Krein Space Quantization and a Spectral Interpretation of the Riemann ξ-Function

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.