Linearized Einstein gravity from Lorentzian Regge calculus on the Kuhn lattice, and the divergence of its one-loop graviton limit
- 1Recognition Physics Institute
Abstract
We study a real-time path sum over squared edge lengths on the Kuhn triangulation of the periodic hypercubic lattice, with the Lorentzian Regge action and the flat metric I − κ11T, at κ = 1, where the edges that advance one coordinate are null. The value κ = 1 is an identification, motivated by an order on those edges. Classically, linearized Einstein gravity is recovered as a long-wavelength limit of lattice solutions. On every torus of odd side not divisible by 105 the flat Hessian has no zero modes beyond constant strains and vertex moves. For band-limited conserved sources off the light cone, the long-wave limit of the linear lattice field is the de Donder solution. With a resting body and a model medium, the unconstrained Regge equations on all edges have exact solutions for small coupling; their weak-field long-wave limit gives Newton's potential with G = 1/(8πs), where s is the action scale. At one loop, in a stated diagram model, the quantum limit fails. Four exact massless doublers at the corners of the Brillouin zone make the zero-momentum anisotropy of the graviton self-energy diverge like (5/(44√335 π2)) log(1/ε) as the regulator ε is removed after the volume, because every other degenerate zero of the operator is non-pinched; the proof's computations are exact. The one-loop two-mass amplitude assembled from the long-wave Feynman rules has the coefficient −(41/10π)G2ℏm1m2/r3 in an infrared-subtracted reading; the lattice path sum at κ = 1 is not shown to inherit it. This record contains the preprint and QGTitle1_computations.zip: the exact data behind every computation on which a theorem depends, with independent programs (Python, exact rational arithmetic) that re-check every identity, table and inequality.
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BibTeX
@misc{washburn2026linearized,
title = {{Linearized Einstein gravity from Lorentzian Regge calculus on the Kuhn lattice, and the divergence of its one-loop graviton limit}},
author = {Jonathan Washburn},
howpublished = {Zenodo (CERN European Organization for Nuclear Research)},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23764490},
url = {https://doi.org/10.5281/zenodo.23764490},
note = {CC BY},
}
RIS
TY - UNPB
TI - Linearized Einstein gravity from Lorentzian Regge calculus on the Kuhn lattice, and the divergence of its one-loop graviton limit
AU - Jonathan Washburn
PY - 2026
DA - 2026/10/29
PB - Zenodo (CERN European Organization for Nuclear Research)
DO - 10.5281/zenodo.23764490
UR - https://doi.org/10.5281/zenodo.23764490
AB - We study a real-time path sum over squared edge lengths on the Kuhn triangulation of the periodic hypercubic lattice, with the Lorentzian Regge action and the flat metric I − κ11T, at κ = 1, where the edges that advance one coordinate are null. The value κ = 1 is an identification, motivated by an order on those edges. Classically, linearized Einstein gravity is recovered as a long-wavelength limit of lattice solutions. On every torus of odd side not divisible by 105 the flat Hessian has no zero modes beyond constant strains and vertex moves. For band-limited conserved sources off the light cone, the long-wave limit of the linear lattice field is the de Donder solution. With a resting body and a model medium, the unconstrained Regge equations on all edges have exact solutions for small coupling; their weak-field long-wave limit gives Newton's potential with G = 1/(8πs), where s is the action scale. At one loop, in a stated diagram model, the quantum limit fails. Four exact massless doublers at the corners of the Brillouin zone make the zero-momentum anisotropy of the graviton self-energy diverge like (5/(44√335 π2)) log(1/ε) as the regulator ε is removed after the volume, because every other degenerate zero of the operator is non-pinched; the proof's computations are exact. The one-loop two-mass amplitude assembled from the long-wave Feynman rules has the coefficient −(41/10π)G2ℏm1m2/r3 in an infrared-subtracted reading; the lattice path sum at κ = 1 is not shown to inherit it. This record contains the preprint and QGTitle1_computations.zip: the exact data behind every computation on which a theorem depends, with independent programs (Python, exact rational arithmetic) that re-check every identity, table and inequality.
ER -