Quantum Observables on a Completely Simple Semigroup

TL;DR

This paper explores how tensor powers of random walks on completely simple semigroups relate to quantum observables and algebraic structures like zeons, revealing deep connections between probabilistic limits and algebraic properties.

Contribution

It introduces a novel approach linking random walk limits on semigroups with quantum observables via zeon powers, enhancing understanding of the kernel structure.

Findings

Tensor powers lead to quantum observables on the kernel

Zeon powers reveal algebraic structure of the kernel

Asymptotic behavior relates to zeon algebra properties

Abstract

Completely simple semigroups arise as the support of limiting measures of random walks on semigroups. Such a limiting measure is supported on the kernel of the semigroup. Forming tensor powers of the random walk leads to a hierarchy of the limiting kernels. Tensor squares lead to quantum observables on the kernel. Recall that zeons are bosons modulo the basis elements squaring to zero. Using zeon powers leads naturally to quantum observables which reveal the structure of the kernel. Thus asymptotic information about the random walk is related to algebraic properties of the zeon powers of the random walk.