Time Quantization and q-Deformations

TL;DR

This paper explores how time quantization and q-deformations can be applied to quantum mechanics, specifically through stochastic subordination and time-changed Brownian motions, with examples involving q-deformed harmonic oscillators.

Contribution

It introduces a novel perspective on q-deformations as different patterns of time quantization in quantum mechanics, extending stochastic subordination techniques.

Findings

q-deformed harmonic oscillators can be interpreted as time-quantized systems

stochastic subordination expresses semi-martingales as time-changed Brownian motions

different q-deformation patterns correspond to distinct time quantization rules

Abstract

We extend to quantum mechanics the technique of stochastic subordination, by means of which one can express any semi-martingale as a time-changed Brownian motion. As examples, we considered two versions of the q-deformed Harmonic oscillator in both ordinary and imaginary time and show how these various cases can be understood as different patterns of time quantization rules.