Remarks on Physics as Number Theory

Scientific Paper
Title	Remarks on Physics as Number Theory
Read in full	Link to paper
Author(s)	Lucian M Ionescu
Keywords	Multiplicative number theory, Riemann zeta function, Fine structure constant, prime numbers, quarks
Published	2012
Journal	Proceedings of the NPA
Volume	9
No. of pages	13
Pages	232-244

Read the full paper here

Abstract

There are numerous indications that Physics, at its foundations, is algebraic Number Theory, starting with solid state physics evidence in the context of the universal model of Quantum Computing and Digital World Theory. Bohr's Model for the Hydrogen atom is the starting point of a quantum computing model on serial-parallel graphs is provided as the quantum system affording the partition function of the Riemann Gas / Primon model. The propagator of the corresponding discrete Path Integral formalism is a fermionic Riemann zeta function value "closely" related to the experimental value of the fine structure constant of QED. The Kleinian geometry of the primary finite fields unravels a rich structure of the set of prime numbers, and logic reasoning as well as quark masses lead to the conjecture that Fermat primes correspond to quarks.

Overview

Lucian M. Ionescu is a mathematician at Illinois State University, and this NPA conference paper — his own description is "the sketched picture with its underlying riddles" — is a programmatic essay rather than a finished derivation. Its thesis is that physics at its foundation is algebraic number theory: that the objects of physics are integers, primes and finite abelian groups, and that the continuum, space-time, fields and particles are all derived approximations. He invokes Plato's priority of number over substance and compares his position to the way Democritus's atomism anticipated the modern atom at the conceptual level.

Three claims carry the paper. First, that the hydrogen atom is better modelled as a quantum computing circuit on serial-parallel graphs — the Primon model or "Riemann gas" — than by Bohr's orbits. Second, that in the discrete path-integral formalism attached to that model the propagator is a value of the fermionic Riemann zeta function, which he claims is "closely" related to the fine structure constant of QED. Third, that the internal symmetry structure of prime numbers, exhibited as what he calls Fermat trees, picks out the Fermat primes as the true elementary objects, and that these correspond one-to-one with the quark flavours. The departure from the mainstream is total rather than incremental: he rejects ambient space-time, rejects particles, rejects fields, and describes the Schrödinger equation as a wrong turn comparable to Heaviside's truncation of Maxwell's theory.

The argument

Three "clues" from non-mainstream research

Ionescu opens with three premises he takes as established outside the mainstream. There is no ambient space-time background: consistent with general relativity's identification of interaction with the energy-momentum tensor, and remediable by modelling geometry with graph representations in the spirit of Feynman diagrams and Kontsevich's Formality theorem. Electromagnetism is not a U(1) theory: he holds that Maxwell's theory "as truncated by Heaviside and others" is an approximation to a non-commutative U(2) theory of quantum information waves, unifying transverse Maxwell–Hertz waves with the longitudinal waves associated with Tesla and Meyl. And there are neither particles nor fields: wave–particle duality is reinterpreted as the local/global dichotomy, formalised as homology–cohomology duality, so the particle/field separation is inappropriate in a theory with no empty space between particles.

The positive proposal is a categorical one. Morphisms define objects; fermions are the objects and bosons the morphisms; SU(2) and SL(2,ℂ) coefficients play the role of local Lorentz transformations and of connections between "pieces of space-time" (qubits). He labels the underlying spinor–quaternion correspondence 2+2* = 3+1 and links it to Penrose's twistor programme. Everything is quantum information processing, an SU(2) analogue of a Markov process. He is dismissive of accelerator physics on the grounds that it "clearly favors particle aspects" and misleads one into believing the electron is pointlike merely because no parts are resolved.

The Primon model of the hydrogen atom

The technical core replaces Bohr's circular orbits with serial-parallel (SP) graphs. A spectral line with principal quantum number n is identified with the SP-graph of the prime factorisation of n: parallel composition is the direct sum ℤ_n × ℤ_m (multiplication), serial composition is the semidirect product for p-groups, ℤ/p^eℤ built from e copies of ℤ/pℤ, which he reads as harmonics k = 1…e of a fundamental frequency 1/p. Bohr's picture becomes a "bouquet of closed paths" whose lengths are the prime exponents. Each edge is assigned an impedance z_e = 1/p; the resulting partition function is then the Riemann zeta function, which is exactly the standard Primon-gas construction with energy levels E_p = log p.

Ionescu presents this as an upgrade of "old quantum mechanics" — Bohr, Sommerfeld, de Broglie — which he regards as the right road, with Heisenberg's matrix mechanics as premature quantum computing and Schrödinger's wave mechanics as a retreat. Bohr–Sommerfeld quantisation ∫p dq = nh becomes a primitive way of introducing homological periods; the modern replacement is proper resonance modes of the SP-graph, treated like an RLC circuit, with Hodge periods on the mathematics side. Coordinates lose intrinsic meaning and arise only as path integrals of momentum; there are no pointlike charges, only periods — electric charge a 2-period, magnetic charge a 1-period.

The propagator and the coupling constant

Rather than treat one hydrogen atom in isolation, Ionescu models two atoms exchanging a quantum of energy as a single system, geometrically a genus-two Riemann surface (the Hopf bundle being a torus fibration). In a poset with a path integral the correlator is given by the Möbius function μ, which is the convolution inverse of the constant function 1 and hence the Green function — the propagator in Feynman's sense. Its Dirichlet series is the "fermionic" zeta function ζ₋ = DS(μ), satisfying ζ₋·ζ₊ = 1 with ζ₊ = ζ the bosonic partition function.

The propagator is then identified with the probability that two SP-graphs are relatively prime, which is the classical number-theoretic result 1/ζ(2) = 6/π² ≈ 0.6079. Ionescu proposes (Conjectures 4.1 and 4.2) that the fine structure constant of the Primon model is this value, and that the rest mass of the charge is related to the prime zeta value Pζ(2). He supports the association by noting that "energy = log(probability)", so a relation between log α and prime zeta values is to be expected, and cites an independent observation relating log α to a prime zeta value that arose "from a totally different perspective (a relation between the masses of quarks and prime numbers!)".

Theorem 4.1 supplies a derivation of the Primon model's energy levels rather than accepting them ad hoc: taking the interaction Hamiltonian to be log δP where P(n) = φ(n)/n is a density of symmetries, the 2-cochain is exact, a potential energy spectrum exists, and δP = log(1 − 1/p) ≈ −1/p. He connects this to information theory — the probability P(n) = |Aut ℤ_n|/|ℤ_n| is an information content, and by Landauer's principle and recent Maxwell-demon experiments information and energy belong in the action on the same footing. His broad conclusion is that there are no fundamental constants: Planck's constant reflects the discreteness of action, c a limit on information-processing capacity, and α is a propagator computable from combinatorics.

Riemann hypothesis as a statement about characters

Section 4 recasts the Riemann hypothesis: rather than saying the non-trivial zeros lie on Re s = ½, one says the poles of the fermionic partition function have the form s_k = ½ + iσ_k, a statement about equivariant characters taking the value p^−s = p^−1/2exp(−iσ_k log p) at primitive elements. Using the classical counting formula N(T) ≈ (T/2π)log T, he argues that the normalised zeros should be linear combinations of "periods" for a homological basis indexed by the primes — consistent with the Hilbert–Pólya expectation that they are eigenvalues of a Hermitian operator, and with Berry's observation that the statistics are those of a quantum system without time-reversal symmetry and with chaotic classical trajectories.

Fermat trees, and primes as quarks

The final section proposes a hidden hierarchy among the primes. Viewing ℤ_n as a Klein geometry acted on by Aut(ℤ_n), the multiplicative structure of p − 1 becomes visible: ℤ₇ has p − 1 = 2·3, giving orbit structure under multiplication by 2. Ionescu defines the hierarchy tree of a prime inductively: if p = 2^k·q₁^e1···q_m^em + 1 the node is labelled k and its descendants are the trees of the q_j. He calls these Fermat trees; Mersenne primes have trees with a single node, and the construction generalises Proth primes k·2ⁿ + 1. Worked examples: the tree 2–(2–2, 1, 1) reconstructs 2²·5·3² + 1 = 181, prime; but 2³·7 + 1 = 393 = 3·131, not prime.

Only the odd primes of the form 2ⁿ + 1 — the leaves of these trees, forcing n = 2^k and hence the Fermat primes — are proposed as "true quarks":

u = 2, d = 3, s = 5, c = 17, b = 257, t = 65537

with 2 admitted as a trivial prime since Aut(ℤ₂) = 0. Ionescu quotes the corresponding rest masses in MeV as 2.4, 4.8, 104, 1270, 4200, 171200. Conjecture 5.1 states that there are only five Fermat primes and therefore only six quark flavours. The proposed replacement for generations is a hierarchy of levels of internal structure; flavour symmetry is dismissed as "too broken to be true in some sense", though SU(3) colour is retained as reflecting the 2+2* = 3+1 correspondence. The trees also induce a partial order on the primes — p → q if q’s tree is a subtree of p’s — under which 7, 11, 13, 19 sit at the first level above the Fermat "atoms", 23 and 29 at the second, and unoriented cycles appear (43, 7, 3), giving the poset the capacity to carry proper modes.

Assessment

The mathematical furniture Ionescu assembles is real and correctly used at the level of citation. The Primon gas is a legitimate construction in which the Riemann zeta function is the partition function of a boson gas with energies log p; the Möbius function genuinely is the convolution inverse of 1 on the divisibility poset and so genuinely is a Green function in Rota's discrete calculus; ζ₋ζ₊ = 1 is the standard Dirichlet-series identity; and the Hilbert–Pólya and Berry–Keating connections between zeta zeros and quantum spectra are active mathematics, not invention. Theorem 4.1 is the paper's best original moment: deriving the Primon model's E_p = log p from an exactness condition on a symmetry-density Hamiltonian, rather than stipulating it, is the kind of thing the programme needs more of. The insistence that "fundamental constants" may be derived rather than given is a defensible and long-standing dissident position, and the information/energy discussion is sober and correctly attributes Landauer.

But the two headline physical claims do not survive arithmetic. The propagator identified with α is 1/ζ(2) = 6/π² ≈ 0.608. The measured fine structure constant is ≈ 1/137.036 ≈ 0.00730. These differ by a factor of about 83, i.e. by nearly two orders of magnitude — a gap the abstract's scare-quoted word "closely" is asked to carry alone. Ionescu is candid that the relation runs through "energy = log(probability)", but no chain of steps is given that actually produces 137 from ζ(2), and Conjecture 4.2 is stated as a conjecture with a "Proof" that reduces to a list of relations still needing "to relate the probabilistic L¹-norm (non-relativistic) picture and the L²-norm relativistic picture". Both the conjecture and its proof are, on the paper's own admission, incomplete. He also acknowledges directly that "the relations between them are not yet clear at this time" regarding his E_p = 1/p versus the Primon model's E_p = log p — two different energy assignments used in different sections.

The Fermat-prime/quark correspondence is weaker still, and the paper supplies the material to check it. Placing the quoted masses beside the primes gives 2.4/2, 4.8/3, 104/5, 1270/17, 4200/257, 171200/65537 — ratios of 1.2, 1.6, 20.8, 74.7, 16.3, 2.6, with no monotone trend and no functional relation offered. The ordering does not even match: the bottom quark is heavier than the charm by a factor of about 3.3, while 257/17 ≈ 15, and the top exceeds the bottom by roughly 41 while 65537/257 ≈ 255. The mapping is an ordering of six labels onto six numbers, and six-element orderings are cheap. Conjecture 5.1 also inverts the logic: whether there are exactly five Fermat primes is a famous open problem in number theory, so deriving "six quark flavours" from it explains a known fact by an unknown one. And the correspondence takes no position on the quantities that actually individuate quarks — charge, colour, weak isospin — assigning content only to mass.

Finally, the paper's rhetorical register works against it. Standard results are dismissed rather than engaged: Schrödinger's equation is "a step back" and "the dead end", accelerator physics is "Don't think; just compute!", the Bohmian programme is "the song of the swan". The endorsement of longitudinal Tesla–Meyl waves as part of a U(2) electromagnetism is asserted with a single citation and no derivation, and it is the one claim in the paper that is directly testable and directly contradicted by the absence of any detected longitudinal electromagnetic mode in vacuum. Readers should treat this as what its title says — remarks — an inventory of suggestive analogies between number theory and physics, some of which (the Primon gas, the Möbius propagator, zeta zeros as a spectrum) are serious mathematics worth pursuing, and none of which is here brought to the point of reproducing a measured quantity.

See also

• Lucian M Ionescu
• Fine Structure Constant
• Quantum Electrodynamics
• Hydrogen Atom
• Niels Bohr
• Quark
• Richard Feynman
• Werner Heisenberg
• Konstantin Meyl
• Oliver Heaviside
• Maxwell's Equations