3 Quantization(s)

The canonical treatment of the symmetry reductions of GR requires the understanding of constrained Hamiltonian systems. In the cases that we are going to discuss (and leaving aside functional analytic issues relevant for field theories [101Jump To The Next Citation Point]), the starting point consists of the classical unconstrained configuration space 𝒞 of the model and the cotangent bundle Γ over 𝒞 endowed with a suitable symplectic form Ω. A dynamical Hamiltonian system is said to be constrained if the physical states are restricted to belonging to a submanifold ¯Γ of the phase space Γ, and the dynamics are such that time evolution takes place within ¯Γ [101]. In the examples relevant for us the space ¯Γ will be globally defined by the vanishing of certain sufficiently regular constraint functions, C = 0 I. In the case of GR these constraint functions are the integrated version of the scalar and vector constraints and the subindex I refers to lapse and shift choices (see, for example, [8]). Notice, however, that there exist infinitely-many constraint equations that define the same submanifold ¯Γ. The choice of one representation or another is, in practice, dictated by the variables used to describe the physical system. We will assume that ¯Γ is a first-class submanifold of Γ. This is geometrical property that can be expressed in terms of the concrete constraint equations describing ¯ Γ as

{C ,C }| = 0. (1 ) I J ¯Γ
The pull-back of the symplectic structure of Γ to ¯Γ is degenerate and the integral submanifolds defined by the degenerate directions are the gauge orbits. The reduced phase space &tidle;Γ is the quotient space whose points are the orbits of the gauge flows. It can be endowed with the natural symplectic structure &tidle;Ω inherited from Ω. If a non-trivial dynamics describes the evolution of the system in &tidle; Γ this will be given by the reduced Hamiltonian &tidle;H obtained by restricting the original one to &tidle;Γ. This restriction is well defined whenever H is gauge invariant and, hence, constant on the gauge orbits.

 3.1 Reduced phase-space quantization
 3.2 Dirac quantization
 3.3 Quantization with partial gauge fixing
 3.4 Path integral quantization
 3.5 Symmetry reductions and quantization

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