Classical and Quantum Dynamics in an Information Theoretic Space

TL;DR

This paper explores classical and quantum dynamics within an information geometric space based on Bernoulli variables, deriving spectral properties and quantizing momentum to analyze quantum oscillators and wave equations.

Contribution

It extends previous work by deriving spectral properties and quantizing momentum in Bernoulli space, linking classical and quantum dynamics with information geometry.

Findings

Derived the spectrum for the Laplace-Beltrami operator in Bernoulli space.

Obtained Green's functions for the Helmholtz equation in this space.

Quantized momentum and found energies and wavefunctions for quantum oscillators.

Abstract

We study elementary classical and quantum dynamics in an information geometric space corresponding to a Bernoulli random variable, extending work by Goehle and Griffin [Chaos, Solitons & Fractals, 188, 115535, (2024)], who study the information theoretic analog of the spring-mass system. Information geometric constructions are useful in both statistical physics and in physical interpretations of Friston's free energy principle, a form of the Bayesian brain hypothesis. In this letter, we derive the spectrum for the Laplace-Beltrami operator in Bernoulli space and find Green's functions for the Helmholtz equation, which provides solutions to the wave, heat, and Poisson equations. We then show how to quantize momentum in Bernoulli space and obtain energies and wavefunctions for both a free particle and a variety of quantum (harmonic) oscillators in this space. In particular, we show that quadratic approximation of the Kullback-Leibler potential used by Goehle and Griffin results in a quantum oscillator in information space that is equivalent to a quantum pendulum in Euclidean space.