The Electromagnetic Origin of Quantization and the Ensuing Changes in Copenhagne Interpretation
| Scientific Paper | |
|---|---|
| Title | The Electromagnetic Origin of Quantization and the Ensuing Changes in Copenhagne Interpretation |
| Read in full | Link to paper |
| Author(s) | Evert Jan Post |
| Keywords | electromagnetic, Zero-Point Energy, Copenhagen, quantum, de Rham cohomology, period integrals, ensemble interpretation, quantum Hall effect |
| Published | 2002 |
| Journal | Annales de la Fondation Louis de Broglie |
| Volume | 27 |
| Number | 2 |
| No. of pages | 24 |
| Pages | 217-240 |
Read the full paper here
Abstract
The pre-1925 quantum prescriptions of Planck, Einstein, Bohr, Sommerfeld and recently Aharonov-Bohm permit a recasting as part of a complete set of electromagnetic residue integrals such as used in a mathematical discipline known as de Rham cohomology. The ensuing spacetime topological reorganization of early quantum aspects seems well supported by Josephson- and quantum Hall effects. This reversal of priorities demands a physical readjustment of standard nonclassical Copenhagen pronouncements. The Schroedinger equation becomes a tool solely applicable to ensembles consisting of single systems of random phase and -orientation. This reorganization is a return to the ensemble initiatives of the Thirties by Slater, Popper, Kemble and others, which now can be given a compelling form by identifying long standing classical counter-examples to Copenhagen's nonclassical propositions. Heisenberg uncertainty and zero-point energy have to yield their pedestal of universal absolute status. They now become manifestations governing order-disorder transitions in ensembles.
Overview
Post's essay is an argument about priority, not about mathematics. He states at the outset that his revision "does not affect established mathematical procedures of quantum mechanics"; what it changes is "a more precise delineation of its objects of description and how they relate to reality." The claim is that the pre-1925 quantization rules — Planck's, Bohr's, Sommerfeld's, and their modern descendant the Aharonov–Bohm phase integral — are not crude approximations superseded by the Schrödinger equation, but exact laws of a different kind: electromagnetic period integrals in the sense of de Rham cohomology, counting quanta of flux, charge and action as spacetime topological invariants.
If that is right, the order of business inverts. The residue integrals become the exact tools for single systems; the Schrödinger–Dirac process becomes a near-exact tool for genuinely statistical ensembles; and the Copenhagen doctrine that the wave function describes a single system is the "insidious single system choice" from which, Post argues, most "nonclassical metaphors" follow. He writes candidly of motive: "This essay is almost an act of despair to prevent physics from being spoiled by a too exclusively ontic modus operandi. Knowing how does not obviate a need later for also knowing why." The programme is explicitly a revival of the ensemble interpretations of the 1930s — Slater, Popper, Kemble — now, in his view, backed by metrology that did not exist then.
The argument
A chronology of three quanta
Post opens with a two-century timeline running from Dalton and Avogadro through Faraday's electrolysis, Thomson's electron, Planck's h, Einstein's photoelectric quantum, Millikan's measurement of e, Bohr's angular-momentum condition, Sommerfeld's cyclic integral, Duane's particle theory of X-ray diffraction, de Broglie, Uhlenbeck–Goudsmit spin, Schrödinger, Davisson–Germer, Heisenberg, Dirac, London's 1932 footnote predicting a flux quantum h/e, the 1961 confirmation by Doll and Fairbanks (with Onsager attributing the factor 2 in h/2e to Cooper pairing), Josephson tunnelling in 1962, and von Klitzing's integer and Bell Labs' fractional quantum Hall effects in 1980 and 1982.
His reading of that history is that attention has been unevenly distributed. Of the three quanta — flux h/e, charge e, action h — only two are independent, since flux × charge = action; yet action received "by far the bulk of attention", becoming "a key to a new so-called nonclassical era", while the flux quantum was unknown until 1932 and unconfirmed until 1961. Physics, he suggests, "has forced itself into premature decisions by focusing too exclusively on the quantum of action."
Quanta as pre-metric, topological invariants
The three quanta are known to nine decimal places, do not depend on place or time, are unaffected by strong gravitational fields, and — via the constancy of the fine structure constant in quasar spectra, which he notes is a ratio of the free-space impedance to the quantum Hall impedance h/e2 — appear universal. Since gravity is a matter of metric structure, Post infers that "the quanta are metric-independent entities". His mathematical warrant is that Gauss's law is a case of the generalized Stokes theorem, valid on differential manifolds with no metric defined at all.
He therefore lists three counting integrals in metric-free form: the London–Aharonov–Bohm integral over a one-cycle, giving n·h/e (half-integer in a self-field, integer in an external field); the Ampère–Gauss integral over a two-cycle, giving s·e (with s even for boson counting); and the Kiehn integral over a three-cycle, giving n·s·h, which for a single charge reduces to the Bohr–Sommerfeld condition. The governing statement is: "The AB, AG and RK integrals can assume a status of exact physical law, iff their integration cycles everywhere reside in domains where the exterior derivatives of their integrands vanish." He credits Robert M Kiehn with the three-dimensional action integral and George de Rham with the cohomological machinery, and stresses the corollary: "all primary quantization is electromagnetic, not mechanical in nature."
Why metrology is the evidence
The best values of h and e come from the Josephson AC effect and the quantum Hall effect, "originally argued from a Schroedinger angle" but, Post contends, really Aharonov–Bohm arguments in disguise: "Josephson's phase single valuedness is for all practical purposes an Aharonov–Bohm argument." He objects that the Schrödinger treatment of the Hall effect requires "fudging", including "artifact distinctions between integer and fractional quantum Hall effects" and "unproven assumptions concerning fractional, elementary, electric charge quanta or compound fermions".
He then confronts the obvious difficulty with his own criterion. Both experiments have their integration loops in regions where fields are manifestly present — the Josephson sandwich carries an E field, the Hall electrons orbit in a strong B field — so the period condition seems violated. His answer is a structural conjecture about charge: "The only way of rescuing an apparent exact applicability of the AB integral in external fields is by endowing charge itself with a natural field-free interior", making the electron an object of "nontrivial one- and two-connectedness". He reports elsewhere-published trefoil-tube modelling that he says accounts simultaneously for pair creation, half-integral spin, magnetic moment and anomaly.
Schrödinger's equation as a derived ensemble tool
Since "the Schroedinger equation is a statistical tool whereas the residue integrals are not, they could not possibly apply to the same physical realm." Post traces Schrödinger's construction — the Hamilton–Jacobi equation with S = h ln ψ, a variational integral, the Euler–Lagrange derivative giving the eigenvalue equation, with single-valuedness of ψ supplying the old quantization and square-integrability the statistics. He reads the extremization over the manifold of Hamilton–Jacobi solutions as "comparable to a maximum probability operation in the sense of statistical mechanics", the manifold "represent[ing] an ensemble of identical systems". So Schrödinger's procedure is promoted from recipe to derivation: exact single-system input, near-exact ensemble output. The division of labour: residue integrals exact for single systems and for ordered arrays such as quantum Hall plateau states, "outside plateau states Schroedinger–Dirac prevails."
The diagnosis of Copenhagen
The central error, on Post's account, is the waiving of a "universe of discourse" for the Schrödinger statistics: Copenhagen assumed an abstract Gibbs ensemble of conceivable states of one system where it should have considered an ensemble of real systems. The distinction is inconsequential for most weakly-interacting calculations, but decisive for zero-point energy. Planck in 1912 introduced hν/2 as an ensemble average needed to keep oscillators phase-random, not as an irreducible floor for each oscillator — so "the need for vacuum infinities did not arise". Post notes that Feynman's own classical orientation-averaging calculation of mean angular momentum (Vol. II and the Vol. III appendix) is an unremarked counter-example to the necessity of nonclassical statistics, and cites Boersma's classical maritime analogue of the Casimir effect as showing that the attraction "is a force differential at the low frequency end of the spectrum, without any infinites whatsoever."
He summarizes the required revisions in five numbered points: Schrödinger and Dirac step back to describing randomized ensembles; zero-point energy and Heisenberg uncertainty lose universal status and become features of positional and phase randomness; the AB, AG and RK integrals are exact single-system tools when period conditions hold; wave–particle duality and complementarity fall with the single-system reading, leaving "at best a wave–many particle duality"; and classical statistical calculations reproducing Schrödinger results invalidate appeals to undefined nonclassical statistics. He adds a group-theoretic coda — descriptions should run from pre-metric topology (Diffeo(4)) down through conformal, Lorentz and rotation subgroups, "yet, false pedagogy suggests an opposite course" — and a footnote asserting, on Kottler and Poincaré, that Diffeo(4)-invariant Maxwell equations are invariant under the Galilei as well as the Lorentz group.
Appendix: hydrogen from flux quantization
To show the Bohr condition is not essentially mechanical, Post applies the Aharonov–Bohm integral to the Rutherford hydrogen atom, the electron orbiting in the proton's external field (hence h/e, not h/2e). Changing variables via the orbital equation and the angular-momentum theorem, and integrating the ellipse r = A + Bcosφ, he finds the angular momentum L drops out of the result, leaving the familiar Bohr formula with the fine structure constant written as the ratio of free-space impedance to the quantum Hall impedance. He is careful to flag the cost: the relativistic version of this route "does not yield an automatic fine structure contribution. Hence spin and magnetic moment are no longer magically created but require instead separate and independent propositions." He suggests this makes the Pauli treatment of spin "more fundamental than the Dirac treatment".
Assessment
The strongest and most durable element is the observation Post builds everything on: the flux, charge and action quanta are counted by integrals that require no metric, which is a real fact about the mathematical structure of electromagnetism and is exactly why Aharonov–Bohm phases and flux quantization are so robust. Framing the Josephson and quantum Hall constants as topological rather than dynamical is not eccentric — the community's own subsequent language of topological invariants and Chern numbers moves in the same direction, and the 2019 redefinition of the SI on fixed h and e rests on precisely the reproducibility Post is pointing at. His historical point that the flux quantum arrived late and was under-weighted in the founding decisions is fair and well documented by his own chronology. And the ensemble reading he advocates is a serious, long-standing minority position with a real pedigree, not an invention; his citation of Planck's 1912 ensemble-average origin of hν/2 is accurate and genuinely awkward for textbook accounts that present the zero-point term as unavoidable per oscillator.
The weaknesses are concentrated at the points where the argument must do real work. The rescue of exactness in the Josephson and Hall cases — where the integration loops demonstrably sit in fields — is handled by postulating that charge has "a natural field-free interior". That is the load-bearing step of the whole paper, and it is asserted, with the supporting trefoil-tube model consigned to a book reference rather than developed here. Without it, Post's own iff condition disqualifies the very experiments he uses as evidence. Similarly, the claim that Schrödinger's variational construction is a maximum-probability operation over a real ensemble is offered as a reading, not a derivation: no explicit measure over the Hamilton–Jacobi solution manifold is constructed, and no demonstration is given that extremizing it reproduces the Born rule. Calling the recipe a derivation is therefore a promise rather than a result.
Several dismissals are too quick to be persuasive. The fractional quantum Hall effect is not a bookkeeping "artifact distinction": the observed fractional plateaus and the quasiparticle charge e/3 measured directly in shot-noise experiments are results the paper does not engage, and asserting that an AB-type argument handles them "without any need whatsoever for taking recourse to fudging" is a claim, not a calculation — none appears in the paper. The proposed demotion of the uncertainty relation to a statement about ensemble randomness has to contend with single-system interferometry and with the fact that the relation is a theorem about non-commuting operators, not an empirical postulate that can be relaxed by reinterpreting ψ. The Casimir argument via Boersma is cited but not reproduced, and the assertion that vacuum infinities "simply cancel" leaves untouched the precision agreement of QED radiative corrections — the electron g−2, measured and computed to better than a part in 1012 — where the same vacuum structure is doing quantitative work that Post's scheme, by his own appendix's admission, must now supply by "separate and independent propositions". Finally, the footnote claiming Maxwell's equations are Galilei- as well as Lorentz-invariant is stated flatly against "textbook assertions" with no derivation; whatever is meant by it in the pre-metric formalism, it is not defended, and a reader has no way to evaluate it.
The tone is also a liability. Copenhagen is described as "theocratic", peer reviewers as treating revision "as an attack on sacrosanct values", the field as afflicted by "nonclassical hype". Post's substantive point — that a single-system reading of ψ was chosen rather than proved, and that a real-ensemble alternative was live in the 1930s — is strong enough not to need this, and the polemic makes it easier for the reader he wants to reach to set the paper aside. Read for its technical core rather than its grievances, the essay is a coherent, unusually well-informed statement of the case that quantization is topological and electromagnetic before it is mechanical, and that at least one thing physics calls a derivation is a choice.