Polymer quantization and Symmetries

TL;DR

This paper investigates how symmetries are realized in polymer quantum mechanics and scalar fields, revealing issues with infinitesimal symmetry generators and proposing the use of distributional states to address symmetry breaking.

Contribution

It analyzes the realization of continuous symmetries in polymer quantum systems and explores the role of distributional states in potentially restoring symmetry.

Findings

Infinitesimal symmetry generators are not supported on the Hilbert space.

Symmetries imply infinite degeneracy in invariant Hamiltonians.

Distributional states may offer a way to address symmetry breaking.

Abstract

Polymer quantization was discovered during the construction of Loop Quantum Cosmology. For the simplest quantum theory of one degree of freedom, the implications for dynamics were studied for the harmonic oscillator as well as some other potentials. For more degrees of freedom, the possibility of continuous, kinematic symmetries arises. While these are realised on the Hilbert space of polymer quantum mechanics, their infinitesimal versions are not supported. For an invariant Hamiltonian, these symmetry realizations imply infinite degeneracy suggesting that the symmetry should be spontaneously or explicitly broken. The estimation of symmetry violations in some cases have been analysed before. Here we explore the alternative of shifting the arena to the distributional states. We discuss both the polymer quantum mechanics case as well as polymer quantized scalar field.