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March 24, 20260 citationsOpen Access

Exact Spectrum of the Regge Hessian for the Equilateral 4-Simplex

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MSMukka Srivatsav

Key Points

  • This research aims to compute the complete spectrum of the Regge Hessian for the equilateral 4-simplex.
  • Utilized symmetry methods and Bose–Mesner algebra of T(5) for analysis.
  • Derived the full eigenvalue structure analytically and verified it numerically.
  • Extended analysis to multi-simplex systems and derived a cubic characteristic equation.
  • Determined the corresponding propagator and evaluated it in the long-wavelength limit.
  • Obtained the exact dispersing sector through a cubic characteristic equation.
  • Showed that the shape-mode sector follows a 1/p² behavior in the long-wavelength limit.
  • Established a fully analytic spectral foundation for simplicial quantum gravity.

Abstract

We compute the complete spectrum of the Regge Hessian for the equilateral 4-simplex using symmetry methods and the Bose–Mesner algebra of the triangular scheme T(5) . The full eigenvalue structure is derived analytically and verified numerically. We extend the analysis to multi-simplex systems, obtain the exact dispersing sector via a cubic characteristic equation, and derive the corresponding propagator. In the long-wavelength limit, the shape-mode sector exhibits a 1/p² behavior consistent with the linearised Einstein action. These results provide a fully analytic spectral foundation for simplicial quantum gravity.

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Cite This Study

Mukka Srivatsav (2026) studied this question.

synapsesocial.com/papers/69c2295caeb5a845df0d3a8ehttps://doi.org/10.5281/zenodo.19163942
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