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Quantum models of the Riemann zeta function, lattice spin models and algebraic models of entanglement

This paper provides a brief overview of the connections between the Hilbert-Pólya conjecture and the Riemann hypothesis, alongside new findings on p-adic quantum computing, quantum entanglement derived from lattice spin models, and algebraic entanglement models, while also briefly presenting the properties of photons and electrons used in quantum computing.

Original authors: Nikolaj M. Glazunov

Published 2026-06-30

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Nikolaj M. Glazunov

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

This paper by Nikolaj M. Glazunov is a high-level tour through a landscape where mathematics, physics, and computer science overlap. Imagine the author is standing at a crossroads, pointing out three different paths that all seem to lead to the same mysterious destination: the Riemann Hypothesis.

Here is a simple breakdown of the paper's main ideas, using everyday analogies.

1. The Big Goal: The "Prime Number Puzzle"

The central character of this story is the Riemann Zeta Function. You can think of this function as a giant, complex machine that processes numbers. When you plug certain numbers into it, the machine sometimes "stops" (the result is zero).

The Mystery: Most of these "stops" happen in predictable places. But there are special "non-trivial" stops that seem to follow a secret rule. The Riemann Hypothesis claims that all these secret stops lie on a single, straight line (like a tightrope) in the mathematical universe.
The Goal: Proving this hypothesis is one of the hardest problems in math. If you prove it, you unlock deep secrets about how prime numbers are distributed.

2. Path One: The Quantum Music Box (Quantum Hamiltonians)

The paper suggests a way to solve the puzzle using Quantum Physics.

The Analogy: Imagine a quantum system (like an electron or a photon) as a musical instrument. In physics, every instrument has a specific set of notes it can play. These notes are called eigenvalues.
The Connection: The Hilbert-Pólya conjecture (mentioned in the paper) suggests that the "notes" played by a specific, yet-to-be-discovered quantum machine are exactly the same as the "secret stops" of the Riemann Zeta function.
The Paper's Contribution: The author reviews different mathematical "instruments" (Hamiltonians) that physicists have built to try and match these notes. If we can build the right machine, its music will prove the Riemann Hypothesis.

3. Path Two: The Non-Commutative Map (Alain Connes' Work)

The paper also looks at the work of mathematician Alain Connes, who uses a branch of math called Non-Commutative Geometry.

The Analogy: Think of a normal map where you can go North then East, or East then North, and end up in the same spot. In "non-commutative" space, the order matters: going North then East gets you to a different place than going East then North.
The Application: Connes treats the Riemann Hypothesis like a map of a strange, twisted city. By using "semilocal" tools (looking at the city from a few specific angles at once), he tries to find a "trace formula." This is like counting the echoes in a cave to figure out the cave's shape. The paper discusses how these echoes (mathematical traces) relate to the zeros of the Zeta function.

4. Path Three: The Lattice Spin Game (Spin Models)

The author shifts gears to talk about Lattice Spin Models.

The Analogy: Imagine a giant checkerboard (a lattice). On every square, there is a tiny magnet (a "spin") that can point either Up or Down.
The Game: These magnets like to talk to their neighbors. If they point the same way, they are happy (low energy). If they point opposite ways, they are unhappy (high energy).
The Random Walk: The paper discusses a "random walker" moving across this board. It calculates the probability of the walker ending up in a specific spot after a certain number of steps.
The Twist: The author connects this physical game of magnets and random walkers to p-adic numbers. Think of p-adic numbers as a different way of measuring distance, where numbers that look very different to us might actually be "close" together. The paper suggests that studying these "spins" in this weird p-adic world might help us understand quantum entanglement.

5. The Final Piece: Algebraic Entanglement

Finally, the paper touches on Entanglement in a purely mathematical sense, not just a physical one.

The Analogy: In physics, "entanglement" is when two particles are linked so that what happens to one instantly affects the other, no matter how far apart they are.
The Mathematical Version: The author defines a similar concept for Algebraic Number Fields (complex systems of numbers). Imagine you have several different "worlds" of numbers. Usually, these worlds are independent. But sometimes, they get "entangled," meaning you can't separate them cleanly.
The Discovery: The paper notes that this mathematical entanglement was first discovered by the mathematician Serre and appears in the study of Elliptic Curves (a specific type of equation that looks like a twisted loop). The author proposes that we can model this "entanglement" using algebraic rules, similar to how we model quantum particles.

Summary

In short, this paper is a survey of connections. It doesn't claim to have solved the Riemann Hypothesis yet. Instead, it says:

"Look at these different tools: Quantum machines, twisted maps, spinning magnets on a grid, and linked number worlds. They all seem to be whispering clues about the same secret rule (the Riemann Hypothesis). By studying how these tools work together, we might get closer to the answer."

The author also briefly mentions p-adic quantum computing, suggesting that if we can build computers that operate using these "weird" number systems, we might be able to simulate these spin models and entanglement effects more effectively.

Technical Summary: Quantum Models of the Riemann Zeta Function, Lattice Spin Models, and Algebraic Models of Entanglement

Author: Nikolaj M. Glazunov
Institution: Glushkov Institute of Cybernetics NASU, Kiev
Date: June 28, 2026 (arXiv preprint)

1. Problem Statement and Scope

This paper provides a brief overview of theoretical connections between the Hilbert-Pólya conjecture and the Riemann Hypothesis (RH), specifically exploring the hypothesis that the non-trivial zeros of the Riemann zeta function correspond to the eigenvalues of a quantum Hamiltonian. The work situates this problem within the broader context of Selberg's trace formula, noncommutative geometry (specifically the work of Alain Connes), and p-adic quantum mathematics.

Furthermore, the paper addresses the physical and mathematical challenges of understanding quantum entanglement. Due to difficulties in establishing a purely physical basis for entanglement, the author investigates algebraic models of entanglement, drawing parallels between quantum information theory and the arithmetic properties of algebraic number fields, specifically regarding Galois representations and elliptic curves.

2. Methodology

The paper employs a multidisciplinary approach combining:

Spectral Methods: Utilizing the spectral theory of operators to link number-theoretic zeros to quantum energy levels.
Adelic Approach: Integrating local fields (real and p-adic) to analyze the Riemann zeta function and related L-functions.
Lattice Spin Models: Applying combinatorial analysis (Pólya enumeration) and statistical mechanics (Ising models) to model particle behavior on lattices.
Algebraic Number Theory: Using Galois theory and the theory of elliptic curves to define and analyze "entanglement" in the context of field extensions.

The review is structured to first establish the quantum mechanical framework (Hamiltonians, angular momentum), move to the analytic number theory of the Riemann zeta function and Selberg classes, and finally bridge these concepts with lattice models and algebraic entanglement.

3. Key Contributions and Results

3.1. Quantum Hamiltonians and the Riemann Hypothesis

The paper reviews the formulation of the Hilbert-Pólya conjecture, which posits that the non-trivial zeros of $\zeta(s)$ lie on the critical line $\Re(s) = 1/2$ if and only if they are the eigenvalues of a self-adjoint (Hermitian) operator.

Berry-Keating Hamiltonian: The author cites the operator $\hat{H}_{BK} = \hat{x}\hat{p} + \hat{p}\hat{x}$ as a candidate Hamiltonian whose spectrum is related to the zeros of the zeta function.
Functional Equations: The paper reiterates the functional equation of the Riemann zeta function, $\xi(s) = \xi(1-s)$ , derived via the Mellin transform of the theta function, and notes A. Speiser's result linking the RH to the absence of zeros of $\zeta'(s)$ to the left of the critical line.

3.2. Selberg Classes and Trace Formulas

The paper discusses Selberg's trace formula and the Selberg class ( $\mathcal{S}$ ) of Dirichlet series.

Axioms: It outlines the five axioms defining the Selberg class (Dirichlet series convergence, analytic continuation, functional equation, Ramanujan conjecture, and Euler product).
Extended Selberg Class ( $\mathcal{S}^\#$ ): The paper references the work of Kaczorowski and Perelli on the extended class where the Ramanujan conjecture and Euler product are relaxed.
Zero Distribution: Theorems regarding the distribution of zeros for functions in $\mathcal{S}^\#$ are presented, specifically relating the number of zeros of a function $F(s)$ to the zeros of its derivative $F'(s)$ . A corollary states that $F(s)$ satisfies the RH if and only if $F'(s)$ has no zeros in the region $\sigma < 1/2$ .

3.3. Noncommutative Geometry and Connes' Approach

The paper summarizes recent work by Alain Connes and collaborators regarding the spectral realization of zeta zeros.

Semilocal Adelic Operators: It highlights the integration of Connes' trace formula with "semilocal adelic operators" and "prolate wave operators."
Spectral Realization: The results are interpreted as describing the infrared (low-lying) and ultraviolet behaviors of the Riemann zeta function zeros.
Hardy–Titchmarsh Transform: The paper notes the definition of a semilocal analogue of the Hardy–Titchmarsh transform and its relation to Weil cohomology.

3.4. Lattice Spin Models and p-adic Mathematics

The author explores the intersection of spin models and non-Archimedean geometry.

Sub-Planckian Scales: The paper proposes that spacetime geometry may be non-Archimedean at scales below the Planck length ( $\ell_P \approx 1.6 \times 10^{-35}$ m), motivating the study of p-adic spin models and the p-adic rotation group $SO(3)_p$ .
Random Walks on Lattices: The paper derives the probability $p_t(m)$ of a particle returning to a lattice point after $t$ steps using integral operators over probabilistic structures defined by exponential functions (Fourier analysis on lattices).
Ising Model: The interaction energy of adjacent spins in the Ising model is defined, noting that parallel spins ( $\uparrow\uparrow$ ) minimize energy ( $-E_s$ ) while antiparallel spins ( $\uparrow\downarrow$ ) increase it ( $E_s$ ).

3.5. Algebraic Models of Entanglement

The paper introduces a novel algebraic definition of entanglement distinct from the standard quantum mechanical density matrix definition.

Definition: A family of Galois extensions $\{K_n\}$ of a field $K$ is defined as entangled if the natural inclusion map from the Galois group of their compositum to the product of their individual Galois groups is not an isomorphism.
Origin: This phenomenon is attributed to Serre and is observed in the division fields of elliptic curves with complex multiplication.
Connection: This algebraic "entanglement" is presented as a model to investigate the physical basis of quantum entanglement through the lens of elliptic curves and Galois representations.

4. Significance and Claims

The paper claims to provide a brief overview of new results (dated 2026) in p-adic quantum computing and algebraic models of entanglement. Its significance lies in:

Synthesizing Disparate Fields: It attempts to unify the spectral approach to the Riemann Hypothesis (Hilbert-Pólya, Connes) with lattice spin models and algebraic number theory.
Algebraic Entanglement: It proposes a rigorous algebraic framework (based on Galois representations) to model entanglement, offering an alternative perspective to purely physical interpretations.
p-adic Quantum Mechanics: It motivates the study of quantum systems at sub-Planckian scales using non-Archimedean (p-adic) mathematics, specifically through the lens of p-adic rotation groups.

The author explicitly states that the examination is conducted on the basis of spectral methods and the adelic approach, continuing a presentation given at a seminar in December 2025. The paper does not claim to have proven the Riemann Hypothesis but rather reviews the theoretical machinery and recent developments connecting quantum Hamiltonians to the distribution of zeta zeros.