Expressions of algebra elements and transcendental noncommutative calculus

TL;DR

This paper explores how deformation quantization can be applied to algebra elements, especially transcendental ones, offering new insights into the structure of the Weyl algebra and potential implications for Dirac's positron theory.

Contribution

It introduces a novel approach to deform algebra elements using ideas from deformation quantization, extending to transcendental elements and revealing a discrete spectrum for certain expressions.

Findings

$rac{1}{i ext{ } ext{h}} ext{uv}$ viewed as an indeterminate in } ext{N}+1/2$ or $- ext{(N}+1/2)$

Provides a mathematical perspective on Dirac's positron theory

Extends deformation quantization to transcendental algebra elements

Abstract

Ideas from deformation quantization are applied to deform the expression of elements of an algebra. Extending these ideas to certain transcendental elements implies that $\frac{1}{i\h}uv$ in the Weyl algebra is naturally viewed as an indeterminate living in a discrete set $\mathbb{N}{+}{1/2}$ {\it or} ${-}(\mathbb{N}{+}{1/2})$ . This may yield a more mathematical understanding of Dirac's positron theory.