Dirac equation in a gauge-field background in the Moyal plane (2501.14508v1)

Abstract: Starting with the Dirac equation for an electron in a constant electromagnetic background on a noncommutative (NC) plane, we obtain a gauge invariant description of the system. Surprisingly, the dynamics of the system is dictated by the standard form of Lorentz force law, once the effective magnetic and electric fields $\left( B^{NC}, \, E^{NC} \right)$ correct up to leading order in the NC parameter are identified. The Hall effect is studied using the NC corrected fields in the non-relativistic (NR) limit. This shows that noncommutativity affects the cyclotron frequency, but leaves the Hall conductivity unaffected at least to first order in the NC parameter. Owing to the NC corrected magnetic field, the hyperfine splitting of Hydrogen atom spectrum also shows a first order correction which helps establish an upper bound on the spatial NC parameter.

• The paper demonstrates that conventional extensions of the Dirac equation fail under gauge-field influences, necessitating noncommutative frameworks like the Moyal star product and Seiberg-Witten maps.
• The analysis shows that, with NC corrections, electron dynamics conform to the traditional Lorentz force law through redefined electric and magnetic fields.
• The work explores practical implications by reexamining the quantum Hall effect and proposing a first-order NC correction to hyperfine splitting in hydrogen, suggesting feasibility for experimental constraints.

Analysis of the Dirac Equation in a Gauge-Field on the Moyal Plane

The paper investigated the dynamics of an electron modeled by the Dirac equation on the noncommutative (NC) Moyal plane within a constant electromagnetic background. By employing a gauge-invariant approach, the authors revealed the necessity of adopting a noncommutative field theory framework to derive consistent physical predictions.

Key Findings

The initial examination uncovers the insufficiencies when conventional quantum mechanical systems are extended to their NC counterparts at the equation level, specifically under gauge-field influences. Attempting to generalize the Dirac equation under these conditions directly leads to gauge-dependent outcomes, compromising the equation's utility in theoretical predictions. This discrepancy is resolved by engaging the Moyal star product framework and Seiberg-Witten maps, achieving a commutative equivalent theory that maintains gauge invariance in NC coordinates.

Lorentz Force and Dynamics

Remarkably, within this refined theoretical structure, the dynamics correspond to the traditional Lorentz force law upon identifying appropriate NC-corrected electric ($\mathbf{E}_{\text{NC}}$) and magnetic ($\mathbf{B}_{\text{NC}}$) fields. This alignment affirms the theoretical consistency across transformations and mappings employed in NC field theories.

Applications: Hall Effect and Hyperfine Splitting

Utilizing the non-relativistic limit of the NC-corrected Dirac Hamiltonian, two important physical phenomena were investigated:

1. Quantum Hall Effect: The analysis shows that while the cyclotron frequency is modulated by the introduction of NC parameters, the Hall conductivity remains unaffected to first order in the NC parameter. This stands in contrast to findings from previous literature which suggested otherwise, highlighting the need for further experimental and theoretical consolidation.
2. Hyperfine Splitting in Hydrogen: The paper investigates the NC effects on the hyperfine structure of hydrogen atoms' spectra. For the first time, a first-order NC correction to hyperfine splitting is proposed, offering a mechanism to set an upper bound on the spatial noncommutative parameter $\theta$. Although found to be a weaker constraint compared to other estimates, it underscores the influence of spatial noncommutativity contingent on background magnetic fields.

Theoretical and Practical Implications

This research demonstrates the intricacy of modeling subatomic systems within NC space and underscores the necessity of NC field theories for capturing modifications arising due to noncommutativity in high-energy scenarios. The gauge-invariant methodology adopted here facilitates clearer physical interpretations and could support experimental endeavors aiming to decipher such NC effects.

Furthermore, this work contributes to the broader discourse on NC geometry's implications in particle physics and quantum field theory, propelling further speculation and exploration in theories that integrate NC parameters with quantum mechanical and relativistic domains alike.

Future Directions

In light of these findings, there is a salient demand for refined experimental techniques capable of detecting subtle influences of NC corrections in electromagnetic systems. Additionally, extending this framework to include other interactions and particle species could deepen understanding and enhance the models' applicability, paving the way for richer insights into quantum behavior at reduced length scales. Continued exploration in this vein can potentially unravel new foundational insights in theoretical physics, particularly in realms marrying quantum mechanics with relativistic and quantum gravity frameworks.