Probability Distributions and Coherent States of $B_r$, $C_r$ and $D_r$ Algebras

TL;DR

This paper introduces new probability distributions derived from coherent states of classical Lie algebras, linking quantum algebraic structures with classical probability distributions and providing new insights into Hermite polynomial addition theorems.

Contribution

It develops a novel approach connecting Lie algebraic coherent states with probability distributions, extending classical distributions with quantum algebraic features.

Findings

Derived new probability distributions from $B_r$, $C_r$, and $D_r$ algebra coherent states.

Provided simple proofs and interpretations of Hermite polynomial addition theorems.

Linked algebraic structures with classical probability and polynomial identities.

Abstract

A new approach to probability theory based on quantum mechanical and Lie algebraic ideas is proposed and developed. The underlying fact is the observation that the coherent states of the Heisenberg-Weyl, $su(2)$, $su(r+1)$, $su(1,1)$ and $su(r,1)$ algebras in certain symmetric (bosonic) representations give the ``probability amplitudes'' (or the ``square roots'') of the well-known Poisson, binomial, multinomial, negative binomial and negative multinomial distributions in probability theory. New probability distributions are derived based on coherent states of the classical algebras $B_r$, $C_r$ and $D_r$ in symmetric representations. These new probability distributions are simple generalisation of the multinomial distributions with some added new features reflecting the quantum and Lie algebraic construction. As byproducts, simple proofs and interpretation of addition theorems of Hermite polynomials are obtained from the `coordinate' representation of the (negative) multinomial states. In other words, these addition theorems are higher rank counterparts of the well-known generating function of Hermite polynomials, which is essentially the `coordinate' representation of the ordinary (Heisenberg-Weyl) coherent state.