On non-commutativity in quantum theory (II): toy models for non-commutative kinematics

TL;DR

This paper explores non-commutativity in quantum theory through toy models, demonstrating how position and velocity operators become non-commutative after removing the random space, highlighting foundational aspects of quantum kinematics.

Contribution

It introduces simplified models that illustrate non-commutative kinematics, extending previous work and providing new insights into quantum non-commutativity.

Findings

Position and velocity operators do not commute after removing the random space.

Toy models effectively demonstrate non-commutative kinematics.

The approach bridges classical probability and quantum operator non-commutativity.

Abstract

In this article, we continue our investigation on the role of non-commutativity in quantum theory. Using the method explained in "On non-commutativity in quantum theory (I): from classical to quantum probability", we analyze two toy models which exhibit non-commutativity between the corresponding position and velocity random variables. In particular, using ordinary probability theory, we study the kinematics of a point-like particle jumping at random over a discrete random space. We show that, after the removal of the random space from the model, the position and velocity of the particle do not commute, when represented as operators on the same Hilbert space.