Hessaby’s Theory of Infinitely Extended Particles

Introduction

In 1947, an Iranian physicist named Seyyed Mahmoud Hessaby proposed a radical departure from the dominant picture of elementary particles. Rather than treating particles as dimensionless points — the assumption underlying most 20th-century quantum theory — Hessaby argued that particles must have infinite spatial extension: their energy and charge spread continuously across all of space, with density approaching zero at infinity but never reaching it at any finite distance.

This idea, developed across three papers spanning thirty years, represents a genuine attempt to unify the fundamental forces of nature and eliminate the mathematical infinities that have long troubled quantum field theory.

The Problem Hessaby Was Trying to Solve

By the 1940s, physicists using quantum electrodynamics (QED) were running into a disturbing problem: calculations of basic quantities — such as the electron’s self-energy — produced infinite results. These infinities arose directly from treating the electron as a point-like object with zero size. When you concentrate all of a particle’s charge and mass into a single point, the electromagnetic field energy around it diverges.

The standard fix, developed by Feynman, Schwinger, and Tomonaga in the late 1940s, is called renormalization: infinities are systematically canceled by redefining the bare parameters of the theory. Renormalization works remarkably well — QED predictions match experiment to twelve decimal places. But many physicists, including Paul Dirac himself, found it philosophically unsatisfying. Dirac wrote: “I must say that I am very dissatisfied with the situation, because this so-called ‘good theory’ does involve neglecting infinities which appear in its equations.”

Hessaby’s approach was more radical: eliminate the point-particle assumption entirely.

Hessaby: The Man and His Work

Seyyed Mahmoud Hessaby (1903–1992) is recognized as the Father of Modern Physics in Iran. His institutional contributions are extraordinary: he founded Tehran University’s physics department, the Iranian Physical Society, the Atomic Energy Organization of Iran, and the Institute for Geophysics, among others.

In the late 1940s, he received an invitation from Albert Einstein to visit Princeton’s Institute for Advanced Study — not as a student, but as a visiting researcher. It was during this period that he developed the initial version of what he called the theory of Continuous Particles.

The Core Idea

Hessaby’s fundamental postulate:

The gravitational, electric, and nuclear fields are all special cases of a single, more general field.

In his framework, a particle is not a point but a field configuration — a spherically symmetric distribution of energy (and charge, for charged particles) that fills all of space. The energy density is highest near the center and falls off smoothly to zero at infinity, but is never zero at any finite distance. The total energy integrated over all space equals the particle’s mass.

Crucially: no infinities appear in any integration, because the energy is never concentrated at a single point.

Mathematical Framework

Hessaby works within general relativity. He writes the spacetime line element in spherical symmetry as:

where α, β, γ, δ depend only on the radial coordinate r. He introduces a generalized four-potential Φ_μ, and defines the field tensor:

The energy-momentum tensor U_μν is constructed from this field, and by identifying it with the Einstein tensor R_μν, he derives that the scalar curvature R = 0 — a self-consistency condition that constrains the form of the potentials.

From these constraints, he derives expressions for three types of potential:

Particle Mass Predictions

When these potentials are inserted into the wave equations of quantum mechanics, Hessaby obtains predictions for particle masses:

The reproducer of the 2011 arXiv paper, Sina Khorasani, notes: “This paper presents the only existing theory which unifies the three forces and also successfully estimates the mass ratios of various elementary particles. At present, there is simply no other theory being capable of estimation of particle mass ratios, at least within the framework of Standard Model.”

Critical Analysis: Where the Theory Stands Today

Hessaby’s theory deserves to be taken seriously as a creative and internally consistent attempt at unification. But from the standpoint of modern physics, it faces several significant challenges:

1. The original problem has been resolved differently

Renormalization, while philosophically questioned, works with extraordinary precision. The Standard Model — built entirely on point-particle quantum field theory — successfully predicted the W boson, Z boson, gluon, and Higgs boson before any of them were observed in experiment. The urgency of Hessaby’s alternative is considerably reduced.

2. Locality and causality

If a particle’s field extends over all space, its “presence” is felt everywhere simultaneously. In relativistic quantum field theory, locality — that interactions happen at a spacetime point — is what guarantees causality (effects cannot precede causes). Infinitely extended particles create immediate tension with this principle. Hessaby’s framework does not address how causality is preserved.

3. Quantum mechanics is absent

The theory is formulated in classical general relativity. There is no quantization procedure, no commutation relations, no spin-statistics theorem, no gauge symmetry in the modern sense. Any complete theory of elementary particles must be fundamentally quantum mechanical — this is not optional.

4. Mass predictions require independent verification

The claim that the theory correctly estimates muon and baryon masses is intriguing, but these results have not been independently reproduced with the precision required to distinguish them from dimensional analysis or order-of-magnitude coincidence. The Standard Model makes the same predictions to far greater accuracy from a well-established foundation.

5. No peer review of the complete theory

The 1977 unified version was never published in a peer-reviewed journal. The 2011 arXiv submission is a reproduction of an unpublished manuscript. While Khorasani notes that “Hessaby’s derivations seem to be completely flawless,” community review — with adversarial scrutiny — is the process that actually tests a theory’s consistency.

Legacy and Modern Relevance

Despite these criticisms, Hessaby’s core intuition was not wrong. The pathology of point particles has driven some of the most productive research programs in modern theoretical physics:

• String Theory replaces point particles with 1-dimensional extended objects (strings), motivated by exactly the same infinity problem Hessaby identified in the 1940s.
• Loop Quantum Gravity proposes that spacetime itself has discrete, granular structure — another way of avoiding the singularities of point particles.
• Non-commutative geometry and other approaches similarly question whether spacetime is well-defined at arbitrarily small scales.

Hessaby was working largely in isolation from the main centers of theoretical physics — in Tehran, in the 1940s–70s — without access to the rapid community feedback that normally sharpens a physical theory. That he produced an internally consistent classical unified field theory with particle mass predictions under these conditions is a remarkable achievement, and a testament to his caliber as a physicist.

His theory remains an important piece of Iranian scientific history and a thought-provoking alternative perspective — best appreciated with clear eyes about both its insights and its limitations.

References

1. Hessaby, S.M. (1947). Proceedings of the National Academy of Sciences (USA).
2. Hessaby, S.M. (1957). Modèle de particule infinie.
3. Khorasani, S. (2011). Theory of Infinitely Extended Particles (reproduction of Hessaby 1977). arXiv:1106.2863 [physics.gen-ph].
4. Mahmoud Hessabi — Wikipedia.