Krein Space Quantization and a Spectral Interpretation of the Riemann $ξ$-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 $ξ$-function in which the Legendre kernel appears naturally. Motivated by this correspondence, we introduce a retarded propagator whose spectral weight is given by the $ξ$-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 $ξ$-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.