krein-space-riemann-xi

name	krein-space-riemann-xi
description	Spectral interpretation of the Riemann xi-function via Krein space quantization in de Sitter QFT. Uses invariant two-point functions, Legendre functions, Lorentzian harmonic analysis, and Mehler-Fock transform to construct a retarded propagator with xi-function spectral weight. Activation: Krein space quantization, Riemann xi-function spectral, de Sitter QFT, Legendre function, Mehler-Fock transform, Hilbert-Polya, critical line zeros
metadata	{"arxiv_id":"2606.13932","published":"2026-06-11","authors":"Multiple authors","tags":["quantum","number-theory","riemann-hypothesis","krein-space","de-sitter","spectral-theory"]}

Krein Space Quantization and Riemann xi-Function

Description

Constructs a geometric and spectral interpretation of the Riemann ξ-function (completed zeta function) restricted to the critical line, using Krein space quantization of a scalar field in de Sitter spacetime. The invariant two-point function expressed via Legendre functions connects to ξ-function through Mehler-Fock transform.

Activation Keywords

• Krein space quantization
• Riemann xi-function spectral
• de Sitter QFT
• Legendre function
• Mehler-Fock transform
• Hilbert-Polya
• critical line zeros
• sign-indefinite spectral measure

Core Concepts

de Sitter → Legendre → ξ-Function Chain

1. de Sitter two-point function → expressed via Legendre functions (Lorentzian harmonic analysis)
2. Mehler-Fock transform → maps Legendre kernel to integral representation of ξ-function
3. Retarded propagator → constructed with ξ-function as spectral weight
4. Krein space → allows sign-indefinite spectral measures (essential for ξ-function zeros)

Krein Space Quantization

Standard Hilbert space requires positive-definite inner products. Krein spaces admit sign-indefinite spectral measures, enabling the construction of a propagator whose spectrum is the ξ-function. This is the key mathematical innovation.

Mass-Time Scaling and Zero Spacing

The asymptotic spacing of ξ-function zeros relates to a mass-time scaling in de Sitter geometry. This provides a physical interpretation of the zero distribution pattern.

Usage Patterns

Pattern 1: Spectral Construction

1. Start with de Sitter scalar field two-point function
2. Express via Legendre functions
3. Apply Mehler-Fock transform → ξ-function integral representation
4. Construct retarded propagator with ξ-function spectral weight
5. Use Krein space framework to handle sign-indefiniteness

Pattern 2: Zero Distribution Analysis

• Analyze zero spacing through de Sitter mass-time scaling
• Geometric interpretation of critical line restriction
• Connect to Hilbert-Polya conjecture via spectral theory

Pitfalls

• Krein spaces are non-standard: Most QFT uses Hilbert spaces — sign-indefinite measures require careful handling
• Not a proof of RH: Provides interpretive framework, not proof of Riemann Hypothesis
• Legendre function asymptotics matter: The Mehler-Fock transform convergence depends on precise Legendre function behavior

Related Skills

• quantum-number-theory-algorithms (quantum number theory)
• quantum-foundations-probability (quantum foundations)
• quantum-models-riemann-zeta-lattice-spin (Riemann zeta quantum models)
• stark-units-sic-overlaps (algebraic number theory + quantum)