Hamiltonian dynamics of linearly polarized Gowdy models coupled to massless scalar fields

Abstract

The purpose of this paper is to analyze in detail the Hamiltonian formulation for the compact Gowdy models coupled to massless scalar fields as a necessary first step toward their quantization. We will pay special attention to the coupling of matter and those features that arise for the S 1 × S 2 and S 3 topologies that are not present in the well-studied T 3 case-in particular the polar constraints that come from the regularity conditions on the metric. As a byproduct of our analysis we will get an alternative understanding, within the Hamiltonian framework, of the appearance of initial and final singularities for these models.

Uniqueness of the Fock representation of the Gowdy S 1 × S 2 and S 3 models

“…The proof of this uniqueness result is an adaptation of the one presented in [4]. In this section we briefly review the quantization of the Gowdy S 1 × S 2 and S 3 models discussed in [12,13].…”

“…More recently, part of the results obtained originally in the context of the T 3 model were extended to the linearly polarized Gowdy S 1 ×S 2 and S 3 models [12,13]. As shown in [12], the local degrees of freedom of these models are effectively described by an axisymmetric scalar field on S 2 [more precisely in a space-time (0, π) × S 2 ], obeying the same field equation in both cases.…”

“…As shown in [12], the local degrees of freedom of these models are effectively described by an axisymmetric scalar field on S 2 [more precisely in a space-time (0, π) × S 2 ], obeying the same field equation in both cases. Starting from this formulation, the issue of unitary evolution was then discussed, restricting the considerations to Fock representations of the scalar field determined by SO(3)-invariant complex structures [13].…”

Uniqueness of the Fock quantization of a free scalar field onS1with time dependent mass

“…Actually, the spatial sections must be homeomorphic to either the three-torus, T 3 , the three-sphere, S 3 , or the three-handle, S 2 × S 1 . For these spacetimes, the gauge N ∼ τ = 1 is indeed allowed [2,10]. In the case of the topology of the three-torus, this gauge is introduced by fixing the freedom associated with the densitized Hamiltonian constraint, what in turn is achieved by choosing the metric function τ (essentially) as the time coordinate, namely, τ = Ct where C is a constant of motion [2,3].…”

Schrödinger quantization of linearly polarized Gowdy \protect\bb{S}^{1}\times\protect\bb{S}^{2} and \protect\bb{S}^{3} models coupled to massless scalar fields

“…The existing literature has been recently extended to the remaining topologies, S 1 × S 2 and S 3 , allowing the coupling of gravity to massless scalar fields (see [15] for a rigorous classical treatment of these models). Here, both gravitational and matter local degrees of freedom can be encoded by massless scalar fields evolving in the same fixed background metric.…”