Abstract
It is shown that quantum mechanics is a plausible statistical description of an ontology described by classical electrodynamics. The reason that no contradiction arises with various no-go theorems regarding the compatibility of QM with a classical ontology, can be traced to the fact that classical electrodynamics of interacting particles has never been given a consistent definition. Once this is done, our conjecture follows rather naturally, including a purely classical explanation of photon related phenomena. Our analysis entirely rests on the block-universe view entailed by relativity theory.
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Notes
Solutions to Maxwell’s equations exist for a conserved source only.
If one accepts that particle-in-a-polarimeter type processes are part of the microscopic description of any complex system, then there is no escape from the conclusion that any macroscopic system classified as ‘chaotic’, is inherently unpredictable—not merely practically or due to the butterfly effect. The radical implications of this realization will be discussed elsewhere.
As two successive decays necessarily correspond to two e-m distributions which are supported on different times, the distributions are defined modulo a time-shift.
In this section the particle index possibly refers to a composite particle, and the particle’s densities to the sum of its constituent’s densities. Note that the constitutive relations are unaltered by this modification.
A typical peak corresponds to particle 1 exiting polarimeter A with ‘spin up’ and particle 2 exiting polarimeter B with ‘spin down’ which has the same integral as the peak corresponding to the above with \(1\leftrightarrow 2\).
A somewhat related picture of an atom is given by stochastic electrodynamics (SED; see e.g. [3]). In SED, point charges generate only retarded waves, self-force effects are (allegedly) absorbed into their dynamics by replacing the Lorentz force equation with the Abraham-Lorentz-Dirac (ALD) equation, and a zero-point-field (ZPF) occasionally compensates the electron for its radiated energy loss, preventing it from spiralling into the nucleus, as otherwise follows from the ALD equation. As the name implies, the source of ZPF are retarded waves generated by the rest of the particles in the universe which are at their ground state.
There are, however, some crucial differences between the two ZPF’s. First, the SED ZPF, on average, must ‘pump’ energy back to the electron to save it from ‘spiralling to death’, whereas in ECD, the electron radiates both advanced and retarded fields and needs no such salvation to begin with. The ECD ZPF is energetically neutral.
The second difference has to do with the magnitude and spectral content of both ZPF’s. The ECD one is assumed feeble compared with the other fields involved in the dynamics of an atomic electron—the self-field and the Coulomb central field. It is also supposed to have an upper frequency cutoff equal to \(\hbar /(m_e c^2)\), inherited from the extended size of electrons, which are the source of the ZPF. This should be contrasted with a point particle spiralling towards the Coulomb center, eventually attaining arbitrarily high frequencies. The compensating SED ZPF, therefore, not only is it not band limited, but it further turns out that it must diverge at high frequencies to balance the ALD dissipation. The justification commonly given to this divergence is that the spectrum of the ZPF must be Lorentz invariant. This reason seems rather contrived, given that every other aspect of our (local) universe is not Lorentz invariant.
As we shall see below, advanced fields are also absolutely mandatory in any classical account of photons. Moreover, as shown in [8], classical electrodynamics based on the ALD equation is inconsistent with the constitutive relations. These are not only necessary in order for a theory to be compatible with the experimental scope of CE but, as we saw in Sect. 4, also to establish the compatibility of an ontology with QM. We conclude that despite being in the spirit of ECD, SED is ‘too simple’ to offer a realistic description of micro-physics.
Nevertheless, SED has had some impressive quantitative success in reproducing certain quantum mechanical results based on the concept of ensemble average, and it is therefore tempting to apply similar methods to ECD. However, the ECD counterparts of those methods are not only infinitely more complicated due to the extended structure of an ECD particle, but they also expose the ‘deception’ inherent in any alleged derivation of a statistical theory from a single system theory: One must postulate an ensemble over which the statistics is to be computed. When the single system equations are sufficiently simple, the postulated ensemble can be compactly defined, camouflaging the fact that critical information besides the single-system equations has been added to the computation. The definition of an ensemble of ECD solutions (more accurately, of ‘segments’ cut from the global ECD solution of the block-universe), each representing a repetition of an experiment, requires an infinity of such postulates, making manifest the status of QM as a fundamental law of nature, on equal footings with the underlying ontology—allegedly ECD—and further explains why QM could have predated ECD (or whatever underlying theory).
A collective linear increase in the mass of particles over time (or a decrease in the value of \(\hbar \)) for example, leads to a Hubble-like relation, as light collected from remote galaxies is emitted at an epoch of lower mass (hence longer wavelength) which is proportional to the distance between the emitter and the observer, for any (co-moving) observer. More accurately put, pure ‘geometric’ expansion of the space-time manifold, normally held responsible for the Hubble relation, is utterly meaningless without the equations for matter, setting the scale to any measurement (e.g. how many electrons can fit between two galaxies). Generally covariant ECD is the only available theory consistently merging those two components which are necessary for expansion/inflation analysis.
The inclusion of advanced fields requires a modification to the notion of ‘complete absorption’ which otherwise means that no retarded flux appears on the surface of a sufficiently large sphere containing the relevant system. The natural generalization is to postulate that no time-averaged imbalance between advanced and retarded fluxes appears across the sphere, thereby guaranteeing the conservation of e-m inside the sphere.
Use of advanced fields in order to explain the non classical statistics exhibited by photons, later receiving the name ‘the transactional interpretation of QM”, was made by Cramer in [2]. The construction of the global e-m distributions in that proposal uses time symmetric action-at-a-distance electrodynamics [15], but with self interaction naturally included (this makes mathematical sense for extended particles only). While not being explicit about the precise physical meaning of “hand shakes” between the detecting particles and the source, it definitely provides a conceptual way of saving locality. However, having a fixed, \(\alpha _\text {ret}=\alpha _\text {adv}={\frac{1}{2}}\) division between advanced and retarded fields, his proposal cannot possibly describe photodetection. The EM field close to a charge is entirely dominated by the self field, hence without a temporary imbalance between advanced and retarded parts, of the type appearing in ECD, no sudden delivery of energy to a charge can occur, which is mandatory in photodetection.
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The author wishes to thank an anonymous referee for his critical remarks on an earlier draft of this paper, which greatly improved the final outcome.
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Knoll, Y. Quantum Mechanics as a Statistical Description of Classical Electrodynamics. Found Phys 47, 959–990 (2017). https://doi.org/10.1007/s10701-017-0096-1
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DOI: https://doi.org/10.1007/s10701-017-0096-1