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

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.

Keywords

• DKP equation
• noncommutative DKP equation
• Schrödinger equation
• noncommutative SchrödingerPauli equation
• phase-space noncommutativity
• Foldy-Wouthuysen transformation
• non-relativistic limit