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Histories That Turn in Time, and What Becomes of the Paradoxes

Laurence A Jacobs, Alejandro Frank

Published Sep 29, 2026• 0 citations
Publisher

Abstract

Feynman's sum admits only histories that cross each instant once, forward in time. We withdraw this condition. A history may turn in time, with an amplitude per unit time fixed by a constant $\Delta$, and the forward direction is defined by the absorbers of each experiment. One conservation law follows. From it the arrow of time, the square in Born's rule, anomalous weak values and the correlations of Einstein, Podolsky and Rosen cease to be paradoxes. The turn hides exactly behind the electron's mass and shows itself only to a reader of the backward strand.

DOI
10.5281/zenodo.23040204

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Feynman's sum admits only histories that cross each instant once, forward in time. We withdraw this condition. A history may turn in time, with an amplitude per unit time fixed by a constant Δ\DeltaΔ, and the forward direction is defined by the absorbers of each experiment. One conservation law follows. From it the arrow of time, the square in Born's rule, anomalous weak values and the correlations of Einstein, Podolsky and Rosen cease to be paradoxes. The turn hides exactly behind the electron's mass and shows itself only to a reader of the backward strand.
Quantum mechanics as usually understood requires every history in Feynman's sum to cross each instant of time exactly once, running forward; the direction of time thus enters as a condition on the histories rather than as a consequence of anything observed. In this paper we examine what follows when the condition is withdrawn. A history may turn in time, with an amplitude per unit time fixed by a constant Δ\DeltaΔ characteristic of the field, and the forward direction of time is defined, operationally, as the direction in which an experiment's absorbers absorb. From these two postulates, for the electron with its spin in three spatial dimensions, one conservation law follows -the signed pairing between the prepared state and the absorbed state is independent of the intermediate time -and from it the two-time probability rule, weak values, and the rule of Born are derived rather than postulated. The turn lowers the electron's mass and hides behind it exactly: the theory at fixed Δ\DeltaΔ is a constant rotation of a two-valued label, carrying the free electron and the electron in any electromagnetic field onto a Dirac electron of mass m2−Δ2\sqrt{m^2-\Delta^2}m2−Δ2​, so that no free measurement bounds Δ\DeltaΔ and every ordinary intensity is even in it. The turn is exposed only to an instrument that is not in its ground state: a population-inverted screen behind a double slit, dark in the ordinary theory, on which the strand of the electron running against time interferes with itself. The correlations between distant measurements are carried by local histories fixed at both ends, which is not action at a distance. Part~I states the theory and its consequences in physical terms; Part~II proves them.
It follows from the standard quantum formalism that in the time between any two noncommuting measurements there are two quantum states of the same system representing two complementary, forward and backward histories, histories that may be observed in two complementary experiments. It follows from this in turn that all interpretations of quantum mechanics based on the idea of a single state between measurements should be rejected. The non-existence of a faster-than-light nonlocality and of paradoxes in EPR-type experiments1 can be proved in the frame of the standard formalism when the reality of backward histories is taken into account.
Three results are new. First, the two-state-vector formalism of Aharonov and collaborators is \emph{derived} rather than postulated: for a Feynman sum whose histories may fold backward in time, with both boundary conditions imposed, the unique slice-invariant pairing between the forward- and backward-evolved boundary data is fixed by a conserved topological charge --- the signed count of slice crossings --- and the two-state description is the native form of the path integral. Second, at nonzero fold coupling Δ\DeltaΔ the two-state vector acquires dynamics: the fold vertex exchanges amplitude between the two states, which in every existing time-symmetric formulation evolve independently. Third, the measurement calculus follows in sequence: weak values, the Aharonov--Bergmann--Lebowitz rule, and, upon summing the future boundary, the Born rule, whose first-quantized deficit is exactly the flux of created pairs. Conservation of the signed count forces the fold vertex to be a hyperbolic SU(1,1)SU(1,1)SU(1,1) mixing, so the coupling enters the dispersion as E2=c2p2+ℏ2a2−Δ2E^2=c^2p^2+\hbar^2a^2-\Delta^2E2=c2p2+ℏ2a2−Δ2, reducing mass and forcing second quantization, under which the fold is a two-mode squeezing with conjugate-amplitude ratio R=tanh⁡(12 artanh Δ/E)R=\tanh(\tfrac12\,\mathrm{artanh}\,\Delta/E)R=tanh(21​artanhΔ/E). The coupling hides behind the mass it shifts and is not yet bounded by experiment; with absorber-defined boundary data every strand-blind observable is even in Δ\DeltaΔ, so first-order access requires strand-odd probes or a reader of the backward strand --- a gain medium for a bosonic field, a post-selection for a fermion. The arrow of time appears nowhere in the equations; it enters through the readiness of absorbers.
In 1927 at the fifth Solvay Council, that reunited all the aristocracy of theoretical physics, Einstein, regarding with solicitude the new-born “quantum mechanics” of Louis de Broglie, Schrödinger, Heisenberg and Dirac, discerned with his usual sagacity an indelible mark that was destined to become, with time, a subject of passionate discussion among those whose vocation is to adulate this enigmatic and capricious personality. In 1926 Born had given the prophetic stroke to the portrait. Turning to probability as to the official factotum of the reconciliation of the continuous and the discontinuous-here, the associated wave and particle-he transmuted the waves of de Broglie and Schr6dinger into an undulatory calculus o f probabilities, deducing, from a surprising principle, consequences that were even more surprising but always verified through experiment. Parting from the idea that the intensity of the wave is the probability of the detection of the particle at a given point and time, Born replaced the classic principle of addition of partial probabilities with his “principle of the addition of partial amplitudes” that are, as in classical optics, represented by “complex” dimensions, with one real part and one imaginary part. In general, the square of the module of the sum of amplitudes will be the probability. This expression contains, of course, the terms “square” and “rectangular.” The first, if they were alone, would give the former law; as for the second, they express the existence of phenomena of interference that are at the origin of the thousand and one well verified paradoxes of the “new mechanics “-the one thousand and first being the one under consideration here.
We always have had a great deal of difficulty in understanding the world view that quantum mechanics represents. At least I do, because I'm an old enough man that I haven't got to the point that this stuff is obvious to me … It has not yet become obvious to me that there is no real problem. I cannot define the real problem, therefore I suspect there is no real problem, but I'm not sure there's no real problem. Richard Feynman Max Born and quantum probabilities In this chapter, we will take a short break from our survey of successful applications of quantum mechanics to take a closer look at the foundations of this great edifice of modern physics. In a sense, we are now disobeying Feynman's warning about the results of the double slit experiment of chapter 1. We shall now ask Feynman's ‘forbidden’ question ‘But how can it be like that?’ Nobody disputes that quantum mechanics has been magnificently successful, enabling us to make correct quantitative calculations of atomic and nuclear effects. But there is a great divergence of opinion about the implications of quantum mechanics for the nature of matter, and indeed, of reality itself. To avoid becoming embroiled in a purely philosophical mire, we will focus on two famous paradoxes. The first of these is the ‘Einstein–Podolsky-Rosen’ paradox, named after its originators Albert Einstein, Boris Podolsky and Nathan Rosen, and usually abbreviated by the initials ‘EPR’. The second paradox is named after ‘Schrödinger's cat’.