Krein Space Quantization and a Spectral Interpretation of the Riemann $\xi$-Function

Abstract

The invariant two-point function of a scalar field in de Sitter spacetime can be expressed in terms of Legendre functions via Lorentzian harmonic analysis. Using this structure together with the Mehler--Fock transform, we obtain an integral representation of the completed Riemann $\xi$-function in which the Legendre kernel appears naturally. Motivated by this correspondence, we introduce a retarded propagator whose spectral weight is given by the $\xi$-function and analyze it within the framework of Krein space quantization, where sign-indefinite spectral measures are admissible. This construction yields a geometric and spectral interpretation of the $\xi$-function restricted to the critical line and relates the asymptotic spacing of its zeros to a mass--time scaling in de Sitter geometry. The results provide a novel interpretive framework linking de Sitter quantum field theory, harmonic analysis, and analytic number theory.

arXiv comment13 pages

References in corpus (11)

• Two-point Functions and Quantum Fields in de Sitter Universe
• Gupta-Bleuler quantization for minimally coupled scalar fields in de Sitter space
• Particle decays and stability on the de Sitter universe
• N=1 de Sitter Supersymmetry Algebra
• Entropy of Quantum Fields in de Sitter Space-time
• Quantum Yang-Mills theory in de Sitter ambient space formalism
• Scalar and vector gauges unification in de Sitter ambient space formalism
• Asymptotic states and $S$-matrix operator in de Sitter ambient space formalism
• "Krein" regularization method
• Axiomatic de Sitter quantum Yang-Mills theory with color confinement and mass gap
• Quantum de Sitter geometry