What question did this study set out to answer?

This research aims to develop a rigorous framework for spectral quantum gravity and address geometric singularities.

July 31, 20260 citationsOpen Access

Rigorous Functional Measure, Non-Singular Spectral Regularization, and Horizon Modular Duality in SQG

Key Points

This research aims to develop a rigorous framework for spectral quantum gravity and address geometric singularities.
Construction of the functional measure dµ SQG over distribution spaces utilizing the Krein-Bochner-Minlos theorem.
Implementation of fundamental scale boundaries, including the Planck and de Sitter horizon scales.
Analysis of the local horizon von Neumann algebra and verification of the KMS condition at Hawking temperature.
Proved that the variance kernel operator is of trace class, ensuring well-defined correlations.
Demonstrated that the Trans-Planckian regime serves as a ghost-regulator to resolve UV divergences.
Established a strictly positive-definite Källén-Lehmann measure post scale-symmetry breaking.

Abstract

We construct a mathematically rigorous functional measure dµ SQG for Spectral Quantum Gravity (SQG) over distribution spaces S ′ utilizing the Krein-Bochner-Minlos theorem. By implementing the fundamental scale boundaries—the Planck scale L Pl and the de Sitter horizon scale L Λ —we prove that the variance kernel operator K −1/2 governing substrate mode correlations is of trace class. The Trans-Planckian regime acts as an intrinsic, indefinite ghost-regulator, resolving geometric spacetime singularities (r → 0) and UV divergences. Upon scale-symmetry breaking, these ghost modes dynamically decouple, leaving a strictly positive-definite Källén-Lehmann measure. Finally, we establish that the induced physical state on the local horizon von Neumann algebra M(L Λ ) satisfies the Kubo-Martin-Schwinger (KMS) condition at Hawking temperature T dS = (2πL Λ ) −1 with respect to the TomitaTakesaki modular automorphism group.

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