The Non-Relativistic Limit of the DKP Equation in Non-Commutative Phase-Space

Ilyas Haouam

doi:10.3390/sym11020223

Abstract

The non-relativistic limit of the relativistic DKP equation for both of zero and unity spin particles is studied through the canonical transformation known as the Foldy–Wouthuysen transformation, similar to that of the case of the Dirac equation for spin-1/2 particles. By considering only the non-commutativity in phases with a non-interacting fields case leads to the non-commutative Schrödinger equation; thereafter, considering the non-commutativity in phase and space with an external electromagnetic field thus leads to extract a phase-space non-commutative Schrödinger–Pauli equation; there, we examined the effect of the non-commutativity in phase-space on the non-relativistic limit of the DKP equation. However, with both Bopp–Shift linear transformation through the Heisenberg-like commutation relations, and the Moyal–Weyl product, we introduced the non-commutativity in phase and space.

Highlights

It has been very interesting to investigate the theoretical basis of the modern physics to explain the nature and the behavior of the matter and energy on the subatomic scale, sometimes referred to as quantum theories, such as the quantum gravity [1,2] or quantum general relativity (QGR) [3], quantum optics and information, the standard model and the gauge theories [4]
This investigation sometimes can be represented in terms of the low-energy regime through the examination of the non-relativistic properties in miscellaneous interactions such as the external electromagnetic fields (EMF), Dirac or DKP oscillator interaction [5,6], Lennard–Jones potential, Coulomb potential, square and step potential; the non-relativistic limit is about low speeds in front of the speed of the light, pc in more detail, it is for the regime of weak-energy in front of the mass-energy mc2 1 [7], where the non-relativistic limit can be realized through numerous methods, among them the Foldy–Wouthuysen (FW) transformation [8], and Eriksen’s method [9] proposed in 1958
We have studied the non-relativistic limit of the DKP equation which provides description of the zero or unity spin particles in the DKP representation using the FW unitary transformation in non-commutative phase space (NCPS), where we introduced the phase-space non-commutativity influence

Summary

Introduction

It has been very interesting to investigate the theoretical basis of the modern physics to explain the nature and the behavior of the matter and energy on the subatomic scale, sometimes referred to as quantum theories, such as the quantum gravity [1,2] or quantum general relativity (QGR) [3], quantum optics and information, the standard model and the gauge theories [4]. The transition to the non-relativistic limit generally is due to a replacement of the operators in our quantum-mechanical systems to its corresponding classical quantities This implicit or explicit replacement used in all calculations was devoted to the Foldy–Wouthuysen transformation. In this work, we investigated the non-relativistic limit of the DKP equation using the FW transformation in a non-commutative phase space (NCPS). Taking into account that we obtained the phase-space non-commutative DKP equation using both Bopp–Shift linear transformation through the Heisenberg- like commutation relations, and the Gronewold–Moyal product (?-product)

Review of the Non-Commutative Geometry

Schrödinger Equation from the DKP Equation in the Non-Commutative Phase

Conclusions