Completing Born and Peng's statistical mechanics of fields ('apeiron' theory): deriving the number of apeirons, particle masses and collision cross-sections, and universal divergence regularization

In a Nature letter of 5 February 1944, Max Born and H. W. Peng announced a reformulation of quantum field theory in which a field confined to a finite volume is quantized through volume-level commutation laws, the wave vector k becomes a dynamical constant of the motion, and groups of pure states sharing one k-eigenvalue — called 'apeirons', after Anaximander's boundless — are occupied according to a Fermi–Dirac-like statistic that gives the field's total energy a natural ultraviolet cutoff. The authors left open: applying the theory to the detailed determination of the masses of elementary particles and their mutual collision cross-sections; deriving the total number n of apeirons (equivalently the cutoff k_m, fixed only empirically through its relation to the electron radius) from first principles; extending the statistics to the electron field, where a negative zero-point energy blocks canonical equilibrium of the apeiron distribution unless interactions are included; and demonstrating, rather than merely expecting, that all divergences of the usual field theories are regularized. The program stopped with Part III, where the authors left these questions open, and it was not continued in the apeiron framework. The problem remains open.

Background

By the early 1940s the quantum theory of fields — the framework in which particles such as photons and mesons are described as quanta of fields extending over space — was formulated in the canonical way introduced by Heisenberg and Pauli: field operators are attached to each space-time point, and the field is expanded in Fourier modes indexed by a wave vector $\mathbf{k}$ . Each mode then behaves as a harmonic oscillator whose lowest state still carries a half-quantum of energy (the 'zero-point energy'), and since infinitely many modes exist in any volume, the formal sum of zero-point energies diverges; the same divergences reappear in computed self-energies (the energy a particle's own field contributes to its mass) and in collision cross-sections (the standard measure of scattering probability). No accepted way to remove these infinities existed before 1947.

Max Born and H. W. Peng, then working in Edinburgh, proposed an alternative construction, announced in the Nature letter 'Statistical Mechanics of Fields and the 'Apeiron'' (5 February 1944) and developed in the three-part paper 'Quantum Mechanics of Fields' in the Proceedings of the Royal Society of Edinburgh. They treat the field in a finite volume $\Omega$ as a single mechanical system described only by its total energy and momentum (the classical energy–momentum densities multiplied by $\Omega$ ); every field component is an operator for the whole volume, not a function of coordinates. The field components satisfy simple commutation laws of the same form as the particle law $pq - qp = \hbar/i$ — for a scalar meson field with potential $v$ , gradient $f$ and time derivative $g$ : $[v, g^*] = [v^*, g] = i\hbar c/\Omega$ and $[g^*, f] = [f^*, g] = \mathbf{k}\hbar c/\Omega$ — in which the wave vector $\mathbf{k}$ appears not as a passive label but as a dynamical variable, a constant of the motion whose eigenvalues, for large $\Omega$ , form a cubic lattice in $\mathbf{k}$ -space.

The eigenstates of a pure field are then grouped by their $\mathbf{k}$ -value; each such group, which is intermediate between a single particle and the whole system (in the phrasing of Part III, it corresponds to one irreducible submatrix of the reducible field matrices, where an irreducible matrix is one that cannot be block-decomposed further), was named an 'apeiron', after Anaximander's boundless substrate. The statistical layer is the new step: ordinary quantum statistics distributes quanta over states, whereas here the $\mathbf{k}$ -points themselves can be occupied or not. For the scalar meson field the meson occupation numbers $N^+_{\mathbf{k}}, N^-_{\mathbf{k}}$ (positive and negative mesons) follow Bose–Einstein statistics (any number of indistinguishable bosons per state), while the apeiron occupation $\Delta_{\mathbf{k}} \in \{0,1\}$ follows a Fermi–Dirac-like distribution (at most one occupant, as for electrons). With $n$ apeirons in thermal equilibrium, the thermal average $\bar{\Delta}_{\mathbf{k}}$ cuts off large $|\mathbf{k}|$ , so the total energy $E = \sum_{\mathbf{k}} \Delta_{\mathbf{k}}\, \varepsilon_{\mathbf{k}}$ , with $\varepsilon_{\mathbf{k}}$ the energy of the apeiron at wave vector $\mathbf{k}$ , converges where the usual theory's zero-point sum diverges — the framework's central promise.

The 1944 letter and the companion papers also record the theory's limitations. The application to the detailed determination of the masses of elementary particles and their mutual collision cross-sections had not been carried out. The number $n$ of apeirons (equivalently the radius $k_m$ of the filled sphere in $\mathbf{k}$ -space, beyond which $\bar{\Delta}_{\mathbf{k}}$ vanishes at low temperature) is not computed from first principles but chosen to reproduce particle self-energies, leading to the empirical relation $k_m r_0 \sim \hbar$ with $r_0$ the electron radius (the letter gives this relation in natural units with $\hbar = c = 1$ ) — leaving $n$ a free parameter. For the electronic (Dirac) field the theory yields a negative zero-point energy; in the words of Part II, 'the application of statistical mechanics to the apeirons of a pure electron (Dirac) field leads to no canonical equilibrium'; the authors set this aside by appeal to interactions (electron ensembles such as those in metals exist only because interaction energy dominates thermal energy), while providing no interaction treatment in the letter. And the claim that all divergent integrals of the usual photon and meson theories are made convergent by apeiron statistics was an expectation, not a demonstrated theorem — Part II states only that the new statistics 'seems to overcome some of the difficulties (divergence of integrals) occurring in the usual quantum theory of fields'. Part III (1946) began the interactive treatment for the electromagnetic and electron fields but reported that this required 'essential modifications' of the theory's basic ideas, and the series ended there.

Problem Statement

Carry out the development of the Born–Peng apeiron theory that its authors left open in the 1944 letter: bring the statistical mechanics of fields into contact with realistic particle physics. Within the framework as defined by the letter and Parts I–III of 'Quantum Mechanics of Fields' (finite-volume commutation laws making the wave vector $\mathbf{k}$ a dynamical constant of the motion; apeirons as groups of pure states sharing a $\mathbf{k}$ -eigenvalue, with Fermi–Dirac-like occupation $\Delta_{\mathbf{k}} \in \{0,1\}$ and Bose–Einstein occupation of meson numbers; total energy $E = \sum_{\mathbf{k}} \Delta_{\mathbf{k}} \varepsilon_{\mathbf{k}}$ , where $\varepsilon_{\mathbf{k}}$ is the energy carried by the apeiron of wave vector $\mathbf{k}$ ), the following are jointly required:

(1) Apply the theory to the detailed determination of the masses (self-energies) of the elementary particles and of their mutual collision cross-sections.

(2) Determine the total number $n$ of apeirons in thermal equilibrium — equivalently the $\mathbf{k}$ -space cutoff $k_m$ , fixed in 1944 only empirically through the self-energy relation $k_m r_0 \sim \hbar$ with $r_0$ the electron radius (in the natural units $\hbar = c = 1$ of that relation) — from first principles.

(3) Determine whether interactions restore canonical equilibrium of the apeiron distribution for the electronic (Dirac) field, and if so establish the resulting statistical treatment. The theory's negative zero-point energy prevents canonical equilibrium of the apeiron distribution for electrons alone, and the authors deferred that case by appealing to interaction energies their treatment omitted.

(4) Settle the universal claim on which the theory rests. The regularization property was asserted by the authors only as an expectation — Part II: the new statistics 'seems to overcome some of the difficulties (divergence of integrals) occurring in the usual quantum theory of fields' — and never demonstrated: demonstrate, or refute within the framework, that all divergent sums and integrals of the usual quantum theory of photons, mesons and electrons are made convergent by the apeiron statistics.

These four items are the one program — the completion of the apeiron statistical mechanics of fields — and are jointly required: a complete solution settles all four, and settling only some of them leaves the program incomplete. Item (2) fixes the framework's only free parameter, item (3) settles the field where its statistics is obstructed, and items (1) and (4) are the confrontation with the physics the theory was built to deliver.

References

Max Born and H. W. Peng, "Statistical Mechanics of Fields and the 'Apeiron'", Nature 153, 164–165 (1944). DOI: 10.1038/153164a0. https://doi.org/10.1038/153164a0
Max Born and H. W. Peng, "VII.—Quantum Mechanics of Fields. I. Pure Fields", Proceedings of the Royal Society of Edinburgh, Section A: Mathematics 62, 40–57 (1944). DOI: 10.1017/S0080454100006415. https://doi.org/10.1017/S0080454100006415
Max Born and H. W. Peng, "XII.—Quantum Mechanics of Fields. II. Statistics of Pure Fields", Proceedings of the Royal Society of Edinburgh, Section A: Mathematics 62, 92–102 (1944). DOI: 10.1017/S0080454100006464. https://doi.org/10.1017/S0080454100006464
Max Born and H. W. Peng, "XV.—Quantum Mechanics of Fields. III. Electromagnetic Field and Electron Field in Interaction", Proceedings of the Royal Society of Edinburgh, Section A: Mathematics 62, 127–137 (1946). DOI: 10.1017/S0080454100006518. https://doi.org/10.1017/S0080454100006518
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