Algebraic structures spanned by differential-like operators on toy Fock spaces

TL;DR

This paper explores algebraic structures generated by differential-like operators on toy Fock spaces, revealing symmetries, group structures, and anticommutative systems related to quantum probability and classical algebraic objects.

Contribution

It introduces new algebraic frameworks for differential-like operators on toy Fock spaces, generalizing classical algebraic structures and identifying symmetries and anticommutative systems.

Findings

Identification of group and algebra structures with varying signatures

Construction of anticommutative Rademacher systems

Generalization of Pauli matrices and quaternions

Abstract

We study multiplicative systems of linear mappings acting on the toy Fock space, a.k.a.\ Rademacher chaos or Walsh-Fourier series, related to the creation, annihilation, and conservation operators in quantum probability. Like differential operators they entail analogs of the Leibnitz Formula and the Chain Rule, derived with the help of Riesz products (or discrete coherent vectors). Two symmetries among these operators entail groups and algebras of varying signatures and mixed commutativity, generalizing Pauli spin matrices and quaternions. In particular, anticommutative Rademacher systems are constructed.