Quantum geometry of gravitons

This paper investigates the quantum geometric foundations of graviton transport in curved spacetime. While previous studies have primarily focused on the Berry curvature to explain phenomena like the gravitational spin Hall effect and the chiral vortical effect (CVE), this work seeks to determine whether a broader set of gauge-invariant geometric structures—well-known in condensed matter physics—also governs graviton dynamics. The author employs quantum field theory in curved spacetime and the spinor-helicity formalism to construct graviton Wigner functions and stress-energy tensors. By performing a gradient expansion, the study maps these physical observables to specific quantum geometric tensors, including the quantum metric, the quantum Levi-Civita connection, and quantum nonmetricity.

The research reveals that graviton transport is fundamentally a manifestation of quantum geometry. The author shows that the polarization mode expansion of the graviton field is entirely captured by these geometric structures. Specifically, the study identifies:

• Zeroth-order: The perfect-fluid stress tensor of a graviton gas arises from the trace of the quantum metric, providing a geometric interpretation for classical results.
• Second-order: A nonchiral quantum gravitomagnetic effect (QGME) emerges, driven by the concerted action of the Berry curvature, quantum metric, and quantum Levi-Civita connection.
• Third-order: An inhomogeneous chiral vortical effect (ICVE) appears, which accounts for local variations in spacetime vorticity and is described by the quantum nonmetricity tensor.

These findings suggest that the spinor-helicity formalism establishes a deep duality between the geometry of quantum state manifolds and the polarization tensors of gravitons, allowing for a unified treatment of bosonic and fermionic systems.

This work bridges the gap between high-energy gravitational physics and the flourishing field of quantum geometry in condensed matter. By demonstrating that graviton transport is entirely quantum-geometric, the paper provides a powerful theoretical framework for analyzing gravitational perturbations in diverse environments, from rotating black holes to the early universe. Furthermore, the framework's natural inclusion of lower-spin excitations offers a versatile tool for future research into relativistic many-body systems and potential applications in straintronics and electron hydrodynamics.