Drift-Governed Spectra and the Riemann Hypothesis: A Gauge-Theoretic Analogy

Abstract

This essay proposes a speculative structural analogy between the noise-independence principle governing exceptional points in Gaussian quantum channels and the spectral theory of the Riemann zeta function.

Overview

This essay mines a recent result in open quantum systems—showing that the spectrum of a Gaussian channel is governed solely by its drift matrix, with diffusion gauged away by a Lyapunov transformation—for structural patterns that might illuminate the Riemann Hypothesis. The speculative bridge treats the "drift" as a universal spectral skeleton (akin to random-matrix statistics) and "diffusion" as arithmetic noise arising from the primes.

The Source Result

Quintela Rodríguez (arXiv:2601.16121v1) proves that for a stable multimode bosonic Gaussian channel Ψ = (X, Y, δ), the eigenvalues and Jordan structure are controlled entirely by the drift matrix X. A Gaussian similarity transformation V_S, constructed from the solution S of a Stein equation, gauges away the diffusion matrix Y. Consequently, exceptional points—parameter values where the channel becomes non-diagonalizable—are determined solely by defectiveness of X, while Y dresses only the eigenoperators and steady states.

The Proposed Analogy

We speculate that the Riemann zeta function ζ(s) admits an operator-theoretic representation in which the non-trivial zeros form the spectrum of a "drift" operator H₀, while the arithmetic information carried by the primes appears as a "diffusion" term that can be formally gauged away without moving the zeros. This mirrors the Hilbert-Pólya conjecture but incorporates the specific structural pattern of noise-independence via similarity transformation.

Assessment

The analogy is rated a formal analogy at best. While the drift-diffusion separation in Gaussian channels is exact and finite-dimensional, the zeta function presents an infinite-dimensional problem where the zeros are fundamentally determined by arithmetic data. The essay details the failure modes—particularly the lack of an exact gauge symmetry for the explicit formula—and proposes numerical experiments on finite-dimensional truncations to test whether the structural pattern persists.

This essay was produced by an automated research pipeline and has not been peer reviewed; conjectures herein are unproven.