Polymer Parametrised Field Theory

TL;DR

This paper develops a polymer quantization of a 2D scalar field theory in a diffeomorphism-invariant framework, replacing smooth geometry with a discrete quantum structure and analyzing semiclassical states without triangulation dependence.

Contribution

It introduces a polymer quantization of scalar field theory that preserves diffeomorphism invariance and avoids triangulation ambiguities, providing a new testing ground for quantum gravity concepts.

Findings

Quantum constraints are solved via group averaging.

Spacetime geometry is represented as a discrete quantum structure.

Semiclassical states are characterized by physical weaves.

Abstract

Free scalar field theory on 2 dimensional flat spacetime, cast in diffeomorphism invariant guise by treating the inertial coordinates of the spacetime as dynamical variables, is quantized using LQG type `polymer' representations for the matter field and the inertial variables. The quantum constraints are solved via group averaging techniques and, analogous to the case of spatial geometry in LQG, the smooth (flat) spacetime geometry is replaced by a discrete quantum structure. An overcomplete set of Dirac observables, consisting of (a) (exponentials of) the standard free scalar field creation- annihilation modes and (b) canonical transformations corresponding to conformal isometries, are represented as operators on the physical Hilbert space. None of these constructions suffer from any of the `triangulation' dependent choices which arise in treatments of LQG. In contrast to the standard Fock quantization, the non- Fock nature of the representation ensures that the algebra of conformal isometries as well as that of spacetime diffeomorphisms are represented in an anomaly free manner. Semiclassical states can be analysed at the gauge invariant level. It is shown that `physical weaves' necessarily underly such states and that such states display semiclassicality with respect to, at most, a countable subset of the (uncountably large) set of observables of type (a). The model thus offers a fertile testing ground for proposed definitions of quantum dynamics as well as semiclassical states in LQG.