Dirac's 1953 Physica paper 'The Lorentz transformation and absolute time' reformulates classical electrodynamics as a Hamiltonian theory of irrotational, separately conserved electron and positron streams evolving through a preferred succession of simultaneity surfaces — an absolute time — while keeping measured physics Lorentz invariant. The construction rests on fixing the electromagnetic gauge by the constraint AμAμ=k2A^\mu A_\mu=k^2 with k=m/ek=m/e, which ties the whole theory to the electron's mass-to-charge ratio, describes spinless matter, excludes pair creation, and defines the absolute time only for irrotational charge flow. Dirac explicitly posed the completion of this primitive starting point as an open program: build the more elaborate absolute-time theory that includes electron spin, accommodates other particles with different mass-to-charge ratios, gives the absolute-time variable (whose conjugate momentum is identified with the total particle density) a physical meaning that experiment could reveal — including its status after quantization — and extends beyond the irrotational approximation, in the hope of improving on quantum electrodynamics.

Background

Between 1951 and 1954 P. A. M. Dirac developed what he called a new classical theory of electrons (parts I–III in the Proceedings of the Royal Society), together with a companion paper in Physica, 'The Lorentz transformation and absolute time' (1953), which is the source of the present problem. The starting point (Dirac 1951) replaces the point electron of classical electrodynamics by a continuous stream of charged matter interacting with the electromagnetic field. The relativistic four-potential AμA^\mu (the combination of the scalar and vector potentials into a four-vector) is not gauge-free as in standard electrodynamics: Dirac fixes the electromagnetic gauge by imposing the constraint AμAμ=k2A^\mu A_\mu=k^2, where k=m/ek=m/e is the electron's mass-to-charge ratio. With this choice the charged matter can be described through the field alone, and all corpuscles in the stream necessarily carry one and the same mass-to-charge ratio.

The 1953 Physica paper turns this into a Hamiltonian theory of the surface-deformation type that Dirac had developed for relativistic fields: the state of the system is specified on a spacelike surface, and the evolution is generated by infinitesimal deformations of that surface, parametrized by arbitrary functions (one for displacements along the normal of the surface, three for tangential displacements), so the formalism provides a whole family of Hamiltonians rather than a single one — the Hamiltonian expression of the absence of a preferred time in relativity. The 1953 construction fixes the normal-deformation function in terms of the theory's own variables, which makes the succession of surfaces into a dynamical, preferred foliation of spacelike simultaneity surfaces. The parameter labelling this succession is what Dirac calls absolute time: the Lorentz transformation (the standard transformation between inertial frames in special relativity) is retained for measured coordinates, but the dynamics singles out one foliation — the simultaneity surfaces orthogonal to the local velocity of the charge stream. Such surfaces exist globally only when the flow is irrotational, meaning the velocity field of the stream has zero vorticity and is the gradient of a scalar (a hypersurface-orthogonal flow); this is why the 1953 theory is restricted to irrotational motion. In this theory electron and positron streams appear as two separately conserved charge densities, so pair creation and annihilation are absent by construction, and the momentum dynamically conjugate to the absolute time has the physical meaning of the total particle density. Upon quantization the absolute time would become an operator subject to uncertainty relations, like any canonical variable.

Dirac closes the 1953 paper by stating that the theory is a primitive starting point and poses its completion as an open program: build a more elaborate theory with absolute time that includes electron spin — the streams are spinless — in the hope of improving on quantum electrodynamics (QED, the standard quantum theory of the electromagnetic field and charged matter, whose perturbative predictions such as the Lamb shift and the electron's anomalous magnetic moment set the benchmark). The obstacles he names, or that are intrinsic to the construction, are: the constant k=m/ek=m/e privileges the electron and must be reconciled with the existence of other particles with different mass-to-charge ratios; the absolute-time variable has no known direct physical meaning that experiment could reveal; and the theory must be extended beyond the irrotational approximation and to other fields and particles.

Contemporaries reviewed the parent theory critically: Schrödinger (1952) recast its field equation as the Klein–Gordon equation (the relativistic scalar wave equation for a charged spinless field) under Dirac's gauge, and Melvin (1953), reviewing the theory in Nature, stated the single mass-to-charge constant and the existence of more than one kind of elementary particle as reasons the theory must be generalized. Dirac's own continuation, part III (1954), achieves the classical multi-stream step but abandons the absolute-time structure and remains unquantized and spinless. Later literature took up Dirac's 'aether' (his name for the underlying medium whose local velocity defines the foliation) in three main ways — a stochastic subquantum medium (Vigier), a geometrical reformulation of electrodynamics in Dirac's gauge (Caser), and a conducting-vacuum cosmological medium (Carvalho and Oliveira) — while preferred-foliation formulations of quantum theory (Holland's Bohmian field theory, Fanchi's invariant-time formulations) and Einstein-aether gravity (Jacobson and Mattingly's dynamical preferred-frame theory) pursued the same motivating idea of a hidden preferred time in different formalisms.

Problem Statement

The problem is Dirac's own completion program for the 1953 absolute-time theory of charged streams, taken at its stated scope. Starting from the 1953 Hamiltonian electrodynamics — gauge fixed by AμAμ=k2A^\mu A_\mu=k^2 with k=m/ek=m/e, irrotational electron and positron streams separately conserved (no pair creation), dynamics on a preferred foliation of simultaneity surfaces labelled by an absolute time whose conjugate momentum is identified with the total particle density — construct the more elaborate theory with absolute time that simultaneously: (a) includes electron spin, and after quantization the correct spin-1/2 degrees of freedom and Fermi statistics for electrons and positrons; (b) reconciles the single mass-to-charge constant k=m/ek=m/e built into the gauge constraint with the existence of other particles having different mass-to-charge ratios; (c) gives the absolute-time variable a physical meaning that could be revealed by experiment — admissible if it yields a measured quantity whose predicted value differs from standard QED in a specified regime — including its status after quantization as an operator subject to uncertainty relations; and (d) removes the irrotational-flow restriction on the streams and extends the framework beyond the electromagnetic electron system to other fields and particles. Dirac's stated motivation for the completion was the hope of improving on quantum electrodynamics. The target is a consistent, quantizable absolute-time electrodynamics satisfying (a)–(d) jointly within its stated scope, whose Lorentz-invariant observable content agrees with established electrodynamics and QED results in the regimes where those are experimentally confirmed, and which makes a definite, testable prediction that discriminates the absolute-time theory from QED.

References

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