Hypercomplex Fock States for Discrete Electromagnetic Schr\"odinger Operators: A Bayesian Probability Perspective

TL;DR

This paper introduces hypercomplex Fock states for discrete electromagnetic Schr"odinger operators, connecting quantum probability with Bayesian methods and deriving new polynomial and distribution frameworks.

Contribution

It develops a novel multivector calculus approach to Fock states, leading to hypercomplex polynomial and distribution generalizations within a Bayesian probability context.

Findings

Derived hypercomplex Poisson-Charlier and Meixner polynomials.

Established complex-valued quasi-probability distributions on lattices.

Connected quantum probability formulations with Bayesian inference.

Abstract

We present and study a new class of Fock states underlying to discrete electromagnetic Schr\"odinger operators from a multivector calculus perspective. This naturally lead to hypercomplex versions of Poisson-Charlier polynomials, Meixner polynomials, among other ones. The foundations of this work are based on the exploitation of the quantum probability formulation '\`a la Dirac' to the setting of Bayesian probabilities, on which the Fock states arise as discrete quasi-probability distributions carrying a set of independent and identically distributed (i.i.d) random variables. By employing Mellin-Barnes integrals in the complex plane we obtain counterparts for the well-known multidimensional Poisson and hypergeometric distributions, as well as quasi-probability distributions that may take negative or complex values on the lattice $h\mathbb{Z}^n$.