The Riemann Hypothesis as a Stability Criterion of a Non-Local Operator on a Primal Manifold: A Novel Approach

To capture this "all primes at once," we defined the manifold $\mathcal{P}$ using the ring of adeles $\mathbb{A}_{\mathbb{Q}}$ We propose that $\mathcal{P}$ is a quotient space of the ideles (invertible adeles), which naturally encodes the multiplicative structure of the primes and possesses a self-similar, fractal topology.

$$ \mathcal{P} \subset \mathbb{A}_{\mathbb{Q}}^{\times} / \mathbb{Q}^{\times} $$

Correlation: Inspired by the work of Tate, Iwasawa algebraic number theory Tate's thesis ("Fourier Analysis in Number Fields and Hecke's Zeta-Functions"), which uses adelic integration to prove the functional equation for general L-functions we look at this algebraic space as a novel geometric 'phase space' for a dynamical system.

The operator $H$ cannot be a local differential operator like the standard Laplacian $\nabla^2$ because the value of $\zeta(s)$ depends on all primes globally. We formalised this by defining $H$ as a non-local integral operator, where the state at point $x$ is influenced by the state at all other points $y$ weighted by an arithmetic kernel $K(x, y)$

$$ (H\psi)(x) = \int_{\mathcal{P}} K(x, y) \psi(y) d\mu(y) $$

Correlation: This is like non-local operators found in fractional quantum mechanics, as in Laskin's "Fractional Quantum Mechanics". Integral operators are fundamental to functional analysis (Fredholm and Hilbert). The challenge is defining the kernel $K(x,y)$ to match the arithmetic of the zeta function.

The functional equation $\xi(s) = \xi(1-s)$ must correspond to a fundamental symmetry of the system. We thought of an involution map $\mathcal{I}: \mathcal{P} \to \mathcal{P}$ like $s \mapsto 1-s$ For $H$ to be self-adjoint (required for real eigenvalues), its kernel must be invariant under this symmetry.

$$ K(x, y) = K(\mathcal{I}x, \mathcal{I}y) $$

Correlation: Linking symmetries to conserved quantities is core to both physics (Noether's Theorem) and mathematics: Alain Connes' work on the "spectral realization" of the Riemann zeros ("Trace formula in noncommutative geometry and the zeros of the Riemann zeta function") constructs a space where the functional equation is a duality whereas this is a direct geometric-dynamical mechanism.

We decompose $H = H_{\text{amp}} - H_{\text{decay}}$. The amplification component $H_{\text{amp}}$ embodies the multiplicative structure of the primes from the Euler product. An "arithmetic force" that consistently drives the amplitude of a wavefunction $\psi$ outwards, away from equilibrium.

$$ (H_{\text{amp}}\psi)(x) \sim \sum_{p \in \text{Primes}} \log(p) \cdot \mathcal{T}_p \psi(x) $$

Here, $\mathcal{T}_p$ is a scaling/translation operator on $\mathcal{P}$ associated with prime $p$

Correlation: Transfer operators in dynamical systems; Ruelle's review "Dynamical Zeta Functions and Transfer Operators". The sum over primes weighted by $\log(p)$ is a like a fragrance of explicit formulas in analytic number theory.

The decay component $H_{\text{decay}}$ acts as the restoring force derived from the global topology and the symmetry $\mathcal{I}$ A stable, finite-energy eigenstate (a zero) can only exist where these two competing forces are in perfect equilibrium: $H_{\text{amp}}\psi = H_{\text{decay}}\psi$ We felt this balance is only achievable for wavefunctions supported on the fixed point set of the symmetry $\mathcal{I}$ the geometric critical line.

An eigenstate off the critical line would have its growth and decay factors unbalanced, causing it to either diverge or collapse: $\int |\psi|^2 d\mu \to \infty$ or $\int |\psi|^2 d\mu \to 0$ Neither is a valid physical state.

Correlation: The novel bit: The RH as a problem of finding ground states in a potential; a common problem in quantum mechanics / QFT. This stability criteria defining fundamental properties of a system is everyhere in physics; stellar evolution, thermodynamics, electricity and magnetism and even cancellations between bosonic and fermionic degrees of freedom for stable vacuum states.