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This research aims to compute the full spectrum of the Regge Hessian for the equilateral 4-simplex.

March 24, 2026Open Access

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

Key Points

This research aims to compute the full spectrum of the Regge Hessian for the equilateral 4-simplex.
Computed the complete spectrum using symmetry methods and Bose–Mesner algebra
Derived full eigenvalue structure analytically and verified it numerically
Extended analysis to multi-simplex systems and obtained exact dispersing sector through cubic characteristic equation
Derived the corresponding propagator.
Full eigenvalue structure derived analytically
Confirmed results numerically
Shape-mode sector exhibits a 1/p² behavior in long-wavelength limit
Provided 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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Mukka Srivatsav

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

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