Dynamics of Scalar Field in Polymer-like Representation

TL;DR

This paper explores the quantum dynamics of a scalar field coupled to gravity within loop quantum gravity, demonstrating well-defined operators and addressing potential anomalies, thereby supporting the consistency of this background independent approach.

Contribution

It constructs self-adjoint, positive Hamiltonian operators for scalar fields in loop quantum gravity, and discusses the implementation of the master constraint to prevent quantum anomalies.

Findings

Hamiltonian operator for scalar field is well-defined and self-adjoint.

Hamiltonian constraint operator is well-defined in the coupled Hilbert space.

The approach shows no divergence issues in matter-gravity coupling.

Abstract

In recent twenty years, loop quantum gravity, a background independent approach to unify general relativity and quantum mechanics, has been widely investigated. We consider the quantum dynamics of a real massless scalar field coupled to gravity in this framework. A Hamiltonian operator for the scalar field can be well defined in the coupled diffeomorphism invariant Hilbert space, which is both self-adjoint and positive. On the other hand, the Hamiltonian constraint operator for the scalar field coupled to gravity can be well defined in the coupled kinematical Hilbert space. There are 1-parameter ambiguities due to scalar field in the construction of both operators. The results heighten our confidence that there is no divergence within this background independent and diffeomorphism invariant quantization approach of matter coupled to gravity. Moreover, to avoid possible quantum anomaly, the master constraint programme can be carried out in this coupled system by employing a self-adjoint master constraint operator on the diffeomorphism invariant Hilbert space.