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March 24, 2026Open Access

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

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.

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

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Abstract

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