Quantum GowdyT3model: A unitary description

Abstract

The quantization of the family of linearly polarized Gowdy T 3 spacetimes is discussed in detail, starting with a canonical analysis in which the true degrees of freedom are described by a scalar field that satisfies a Klein-Gordon type equation in a fiducial time-dependent background. A time-dependent canonical transformation, which amounts to a change of the basic (scalar) field of the model, brings the system to a description in terms of a Klein-Gordon equation on a background that is now static, although subject to a time-dependent potential. The system is quantized by means of a natural choice of annihilation and creation operators. The quantum time evolution is considered and shown to be unitary, so that both the Schrödinger and Heisenberg pictures can be consistently constructed. This has to be contrasted with previous treatments for which time evolution failed to be implementable as a unitary transformation. Possible implications for both canonical quantum gravity and quantum field theory in curved spacetime are noted.

Quantum GowdyT3model: Schrödinger representation with unitary dynamics

“…With respect to some reference (''initial'') time t t 0 , all smooth solutions can then be written as [8] t;…”

“…after a gauge fixing procedure which removes all the gauge degrees of freedom except for a homogeneous one [8]. The spatially homogeneous variable p is a positive constant of motion.…”

“…In Sec. II we recall the quantization of the linearly polarized Gowdy T 3 model constructed by Corichi, Cortez and Mena Marugán, in which the time evolution is implemented unitarily [7,8]. In Sec.…”

“…Therefore, it is important to have a Schrödinger functional description of any symmetry reduced model, such as the Gowdy T 3 model. The purpose of this paper is to present this description for the quantization which admits a unitary time evolution [7,8], and analyze the implementation of such a unitary evolution in this framework, both before and after imposing the remaining constraint of the theory.…”

“…[7,8]: instead of working with a fixed quantum representation and considering the unitary implementability of the family of symplectic transformations defined by the evolution (together with the corresponding unitary evolution operator), we will construct the associated 1-parameter family of representations. Notice that this is precisely the family of representations which is obtained by ''evolving in time'' a fixed GNS state (Gelfand-Naimark-Segal state), and hence the complex structure defining the FR.…”