Spin in Uniform Gravity, Hidden Momentum, and the Anomalous Hall Effect

Based on the paper "Spin in Uniform Gravity, Hidden Momentum, and the Anomalous Hall Effect" by Andrzej Czarnecki and Ting Gao, here is a detailed technical summary covering the problem, methodology, key contributions, results, and significance.

1. Problem Statement

The paper addresses a recent controversy regarding the existence of a "gravitational spin Hall effect" (SHE) in a uniform gravitational field.

• The Claim: A recent study [8] asserted that Dirac wave packets in a uniform gravitational field ($g$) exhibit a polarization-dependent transverse deflection (a spin Hall effect) even without an external transport current or transport geometry.
• The Conflict: This claim suggests a mechanism analogous to the Anomalous Hall Effect (AHE) in ferromagnetics, where spin-orbit coupling leads to transverse currents. However, the authors argue that the physical conditions in a uniform gravitational field differ fundamentally from those in crystalline solids, and the claimed effect may be an artifact of incorrect initial state preparation.

2. Methodology

The authors employ a multi-faceted approach combining classical mechanics, relativistic quantum mechanics, and perturbation theory:

• Hidden Momentum Model: They utilize a classical "circulating mass" model (a rectangular pipe with moving masses) to derive the concept of hidden momentum ($p_{hidden}$) for a spinning object in a gravitational field. This involves analyzing relativistic time dilation effects ($\gamma$) between the upper and lower sections of the loop.
• Foldy–Wouthuysen (FW) Transformation: They apply the FW transformation to the Dirac Hamiltonian in a linear gravitational potential. This allows them to derive a non-relativistic single-particle Hamiltonian ($H_{FW}$) valid to leading order in $g$ and all orders in $1/c$.
• Wave Packet Evolution: They analyze the time evolution of Gaussian wave packets using the derived Hamiltonian, specifically examining the relationship between canonical momentum ($p$), velocity ($v$), and spin ($\sigma$).
• Comparative Analysis: They contrast the gravitational scenario with the Karplus-Luttinger (KL) mechanism for the AHE in ferromagnetics, specifically looking at the role of lattice periodicity and interband matrix elements.

3. Key Contributions and Theoretical Framework

A. The Hidden Momentum Mechanism

The authors demonstrate that a spinning object in a uniform gravitational field acquires a hidden momentum given by:
$$\mathbf{p}_{hidden} \sim \frac{\mathbf{S} \times \mathbf{g}}{c^2}$$
This momentum arises because the relativistic mass-energy of the circulating internal degrees of freedom (spin) interacts with the gravitational potential. Crucially, this hidden momentum modifies the relationship between the canonical momentum and the physical velocity.

• The velocity operator contains a spin-dependent term: $\mathbf{v} \propto \mathbf{p} - \mathbf{p}_{hidden}$.
• Consequently, a state prepared with zero canonical momentum ($\langle \mathbf{p} \rangle = 0$) is not at mechanical rest; it possesses a non-zero initial velocity due to the hidden momentum.

B. Resolution of the "Deflection" Paradox

The authors re-evaluate the claim of transverse deflection found in [8]:

• The Artifact: The previous claim of deflection assumed an initial state with $\langle \mathbf{p} \rangle = 0$. However, because of the hidden momentum, this state actually has an initial transverse velocity $v_x \propto \sigma_y$. The observed "deflection" is simply the particle moving at this initial velocity, not a dynamical acceleration caused by gravity.
• True Rest Condition: To prepare a particle truly at rest (zero physical velocity), one must introduce a spin-dependent phase in the initial wave packet to cancel the hidden momentum.
• Result: When the particle is prepared in a state of true mechanical rest, the $O(g)$ transverse drift vanishes.

C. Absence of Gravitational SHE

Using the FW transformation, the authors prove that for a broad class of states, spin-dependent transverse motion cannot appear earlier than $O(g^2)$. Therefore, a linear-order gravitational spin Hall effect does not exist in a uniform field.

D. Contrast with Anomalous Hall Effect (AHE)

The paper highlights why the gravitational case differs from the AHE in ferromagnetics:

• Ferromagnetics (KL Mechanism): The AHE relies on lattice periodicity ($U(\mathbf{r})$). The periodic potential creates a non-trivial Bloch eigenbasis, leading to non-zero interband matrix elements (Berry curvature) that generate an intrinsic transverse current ($J \propto \mathbf{M} \times \mathbf{E}$).
• Uniform Gravity: There is no lattice periodicity; the natural basis is plane waves. Consequently, the "regular periodic perturbation" term responsible for the intrinsic Hall current in the KL mechanism has no counterpart in a uniform gravitational field. The essential ingredient (the crystal lattice) is missing.

4. Key Results

1. No Linear Gravitational SHE: There is no spin-dependent transverse deflection of free Dirac particles in a uniform gravitational field at linear order in $g$ ($O(g)$).
2. Hidden Momentum is Real: Spinning objects in gravity carry hidden momentum ($\mathbf{p}_{hidden} = \mathbf{L} \times \mathbf{g} / c^2$), which decouples canonical momentum from physical velocity.
3. Trajectory Independence: If particles (scalar and spin-1/2) are prepared with identical initial physical velocities and parallel trajectories, their subsequent trajectories in a uniform gravitational field are identical regardless of spin. The spin-dependent deflection claimed in previous literature is an artifact of failing to account for hidden momentum in the initial state preparation.
4. Fundamental Distinction: The mechanism driving the Anomalous Hall Effect in solids (Berry curvature arising from lattice periodicity) is fundamentally absent in free-fall scenarios in uniform gravity.

5. Significance

• Theoretical Clarification: The paper resolves a significant tension in the literature regarding gravitational spin effects, correcting a misinterpretation of the Foldy–Wouthuysen Hamiltonian and initial state preparation.
• Experimental Implications: It suggests that experiments searching for a "gravitational spin Hall effect" in uniform fields (e.g., free-fall experiments) will not observe a linear-order spin-dependent deflection, provided the particles are correctly prepared in a state of mechanical rest.
• Conceptual Insight: It reinforces the distinction between transport phenomena in condensed matter (driven by lattice symmetries and Berry curvature) and the behavior of free particles in external fields. It clarifies that while the Hamiltonians may look superficially similar, the physical origins of transverse transport are distinct.
• Hidden Momentum Validation: It provides a robust quantum mechanical confirmation of the hidden momentum concept in the context of gravity, extending its relevance beyond electromagnetism.

In summary, Czarnecki and Gao demonstrate that the "gravitational spin Hall effect" is a non-existent phenomenon at linear order in uniform fields, primarily due to the cancellation of effects by hidden momentum and the lack of the lattice periodicity required for intrinsic Hall transport.