Core Concepts

This paper presents a detailed derivation of the Pauli equation for a charged spin-1/2 particle in a weak gravitational field using a direct non-relativistic approximation of the Dirac equation, comparing the results with previous approaches relying on Foldy-Wouthuysen transformations and highlighting discrepancies in the Newtonian limit.

Abstract

**Bibliographic Information:**Oliveira, S. W. P., Oyadomari, G. Y., & Shapiro, I. L. (2024). Pauli equation and charged spin-1/2 particle in a weak gravitational field. arXiv preprint arXiv:2301.10848v2.**Research Objective:**To derive the Pauli equation for a charged spin-1/2 particle in a weak gravitational field using a direct non-relativistic approximation of the Dirac equation and compare the results with previous approaches.**Methodology:**The authors employ a direct non-relativistic approximation of the Dirac equation in curved spacetime, expanding the metric tensor around a flat background and performing a power series expansion in the inverse mass of the particle. They then impose a synchronous gauge condition to simplify the resulting equations and derive the Pauli equation and the equations of motion for the particle. The results are compared with those obtained in previous works using Foldy-Wouthuysen transformations.**Key Findings:**- The authors successfully derive the Pauli equation for a charged spin-1/2 particle in a weak gravitational field using a direct non-relativistic approximation.
- The derived equation is consistent with previous results obtained using perturbative and exact Foldy-Wouthuysen transformations in the case of a plane gravitational wave background.
- However, discrepancies are found in the Newtonian limit compared to previous works employing Foldy-Wouthuysen transformations, particularly in the numerical coefficients of certain terms.
- The authors suggest that these discrepancies might arise from the potential energy being proportional to the mass of the test particle in the Newtonian limit, leading to ambiguities in the 1/m expansion.

**Main Conclusions:**The direct non-relativistic approximation provides a consistent alternative approach to deriving the Pauli equation in curved spacetime. However, the discrepancies observed in the Newtonian limit highlight potential ambiguities and scheme-dependent results when dealing with gravitational fields where the potential energy is proportional to the particle's mass.**Significance:**This work contributes to the understanding of the non-relativistic limit of quantum mechanics in curved spacetime and provides a valuable comparison of different approaches to this problem. The highlighted discrepancies in the Newtonian limit raise important questions about the validity and interpretation of different approximation schemes in this regime.**Limitations and Future Research:**The study is limited to weak gravitational fields and a specific gauge condition. Further research could explore the non-relativistic limit in more general spacetimes and gauge choices. Additionally, investigating the observed discrepancies in the Newtonian limit and their potential implications for experimental tests of gravity would be of significant interest.

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by Samuel W. P.... at **arxiv.org** 10-15-2024

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The choice of gauge condition significantly affects the form of the Pauli equation and the equations of motion in curved spacetime. This is a manifestation of the fundamental principle of general covariance in general relativity, which states that the laws of physics should be independent of the choice of coordinates.
Here's a breakdown of how the gauge condition influences the results:
Simplification of Equations: Choosing a specific gauge can significantly simplify the mathematical expressions involved. In the provided context, the authors employ the synchronous gauge condition (h_0k = 0), which eliminates several terms in the derived equations. This simplification makes the analysis more manageable.
Physical Interpretation: Different gauge choices correspond to different choices of reference frames. While the underlying physics remains the same, the interpretation of individual terms in the equations can change. For instance, certain terms might be attributed to inertial forces in one gauge and to gravitational forces in another.
Comparison with Other Results: When comparing results derived using different gauge conditions, it's crucial to ensure consistency. Direct comparison is only meaningful if the results are transformed to a common gauge.
Limitations: Choosing a specific gauge might obscure certain physical effects or introduce coordinate singularities. It's essential to be aware of the limitations imposed by the chosen gauge and consider alternative gauges if necessary.
In summary, the choice of gauge condition is not merely a mathematical convenience but has profound implications for the interpretation and analysis of the Pauli equation and equations of motion in curved spacetime.

The discrepancies observed in the Newtonian limit, particularly the differing numerical coefficients in the Hamiltonian and equations of motion, might indeed be resolved or at least better understood by employing a different regularization scheme or a more sophisticated treatment of the 1/m expansion. Here's why:
Ambiguities in 1/m Expansion: As the authors point out, the Newtonian limit presents a unique challenge because the gravitational potential couples to the mass of the test particle, leading to terms proportional to mΦ. Since the non-relativistic expansion is in inverse powers of m, this situation can introduce ambiguities.
Regularization Scheme Dependence: The choice of regularization scheme, which essentially dictates how one handles divergent or ill-defined expressions during the derivation, can influence the final result. Different schemes might lead to different finite contributions, potentially explaining the observed discrepancies in the coefficients.
Higher-Order Terms: The discrepancies might arise from neglecting higher-order terms in the 1/m expansion. A more accurate treatment might involve retaining these terms or employing resummation techniques to capture their cumulative effect.
Alternative Expansion Parameters: Instead of solely relying on the 1/m expansion, exploring alternative expansion parameters, such as the ratio of the particle's Compton wavelength to the characteristic length scale of the gravitational field, might provide a more accurate description in the Newtonian limit.
In conclusion, while the discrepancies observed in the Newtonian limit are not uncommon in such derivations, exploring alternative regularization schemes and refining the 1/m expansion could lead to a more consistent and accurate description of the non-relativistic limit of the Dirac equation in a weak gravitational field.

The spin-gravity coupling terms derived in the paper, while small, have intriguing potential experimental implications. These terms suggest that the spin of a particle is not merely a passive property but dynamically interacts with the gravitational field. Here are some potential experimental avenues to explore:
Precision Measurements of Spin Precession: The spin-gravity coupling could manifest as subtle shifts in the precession frequency of particles with spin in a gravitational field. Experiments using ultra-sensitive atom interferometers or neutron interferometers could potentially detect these minute variations.
Spin-Dependent Gravitational Force: The equations of motion suggest a possible spin-dependent gravitational force. This force, though weak, might be detectable by precisely measuring the motion of polarized masses or by looking for spin-dependent deflections of particles in a gravitational field.
Macroscopic Quantum Effects: For macroscopic objects with significant net spin, the spin-gravity coupling might lead to observable quantum effects. Experiments involving levitated micro- or nano-scale objects with controlled spin could probe this regime.
Astrophysical Observations: While challenging, astrophysical observations of the motion of spinning objects in strong gravitational fields, such as pulsars or black hole binaries, could provide indirect evidence of spin-gravity coupling.
Modified Gravity Tests: The spin-gravity coupling terms could be a signature of physics beyond general relativity. Experiments designed to test alternative theories of gravity, such as those searching for Lorentz violation or torsion, might be sensitive to these effects.
Realizing these experiments would require pushing the boundaries of current technology, demanding exceptional sensitivity and control over experimental parameters. However, the potential payoff in terms of understanding the interplay between spin and gravity, and possibly uncovering new physics, makes pursuing these experimental avenues a worthwhile endeavor.

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