3.3 Quantization with partial gauge fixing

As mentioned above the reduced phase space is the space of gauge orbits endowed with a symplectic structure &tidle;Ω inherited from the original one Ω in the full phase space. A strategy that is useful in the context of midisuperspaces is to partially fix the gauge. In practice this means that the dimensionality of the constraint hypersurface (and, as a consequence, of the gauge orbits) is reduced. This may be useful if one is interested in leaving some residual gauge symmetry in the model on purpose (such as radial diffeomorphisms in spherically-symmetric models [54Jump To The Next Citation Point]) to check if one can deal with it in some quantization scheme. In other situations the natural gauge fixing conditions simply fail to fix the gauge completely; this happens, for example, in the compact Gowdy models [165Jump To The Next Citation Point]. In such cases the residual gauge invariance is usually treated by employing Dirac’s procedure, though other approaches are, of course, possible. A very attractive feature of the resulting formulation is that the quantum dynamics of the model is given by a “time” dependent Hamiltonian that can be studied in great detail due to its relatively simple structure. This is possible because its meaning can be understood by using results developed in the study of the time-dependent harmonic oscillator (see [100Jump To The Next Citation Point] and references therein).
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