Non-commutativity: Unusual View

TL;DR

This paper explores the implications of non-commutativity in derivatives and operators within quantum mechanics and field theory, revealing potential mass splitting and non-commuting limits that challenge conventional assumptions.

Contribution

It introduces the concept of non-commutativity of 4-momenta and demonstrates its effects on Dirac equation and quantum field theory calculations.

Findings

Non-commuting commutators of space-time coordinates and momenta.

Mass splitting derived from non-commutative 4-momenta.

Non-commuting limits affecting massless cases in quantum field theory.

Abstract

Some ambiguities have recently been found in the definition of the partial derivative (in the case of presence of both explicit and implicit dependencies of the function subjected to differentiation). We investigate the possible influence of this subject on quantum mechanics and the classical/quantum field theory. Surprisingly, some commutators of operators of space-time 4-coordinates and those of 4-momenta are {\it not} equal to zero. We postulate the non-commutativity of 4-momenta and we derive mass splitting in the Dirac equation. Moreover, two iterated limits may not commute each other, in general. Thus, we present an example when the massless limit of the function of $E, {\bf p}, m$ does not exist in some calculations within quantum field theory. KEYWORDS: Non-commutativity, quantum mechanics, whole-partial derivatives. PACS: 04.62.+v 02.40.Gh 02.30.-f