Optics, Mechanics and Quantization of Reparametrization Systems

TL;DR

This paper explores the quantization of reparametrization-invariant systems, drawing parallels between classical mechanics, quantum mechanics, and quantum geometric optics, and clarifies their underlying Hamiltonian constraints.

Contribution

It introduces a novel perspective on quantizing reparametrization-invariant systems and clarifies the relationship between quantum geometric optics and Maxwell theory.

Findings

Quantum geometric optics is not a field theory in curved space.

The dynamics of quantum theories with reparametrization invariance are governed by a Hamiltonian constraint.

Parallelism between quantum mechanics and quantum geometric optics is established.

Abstract

In this paper we regard the dynamics obtained from Fermat principle as begin the classical theory of light. We (first-)quantize the action and show how close we can get to the Maxwell theory. We show that Quantum Geometric Optics is not a theory of fields in curved space. Considering Classical Mechanics to be on the same footing, we show the parallelism between Quantum Mechanics and Quantum Geometric Optics. We show that, due to the reparametrization invariance of the classical theories, the dynamics of the quantum theories is given by a Hamiltonian constraint. Some implications of the above analogy in the quantization of true reparameterization invariant systems are discussed.