pith. sign in

arxiv: 2605.19817 · v1 · pith:V6QRYTADnew · submitted 2026-05-19 · ✦ hep-th · cond-mat.mes-hall· gr-qc· physics.optics

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

Ritsuki Ito , Kazuya Mameda , Naoki Yamamoto This is my paper

Pith reviewed 2026-05-20 04:12 UTC · model grok-4.3

classification ✦ hep-th cond-mat.mes-hallgr-qcphysics.optics
keywords spin Hall effectBerry curvaturegravitonsWigner functionlinearized gravityEinstein-Hilbert actionhelicitycurved spacetime
0
0 comments X

The pith

Gravitons in curved spacetime show a spin Hall effect from their Berry curvature, splitting their energy current twice as much as photons do.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper builds a quantum field theory description of linearized gravity to define Wigner functions for right- and left-handed gravitons. It applies the Wigner transformation to second-order metric perturbations inside the graviton energy-momentum tensor that comes from the Einstein-Hilbert action. This step produces a Berry curvature whose sign reverses between the two helicities and generates a helicity-dependent spin Hall effect for the graviton energy current in curved spacetime. The size of the resulting splitting is exactly twice the size of the analogous splitting for photons. A sympathetic reader would care because the result supplies a concrete quantum signature that distinguishes gravitational fields from electromagnetic ones at the level of energy transport.

Core claim

Based on quantum field theory of linearized gravity, the Wigner function for right- and left-handed gravitons is formulated. By applying the Wigner transformation to the second-order metric perturbations in the graviton energy-momentum tensor obtained from the Einstein-Hilbert action, the spin Hall effect of gravitons emerges in curved spacetime. This effect arises from the Berry curvature of gravitons, which has opposite signs for the two helicities, and produces a helicity-dependent splitting of the graviton energy Hall current whose magnitude is exactly twice that of the spin Hall current for photons.

What carries the argument

Berry curvature of right- and left-handed gravitons obtained by Wigner transformation of the energy-momentum tensor in linearized gravity

If this is right

  • Right- and left-handed gravitons acquire Berry curvatures of opposite sign.
  • The graviton energy Hall current develops a helicity-dependent splitting in curved spacetime.
  • The magnitude of this splitting is exactly twice the size of the corresponding splitting for photons.
  • The entire effect follows directly from the Einstein-Hilbert action at second order in metric perturbations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same Berry-curvature mechanism could be examined for gravitational waves traveling through regions of strong spacetime curvature.
  • The framework may connect to classical descriptions of helicity-dependent transport in general relativity.
  • Analogous splittings might appear in other spin-2 fields or in effective theories that include gravitational degrees of freedom.

Load-bearing premise

The Wigner transformation applied to the second-order metric perturbations in the graviton energy-momentum tensor obtained from the Einstein-Hilbert action yields a well-defined Berry curvature for gravitons.

What would settle it

A calculation of the helicity-dependent splitting of the graviton energy Hall current in a concrete curved spacetime, such as near a black hole, that yields a factor other than exactly two relative to the photon case.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript formulates the Wigner function for right- and left-handed gravitons within the quantum field theory of linearized gravity. Applying the Wigner transformation to the second-order metric perturbations in the graviton energy-momentum tensor obtained from the Einstein-Hilbert action, the authors demonstrate the emergence of the spin Hall effect for gravitons in curved spacetime. This effect is traced to the Berry curvature, which carries opposite signs for the two helicities and produces a helicity-dependent splitting of the graviton energy Hall current whose magnitude is stated to be exactly twice the corresponding splitting for photons.

Significance. If the central derivation is confirmed, the work supplies a first-principles QFT route to the Berry curvature and spin Hall effect of spin-2 particles, extending the photon case in a controlled manner. The exact factor-of-two relation is a sharp, falsifiable prediction that could be relevant for gravitational-wave propagation in inhomogeneous or curved backgrounds.

major comments (1)
  1. The headline result that the splitting magnitude is exactly twice the photon value rests on the contraction of the spin-2 polarization tensors with the second-order graviton EMT inside the Wigner function. The tensorial weighting and the projection onto physical degrees of freedom must be shown to contribute only the helicity factor of 2 with no residual gauge- or normalization-dependent coefficients; an explicit parallel calculation with the Maxwell EMT would be required to establish this.
minor comments (1)
  1. The notation used for the right- and left-handed projectors in the graviton Wigner function should be defined more explicitly, including the precise relation to the transverse-traceless gauge conditions.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive summary and the detailed major comment. We address the concern regarding the precise factor-of-two relation between the graviton and photon spin Hall splittings below.

read point-by-point responses

  1. Referee: The headline result that the splitting magnitude is exactly twice the photon value rests on the contraction of the spin-2 polarization tensors with the second-order graviton EMT inside the Wigner function. The tensorial weighting and the projection onto physical degrees of freedom must be shown to contribute only the helicity factor of 2 with no residual gauge- or normalization-dependent coefficients; an explicit parallel calculation with the Maxwell EMT would be required to establish this.

    Authors: We agree that an explicit side-by-side comparison strengthens the claim. In the derivation, the second-order graviton EMT is obtained by varying the Einstein-Hilbert action to quadratic order in h_{μν}; after Wigner transformation and projection onto the transverse-traceless physical modes, the contraction with the spin-2 polarization tensors ε_{μν}^± (helicity ±2) produces an effective Berry curvature term whose magnitude is twice that obtained from the analogous contraction for photons. The extra factor originates solely from the helicity eigenvalues, with all gauge-dependent pieces eliminated by the same transverse-traceless conditions used for the Maxwell case. To make this transparent, the revised manuscript will add a short parallel calculation using the Maxwell EMT and photon polarization vectors, confirming that no additional normalization or tensorial coefficients survive. This will appear as a new paragraph in Section 3 together with the relevant algebraic steps. revision: yes

Circularity Check

0 steps flagged

Derivation of graviton spin Hall effect from Einstein-Hilbert action via Wigner function shows no circularity

full rationale

The paper presents a derivation starting from the Einstein-Hilbert action for linearized gravity, constructing the graviton energy-momentum tensor at second order, applying the Wigner transformation to obtain the Wigner function for right- and left-handed gravitons, and extracting the Berry curvature and resulting helicity-dependent splitting. The factor-of-two relation to the photon case is stated as an outcome of this calculation rather than an input or fitted parameter. No self-citations are invoked as load-bearing for the central result, no ansatz is smuggled in, and no renaming of known results occurs. The derivation chain remains self-contained against the provided abstract and context, with the exact twice factor emerging from the tensor structure and helicity projections rather than by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the linearized-gravity approximation, the validity of the Wigner-function formalism for gravitons, and the extraction of Berry curvature from second-order metric perturbations; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption Linearized gravity around a background metric is sufficient to capture the leading spin Hall effect.
    Invoked when the Einstein-Hilbert action is expanded to second order in metric perturbations.
  • domain assumption The Wigner transformation applied to the graviton energy-momentum tensor produces a well-defined Berry curvature.
    Central step stated in the abstract.

pith-pipeline@v0.9.0 · 5646 in / 1264 out tokens · 33882 ms · 2026-05-20T04:12:54.762410+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    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... magnitude... exactly twice that of the corresponding spin Hall current for photons.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

39 extracted references · 39 canonical work pages · 16 internal anchors

  1. [1]

    Spin Hall effect of gravitational waves

    N. Yamamoto,Spin Hall effect of gravitational waves,Phys. Rev. D98(2018) 061701 [1708.03113]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  2. [2]

    Oancea, R

    M.A. Oancea, R. Stiskalek and M. Zumalac´ arregui,Frequency- and polarization-dependent lensing of gravitational waves in strong gravitational fields,Phys. Rev. D109(2024) 124045 [2209.06459]

    work page arXiv 2024

  3. [3]

    Andersson, J

    L. Andersson, J. Joudioux, M.A. Oancea and A. Raj,Propagation of polarized gravitational waves,Phys. Rev. D103(2021) 044053 [2012.08363]

    work page arXiv 2021

  4. [4]

    Andersson and M.A

    L. Andersson and M.A. Oancea,Spin Hall effects in the sky,Class. Quant. Grav.40(2023) 154002 [2302.13634]

    work page arXiv 2023

  5. [5]

    Kubota, S

    K.-i. Kubota, S. Arai and S. Mukohyama,Spin optics for gravitational waves lensed by a rotating object,Phys. Rev. D109(2024) 044027 [2309.11024]

    work page arXiv 2024

  6. [6]

    Frolov and A

    V.P. Frolov and A. Koek,Spinoptics in the Kerr spacetime: Polarized wave scattering,Phys. Rev. D111(2025) 104081 [2503.19208]

    work page arXiv 2025

  7. [7]

    Kubota, S

    K.-i. Kubota, S. Arai, H. Motohashi and S. Mukohyama,Spin wave optics for gravitational waves lensed by a Kerr black hole,Phys. Rev. D110(2024) 124011 [2408.03289]

    work page arXiv 2024

  8. [8]

    Relativistic Chiral Kinetic Theory from Quantum Field Theories

    Y. Hidaka, S. Pu and D.-L. Yang,Relativistic Chiral Kinetic Theory from Quantum Field Theories,Phys. Rev. D95(2017) 091901 [1612.04630]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  9. [9]

    Huang, P

    X.-G. Huang, P. Mitkin, A.V. Sadofyev and E. Speranza,Zilch Vortical Effect, Berry Phase, and Kinetic Theory,JHEP10(2020) 117 [2006.03591]

    work page arXiv 2020

  10. [10]

    Hattori, Y

    K. Hattori, Y. Hidaka, N. Yamamoto and D.-L. Yang,Wigner functions and quantum kinetic theory of polarized photons,JHEP02(2021) 001 [2010.13368]. – 15 –

    work page arXiv 2021

  11. [11]

    Chiral kinetic theory in curved spacetime

    Y.-C. Liu, L.-L. Gao, K. Mameda and X.-G. Huang,Chiral kinetic theory in curved spacetime,Phys. Rev. D99(2019) 085014 [1812.10127]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  12. [12]

    Mameda, N

    K. Mameda, N. Yamamoto and D.-L. Yang,Photonic spin Hall effect from quantum kinetic theory in curved spacetime,Phys. Rev. D105(2022) 096019 [2203.08449]

    work page arXiv 2022

  13. [13]

    DeWitt,Quantum theory of gravity

    B.S. DeWitt,Quantum theory of gravity. i. the canonical theory,Phys. Rev.160(1967) 1113

    work page 1967

  14. [14]

    Henneaux and C

    M. Henneaux and C. Teitelboim,Quantization of gauge systems, Princeton University Press, Princeton (1992)

    work page 1992

  15. [15]

    Kleiss and W.J

    R. Kleiss and W.J. Stirling,Spinor Techniques for Calculating p anti-p —>W+- / Z0 + Jets,Nucl. Phys. B262(1985) 235

    work page 1985

  16. [16]

    Gunion and Z

    J.F. Gunion and Z. Kunszt,Improved Analytic Techniques for Tree Graph Calculations and the G g q anti-q Lepton anti-Lepton Subprocess,Phys. Lett. B161(1985) 333

    work page 1985

  17. [17]

    Xu, D.-H

    Z. Xu, D.-H. Zhang and L. Chang,Helicity Amplitudes for Multiple Bremsstrahlung in Massless Nonabelian Gauge Theories,Nucl. Phys. B291(1987) 392

    work page 1987

  18. [18]

    Gupta,Quantization of Einstein’s gravitational field: general treatment,Proc

    S.N. Gupta,Quantization of Einstein’s gravitational field: general treatment,Proc. Phys. Soc. A65(1952) 608

    work page 1952

  19. [19]

    Weinberg,Photons and Gravitons inS-Matrix Theory: Derivation of Charge Conservation and Equality of Gravitational and Inertial Mass,Phys

    S. Weinberg,Photons and Gravitons inS-Matrix Theory: Derivation of Charge Conservation and Equality of Gravitational and Inertial Mass,Phys. Rev.135(1964) B1049

    work page 1964

  20. [20]

    Nonlinear Responses of Chiral Fluids from Kinetic Theory

    Y. Hidaka, S. Pu and D.-L. Yang,Nonlinear Responses of Chiral Fluids from Kinetic Theory, Phys. Rev. D97(2018) 016004 [1710.00278]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  21. [21]

    Bellac,Thermal Field Theory, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge (2011)

    M.L. Bellac,Thermal Field Theory, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge (2011)

    work page 2011

  22. [22]

    The Zilch Vortical Effect

    M.N. Chernodub, A. Cortijo and K. Landsteiner,Zilch vortical effect,Phys. Rev. D98 (2018) 065016 [1807.10705]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  23. [23]

    Higher spin vortical Zilches from Kubo formulae

    C. Copetti and J. Fern´ andez-Pend´ as,Higher spin vortical zilches from Kubo formulas,Phys. Rev. D98(2018) 105008 [1809.08255]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  24. [24]

    Chiral Vortical Effect for Bosons

    A. Avkhadiev and A.V. Sadofyev,Chiral vortical effect for bosons,Phys. Rev. D96(2017) 045015 [1702.07340]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  25. [25]

    Photonic chiral vortical effect

    N. Yamamoto,Photonic chiral vortical effect,Phys. Rev. D96(2017) 051902 [1702.08886]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  26. [26]

    Zyuzin,Landau levels for an electromagnetic wave,Phys

    V.A. Zyuzin,Landau levels for an electromagnetic wave,Phys. Rev. A96(2017) 043830

    work page 2017

  27. [27]

    Chiral Vortical Effect For An Arbitrary Spin

    X.-G. Huang and A.V. Sadofyev,Chiral vortical effect for an arbitrary Spin,JHEP03 (2019) 084 [1805.08779]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  28. [28]

    Prokhorov, O.V

    G.Y. Prokhorov, O.V. Teryaev and V.I. Zakharov,Chiral vortical effect: Black-hole versus flat-space derivation,Phys. Rev. D102(2020) 121702 [2003.11119]

    work page arXiv 2020

  29. [29]

    Weinberg,Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, John Wiley and Sons, New York (1972)

    S. Weinberg,Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, John Wiley and Sons, New York (1972)

    work page 1972

  30. [30]

    Second Order Perturbation Theory in General Relativity: Taub Charges as Integral Constraints

    E. Altas and B. Tekin,Second Order Perturbation Theory in General Relativity: Taub Charges as Integral Constraints,Phys. Rev. D99(2019) 104078 [1903.11982]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  31. [31]

    Kinetic theory with Berry curvature from quantum field theories

    D.T. Son and N. Yamamoto,Kinetic theory with Berry curvature from quantum field theories,Phys. Rev. D87(2013) 085016 [1210.8158]. – 16 –

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  32. [32]

    Lorentz Invariance in Chiral Kinetic Theory

    J.-Y. Chen, D.T. Son, M.A. Stephanov, H.-U. Yee and Y. Yin,Lorentz Invariance in Chiral Kinetic Theory,Phys. Rev. Lett.113(2014) 182302 [1404.5963]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  33. [33]

    Collisions in Chiral Kinetic Theory

    J.-Y. Chen, D.T. Son and M.A. Stephanov,Collisions in Chiral Kinetic Theory,Phys. Rev. Lett.115(2015) 021601 [1502.06966]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  34. [34]

    Geometrical Aspects in Optical Wavepacket Dynamics

    M. Onoda, S. Murakami and N. Nagaosa,Geometrical aspects in optical wavepacket dynamics,Phys. Rev. E74(2006) 066610 [physics/0606178]

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  35. [35]

    Nishida,Optical Magnus effect on gravitational lensing,2603.15697

    Y. Nishida,Optical Magnus effect on gravitational lensing,2603.15697

    work page arXiv

  36. [36]

    Quantum Metric Corrections to Liouville's Theorem and Chiral Kinetic Theory

    K. Mameda and N. Yamamoto,Quantum Metric Corrections to Liouville’s Theorem and Chiral Kinetic Theory,2509.15731

    work page internal anchor Pith review Pith/arXiv arXiv

  37. [37]

    Quantum Metric and Nonlinear Hall Effect of Photons

    K. Akiba and N. Yamamoto,Quantum Metric and Nonlinear Hall Effect of Photons, 2604.27751

    work page internal anchor Pith review Pith/arXiv arXiv

  38. [38]

    Provost and G

    J.P. Provost and G. Vall´ ee,Riemannian structure on manifolds of quantum states,Commun. Math. Phys.76(1980) 289

    work page 1980

  39. [39]

    Hayata, Y

    T. Hayata, Y. Hidaka and K. Mameda,Second order chiral kinetic theory under gravity and antiparallel charge-energy flow,JHEP05(2021) 023 [2012.12494]. – 17 –

    work page arXiv 2021