Classical Limit and Ehrenfest’s Theorem Versus Non-relativistic Limit of Noncommutative Dirac Equation in the Presence of Minimal Uncertainty in Momentum

Abstract

In this article, we discuss the classic limit and investigate the Ehrenfest’s theorem of the Dirac equation in the context of minimal uncertainty in momentum within a noncommutative setting, and examine its $\mathcal {CPT}$ symmetry and Lorentz symmetry violation. Also, we study the non-relativistic limit of this Dirac system, which leads to obtain a deformed Schrödinger–Pauli equation. Besides we check if this obtained equation still show explicitly the gyromagnetic factor of the electron. Interestingly, the overlap and congruence aspects of the classical and non-relativistic limits of the Dirac equation are clarified. The effects of both minimal uncertainty in momentum and noncommutativity on the Ehrenfest’s theorem and non-relativistic limit are well examined. Knowing that with both the linear Bopp–Shift and $\star$product, the noncommutativity is inserted.

Appendix A: \(\mathcal {C}\), \(\mathcal {P}\) and \(\mathcal {T}\) Discrete Symmetries and \(\mathcal {CPT}\) Symmetry

Discrete symmetries play a fundamental role in modern theoretical physics, providing insights into the underlying structure of the universe. Among these symmetries, $\mathcal {C}$, $\mathcal {P}$ and $\mathcal {T}$ are particularly significant and basic. The $\mathcal {C}$ symmetry, or charge conjugation, involves swap** particles with their corresponding antiparticles while reversing their charges, e.g., $e\rightarrow -e$ and $i\rightarrow -i$. The $\mathcal {P}$ symmetry, or parity, reflects the spatial inversion of a physical system, interchanging left and right, e.g., $\overrightarrow{r}\rightarrow -\overrightarrow{r}$. The $\mathcal {T}$ symmetry, or time reversal, entails reversing the direction of time in a process, e.g., $t\rightarrow -t$. These symmetries were once believed to be individually conserved in all physical interactions. However, the discovery of certain particle decays and weak interactions violating these symmetries led to a deeper understanding of their interconnections. The $\mathcal {CPT}$ symmetry combines charge conjugation ($\mathcal {C}$ ) (particle antiparticle exchange), parity inversion ($\mathcal {P}$), and time reversal ($\mathcal {T}$) into a more encompassing symmetry. The combined operation of $\mathcal {C}$, $\mathcal {P}$ and $\mathcal {T}$ must be an exact symmetry. It ensures that the laws of physics remain unchanged when particles are replaced by their antiparticles, space is inverted, and time flows backward simultaneously. Lorentz symmetry, on the other hand, guarantees that physical laws are the same for all observers in inertial reference frames. CPT symmetry is a powerful concept that underlies our understanding of the fundamental symmetries of the universe. Both $\mathcal {CPT}$ and Lorentz symmetries hold a crucial role in modern quantum field theory, including the Standard Model of particle physics, and its potential violation could have profound implications for our understanding of fundamental physics and the nature of spacetime. Ongoing experimental efforts aim to test the $\mathcal {CPT}$ symmetry with increasing precision, providing valuable insights into the symmetrical underpinnings of the universe.

Furthermore, as known in the literature, the following table shows some of the discrete $\mathcal {C}$, $\mathcal {P}$ and $\mathcal {T}$ symmetries operations

Table 1 Summary of discrete symmetry operations