Quantization of the ModMax Oscillator

TL;DR

This paper quantizes the ModMax oscillator, deriving its propagator, partition functions, and establishing the equivalence of quantization methods for deformed quantum theories, thus extending understanding of deformations in quantum mechanics.

Contribution

It introduces a framework for quantizing deformed quantum mechanical theories based on conserved charges, with explicit results for the ModMax oscillator.

Findings

Propagator satisfies a Laplace-related differential equation.

Classical and quantum partition functions are derived.

Canonical and path integral quantizations are equivalent with phase space approach.

Abstract

We quantize the ModMax oscillator, which is the dimensional reduction of the Modified Maxwell theory to one spacetime dimension. We show that the propagator of the ModMax oscillator satisfies a differential equation related to the Laplace equation in cylindrical coordinates, and we obtain expressions for the classical and quantum partition functions of the theory. To do this, we develop general results for deformations of quantum mechanical theories by functions of conserved charges. We show that canonical quantization and path integral quantization of such deformed theories are equivalent only if one uses the phase space path integral; this gives a precise quantum analogue of the statement that classical deformations of the Lagrangian are equivalent to those of the Hamiltonian.