tidle;, H&tidle;) of the constrained Hamiltonian system whenever this is possible. Notice that the process of taking quotients is highly non-trivial and many desirable regularity properties need not be preserved. In the models that we consider in this paper we will suppose that no obstructions appear so that the reduced phase space is well defined. Even in this case some difficulties may (and in practice do) arise, in particular:

The reduced phase-space description amounts to the identification of the true physical degrees of freedom of the system. As a rule, for many physical models (and certainly in the case of gravitational theories) this description is either unavailable or extremely difficult to handle. In these cases one is forced to learn how to live with the redundant descriptions provided by gauge theories and how to handle the constraints both at the classical and quantum levels. Finally, it is important to notice that whenever the reduced phase space can be characterized by means of a gauge fixing, the quantization ambiguities that may arise do not originate in the different gauge choices – as long as they are acceptable – but rather in the possibility of having different quantizations for a given classical model. This is so because from the classical point of view they are explicit representations of the same abstract object: the reduced phase space [113Jump To The Next Citation Point]. There are several approaches to the quantization of gauge systems that we will briefly discuss next.

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