Spin Hall effect and Berry curvature of gravitons from quantum field theory

Abstract: Based on quantum field theory of linearized gravity, we formulate the Wigner function for right- and left-handed gravitons. By applying the Wigner transformation to the second-order metric perturbations in the graviton energy-momentum tensor obtained from the Einstein-Hilbert action, we demonstrate the emergence of the spin Hall effect of gravitons in curved spacetime. This effect originates from the Berry curvature of gravitons, which has opposite signs for right- and left-handed helicities, and leads to a helicity-dependent splitting of the graviton energy Hall current. The magnitude of this splitting is found to be exactly twice that of the corresponding spin Hall current for photons.

• The paper introduces a quantum-field theoretical framework that constructs a graviton Wigner function for helicity-dependent transport analysis.
• It finds that graviton spin Hall currents are precisely twice those of photons, attributed to their spin-2 nature and associated Berry curvature effects.
• The study extends kinetic theory to curved spacetimes, rigorously addressing diffeomorphism invariance and anomalous energy-momentum currents.

Spin Hall Effect and Berry Curvature of Gravitons from Quantum Field Theory

Overview

The paper presents a quantum field theoretical (QFT) framework for analyzing the spin Hall effect (SHE) and Berry curvature associated with gravitons—massless spin-2 quanta of the linearized gravitational field—in both flat and curved spacetime. By constructing a graviton Wigner function, the analysis rigorously formulates helicity-dependent transport phenomena for right- and left-handed gravitons, generalizing structures previously known for chiral fermions and photons. Notably, the magnitude of the graviton spin Hall current is found to be consistently twice that of photons, reflecting the higher spin of gravitons and consistent with single-particle Berry curvature arguments.

Graviton Wigner Function Formalism

The construction builds on the quantization of linearized gravity in the transverse-traceless (TT) gauge. The graviton field operator, expanded in terms of creation and annihilation operators and polarization tensors, is used to define a helicity-resolved Wigner function for gravitons. The Wigner function captures quantum interference effects up to $\mathcal{O}(\hbar)$ and distinguishes right- and left-handed modes, paralleling analogous treatments for photons and chiral fermions.

The formalism is developed in both flat and curved backgrounds. In the latter, covariant generalizations employ parallel transport and tangent bundle connections to define phase-space distributions in a generally covariant way, preserving diffeomorphism invariance.

Spinor-Helicity Representation

Adopting the spinor-helicity formalism, the graviton polarization tensor is expressed as the symmetrized tensor product of photon polarization vectors. This allows for a systematic derivation of the projection operators and gauge-fixed polarization sums needed for the kinetic description, ensuring elimination of unphysical degrees of freedom in compliance with the TT gauge.

Berry Curvature and Kinetic Theory

By analyzing the graviton Wigner function, the paper identifies explicit helicity-dependent terms in the kinetic equations analogous to Berry curvature corrections established in lower-spin systems. The quantum corrections in curved spacetime are extracted by systematic expansion in $\hbar$, leading to kinetic equations that capture the effects of background gravitational fields on graviton dynamics.

The equations of motion for a single graviton, derived from the transport part of the energy current, include Berry curvature-induced “side-jump” terms in analogy with the well-studied SHE for photons. The magnitude of these corrections is found to be twice that of the photon, consistent with expectations based on graviton helicity $\pm2$.

Energy-Momentum Tensor and Anomalous Currents

The energy-momentum tensor for gravitons is obtained by Wigner-transforming the Einstein-Hilbert action up to second order in metric fluctuations. The resulting tensor is expressed in terms of the Wigner function, facilitating calculation of anomalous contributions such as the chiral vortical effect (in rotating backgrounds) and the graviton SHE (in weak gravitational potentials).

For a rotating metric, the chiral vortical energy current for gravitons (in local thermal equilibrium) is derived:

$$J^{h,\text{vort}}_{i} = \pm \frac{2\zeta(3)}{\pi^2\hbar} T^{3} \omega_{i}$$

where $\omega_{i}$ is the fluid vorticity. For a weak gravity potential, the graviton SHE yields an energy current:

$$J^{h, \text{Hall}}_i = \mp \frac{4\zeta(3)}{\pi^2\hbar} T^3 (\nabla\phi \times \hat{\mathbf{q}})_i$$

where the splitting between right- and left-handed gravitons is equal to twice that for photons, with opposing directions.

Theoretical analysis confirms that these results are frame vector-independent, consistent with generally covariant expectations and paralleling previous findings for fermions and photons.

Contrasts and Strong Claims

• The graviton spin Hall current is precisely twice the photonic value in the weak gravity regime, in both the magnitude and in the structure of the Berry curvature-induced transport, due to the helicity structure of gravitons.
• Frame independence of the energy-momentum tensor is demonstrated up to order $\mathcal{O}(\hbar)$, despite the construction relying on a specific frame vector for defining polarization.

These claims extend and consolidate previous results from one-particle formalisms and non-covariant approaches by embedding them in a fully quantum field theoretical context.

Implications and Outlook

This work bridges the gap between single-particle geometric analysis and QFT in the study of graviton transport phenomena. The formulation enables the extraction of chiral and spin Hall effects in scenarios governed by the Einstein-Hilbert action, with manifest inclusion of quantum geometric phases.

Several implications and future directions arise:

• The results suggest a potentially classical origin for the graviton SHE and Berry curvature effects, just as similar effects for photons can be derived within classical Maxwell theory. An explicit derivation from classical general relativity remains an open problem.
• Extension to $\mathcal{O}(\hbar^2)$ corrections would likely reveal contributions from the quantum metric—an emergent geometric structure identified in recent quantum kinetic theories for chiral fermions and photons.
• The analysis points to a connection with classical gravito-optic effects observed in strong field gravitational backgrounds (such as Kerr spacetime), with possible unification of the quantum and classical treatments.

The practical observation of such effects in realistic astrophysical settings—especially with regards to thermal graviton equilibrium and detectable graviton currents—remains challenging.

Conclusion

This paper provides a comprehensive QFT-based kinetic theory for helicity-dependent graviton transport in curved spacetime, systematically deriving the graviton Berry curvature and associated spin Hall transport terms. The quantized linearized gravity framework robustly generalizes lower-spin kinetic formalisms and clarifies the geometric underpinnings of graviton dynamics through explicit derivations of chiral and anomalous energy currents. The universality of Berry curvature effects across massless particles of different spin is quantitatively affirmed, setting the stage for further exploration of quantum geometric phenomena in gravitational contexts.