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)