Born Reciprocal Representations of the Quantum Algebra

Abstract

This thesis addresses the representations of the Weyl algebra of quantum mechanics with an explicitly Born reciprocal structure and the dynamics of physical systems in these representations. In this context, we discuss both the regular modular representations based on Aharonov's modular variables, as well as the irregular modular polymer representations inspired by the spin network representation in loop quantum gravity.

We introduce the modular space as the quantum configuration space corresponding to the modular representation and treat it as an alternative to the classical configuration space. The modular representations have a built-in Abelian gauge symmetry. We discuss the singular limits in which the modular representation converges to the Schrodinger and momentum representations, and show that these limits require two distinct fixings of the modular gauge symmetry.

In order to explore the propagation of quantum states in the modular space, we construct a Feynman path integral for the harmonic oscillator explicitly as the transition amplitude between two modular states. The result is a functional integral over the compact modular space, which shows novel features of the dynamics such as winding modes, an Aharonov-Bohm phase, and a new action on the modular space. We compare the stationary trajectories that extremize this modular action to the phase space solution of Hamilton's equations in the Schrodinger representation and find a new translation symmetry. We also identify the other symmetries of the harmonic oscillator in the modular action and show their correspondence to the classical case.

We generalize the relationship between the classical Hamiltonian function and the modular action by postulating a new modular Legendre transform. For demonstration we apply this transformation on the Kepler problem and reformulate it in terms of modular variables.

We then switch our focus to the representations of the Weyl algebra that violate the assumptions of the Stone-von Neumann theorem. In a similar fashion to the polymer representations studied as a toy model for quantum gravity, we polymerize the modular representation to obtain an inequivalent representation called the modular polymer (MP) representation.

The new MP representation lacks both position and momentum operators, therefore the Hamiltonian of the harmonic oscillator is regularized with the Weyl operators and the non-separable MP Hilbert space is split into superselection sectors. The solutions to the Schrodinger equation fall into two categories depending on whether the ratio between the modular scale and the polymeric regularization scale is a rational or irrational number. We find a discrete and finite energy spectrum in each superselection sector in the former case, and a continuous, bounded one in the latter.