Classical and Quantum Dynamics in an Information Theoretic Space

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.