Between 1944 and 1946 Max Born and H. W. Peng published a three-part 'quantum mechanics of fields' in which the components of a field in a finite volume are single reducible operators ('total matrices') whose irreducible diagonal blocks -- called apeirons, after Anaximander's boundless primordial substrate -- are labelled by eigenvalue pairs k′k', l′l' of wave-vector operators kk (electromagnetic) and ll (electronic). Unlike the Heisenberg-Pauli Fourier quantization, which automatically includes all wave-vectors, the formalism admits an arbitrary distribution of apeirons, and this freedom was proposed as a cure for the divergent integrals of quantum field theory. Part III (1946), extending the method to the interacting Maxwell-Dirac field, proves that kk and ll commute with the total energy and momentum, so the apeiron distribution is a fixed constant of motion -- yet it is left undetermined: the statistical route of Part II is declared inapplicable to the provisional interaction theory, and the authors suggest the distribution may be fixed by its effects on the self-energies of quanta and on the transition probabilities of collision processes, deferring grander questions about the connection of ultimate particles. The motivating divergence problem was subsequently handled by renormalization -- absorbing the divergent quantities into redefined finite physical parameters -- in mainstream quantum electrodynamics rather than by any apeiron mechanism.

Background

Before the completion of renormalized quantum electrodynamics around 1950, the divergences of quantum field theory were widely regarded as a defect of the quantization procedure itself. In the canonical method of Heisenberg and Pauli (1929), a field in a box of volume Ω\Omega is expanded in a Fourier series; the Fourier coefficients become non-commuting operators, and every wave-vector contributes a 'radiation oscillator' with a zero-point energy. Sums over the infinitely many wave-vectors -- for instance in the self-energy of the electron -- then diverge. The fermionic (electron) field is quantized with anticommutation rules of the type introduced by Jordan and Wigner (1928), but the mode structure is the same: all wave-vectors are present.

In 1944-1946 Max Born and H. W. Peng, then at Edinburgh, proposed an alternative quantization in three papers in the Proceedings of the Royal Society of Edinburgh, Section A. Each field component (the vector potential, the transverse electric and magnetic fields, the Dirac spinor field, and auxiliary fields) is taken to be a single operator, called a total matrix, belonging to the whole volume Ω\Omega rather than to points in space. The postulated commutation and anticommutation laws for these total matrices introduce self-adjoint wave-vector operators, kk for the electromagnetic field and ll for the electron field, which commute with all field variables. Consequently the total matrices are reducible: in the representation where kk and ll are diagonal, each decomposes into irreducible blocks indexed by eigenvalue pairs (k′k', l′l'). Each block is an apeiron (ἄπειρον, 'the boundless' -- Anaximander of Miletos' term for the primordial substrate, borrowed by the authors). In the published words of Part III, the eigenvalues of the energy and momentum 'represent neither the states of single particles nor of a system of particles, but of something intermediate which corresponds to the simple oscillators of Heisenberg-Pauli and which we have called apeirons'. The total energy and momentum of the field are defined as traces over the apeiron blocks rather than as sums over all wave-vectors.

Part II (1944) developed the statistical mechanics of an assembly of apeirons in the hope of determining the distribution statistically; a published footnote records that a pure Dirac field of apeirons admits no canonical equilibrium. Part III (1946) extended the formalism to the interacting Maxwell-Dirac field: it postulated the total-matrix laws for the coupled fields, showed that the diagonal blocks satisfy exactly the Heisenberg-Pauli commutation laws for the Fourier coefficients, defined the charge and current densities by a trace construction with a shift matrix, and established that kk and ll commute with the total energy EE and momentum pp. The physical picture is of a field as an assembly of apeirons whose wave-vectors are fixed once and for all; the interaction changes only the occupation numbers of quanta (photons and electrons) within each apeiron.

The advertised advantage over Heisenberg-Pauli was the freedom of the distribution: where the Fourier method automatically includes every wave-vector -- a uniform distribution over momentum space -- the new theory admits a selection, and the divergent sums over all wave-vectors become sums over whatever apeirons actually exist, with the divergent integrals becoming convergent in principle while all distribution-independent predictions are preserved. The corresponding gap is the subject of this problem: nothing in the formalism says which selection is realized. The distribution is a constant of motion, not fixed by the dynamics; the Part II statistical machinery is declared not to carry over to the provisional interaction theory; and the authors defer how the distribution might be determined, suggesting the self-energies of quanta and the transition probabilities of collision processes as the observables that could pin it down, together with anticipated consequences for the connection of ultimate particles that they reserve for future work in relation to the most general principles of quantum theory.

Problem Statement

In the Born-Peng total-matrix quantization of the interacting Maxwell-Dirac field (Part III, 1946), the theory fixes the operator algebra of the field components, the trace definitions of the total energy and momentum, and the fact that the wave-vector operators kk and ll are constants of motion; it does not fix which eigenvalue pairs (k′k', l′l') -- which apeirons -- actually occur, admitting an arbitrary distribution where the Heisenberg-Pauli theory automatically includes all wave-vectors uniformly. A distribution is here understood as a selection of apeirons -- eigenvalue pairs (k′k', l′l') of the wave-vector operators kk and ll -- whose contributions enter the sums/traces by which Part III defines the total energy and momentum; the Heisenberg-Pauli limit corresponds to the uniform selection of all wave-vectors. Whether the source's intended class includes infinite apeiron assemblies whose defining sums converge must be settled by the solver from the papers' own formalism.

Determine the apeiron distribution: state what selects the realized set of apeiron wave-vectors -- a principle inside the theory, a consistency condition on the formalism, or confrontation with observation.

The question may admit no answer within the formalism: the theory may be inconsistent, its observables may be independent of the distribution so that no experiment can select it, or the formalism may admit no distribution for which its defining sums/traces of total energy and momentum, and hence its observables, are well defined. The source additionally anticipates consequences of the theory for the connection of ultimate particles but explicitly defers that program to future work in relation to the most general principles of quantum theory.

References

  1. 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(2), 127-137 (1946). DOI: 10.1017/S0080454100006518, https://doi.org/10.1017/S0080454100006518
  2. 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(1), 92-102 (1944). DOI: 10.1017/S0080454100006464, https://doi.org/10.1017/S0080454100006464
  3. 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(1), 40-57 (1944). DOI: 10.1017/S0080454100006415, https://doi.org/10.1017/S0080454100006415
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