Some Aspects of Noncommutativity on Real, p-Adic and Adelic Spaces

TL;DR

This paper explores noncommutativity in classical and quantum mechanics across real, p-adic, and adelic spaces, showing how transformations affect algebraic structures and path integrals, with a focus on quadratic systems.

Contribution

It introduces a linear transformation approach to remove noncommutativity from the algebra and examines p-adic and adelic aspects within a unified matrix formalism.

Findings

Transformation preserves quadratic form of Hamiltonian and Lagrangian.

Path integral remains exact for quadratic systems after transformation.

p-adic and adelic noncommutativity aspects are analyzed.

Abstract

Classical and quantum mechanics for an extended Heisenberg algebra with canonical commutation relations for position and momentum coordinates are considered. In this approach additional noncommutativity is removed from the algebra by linear transformation of phase space coordinates and transmitted to the Hamiltonian (Lagrangian). This transformation does not change the quadratic form of Hamiltonian (Lagrangian) and Feynman's path integral maintains its well-known exact expression for quadratic systems. The compact matrix formalism is presented and can be easily employed in particular cases. Some p-adic and adelic aspects of noncommutativity are also considered.