A Note on Non-commutativity

TL;DR

This paper explores the implications of non-commutativity in derivatives and operator commutators in quantum mechanics and field theory, revealing potential ambiguities and non-zero commutators that impact theoretical calculations.

Contribution

It highlights the effects of non-commutativity in derivatives and operators, providing examples where limits and operator relations differ from classical assumptions.

Findings

Certain operator commutators are non-zero.

Limits involving massless particles may not exist.

Ambiguities in derivatives affect quantum field theory calculations.

Abstract

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 not equal to zero. Notes added in the Abstract: Two iterated limits are not equal each other, in general. Thus, we present an example when the massless limit of the function of E, p, m does not exist in some calculations within quantum field theory.