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The Fold Is a Grating: Folded Histories in the Laboratory

Laurence A Jacobs, Alejandro Frank

Published Sep 30, 2026• 0 citations
Publisher

Abstract

Four results are new. A massless field cannot fold: infrared stability forces the photon's fold coupling to vanish, which makes the electron, the lightest charged massive field, and nature's most fold-susceptible object. A static fold is structurally hidden: its dispersion shift is reabsorbed into the measured mass, folded stationary states are exactly stationary, and its first-order coherences beat at $2mc^2/\hbar\sim10^{21}$~Hz. The hiding is exact: the covariant theory shows the fold at fixed coupling to be a constant rotation of the strand label; so no free measurement, vacuum stability included, bounds $\Delta$; it is constrained only through the windows the protections leave open (creation transients, coherent conjugate detection, engineered readers of the backward strand), which we quantify in a sensitivity roadmap from electronvolts to femto-electronvolts. In the folded double slit the familiar fringes survive with a pedestal at second order, while a reader of the backward strand a gain medium for a matter wave, a post-selection for the electron -displays, where ordinary quantum mechanics records nothing, a faint pattern of near-unit contrast: the backward-in-time branch interfering with itself. And the inverse-scattering algorithms of grating engineering transcribe verbatim into fold-profile tomography. Around these results stands a dictionary: the mathematics of folded histories has run in laboratories since 1914 under five names- the Airy sum of a Fabry--P\'erot cavity, the Bragg grating with its net power flux as the signed crossing count, the Manley--Rowe invariant of a parametric amplifier, the Bogoliubov normalization $|u|^2-|v|^2=1$ of a quantum gas. Nothing in it changes the classical answers; it changes the questions, defining an observable the classical theories could not write down: how many times did the light turn around? -whose mean is a weak value, complex and anomalous.

DOI
10.5281/zenodo.23057841

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Four results are new. A massless field cannot fold: infrared stability forces the photon's fold coupling to vanish, which makes the electron, the lightest charged massive field, and nature's most fold-susceptible object. A static fold is structurally hidden: its dispersion shift is reabsorbed into the measured mass, folded stationary states are exactly stationary, and its first-order coherences beat at 2mc2/ℏ∼10212mc^2/\hbar\sim10^{21}2mc2/ℏ∼1021~Hz. The hiding is exact: the covariant theory shows the fold at fixed coupling to be a constant rotation of the strand label; so no free measurement, vacuum stability included, bounds Δ\DeltaΔ; it is constrained only through the windows the protections leave open (creation transients, coherent conjugate detection, engineered readers of the backward strand), which we quantify in a sensitivity roadmap from electronvolts to femto-electronvolts. In the folded double slit the familiar fringes survive with a pedestal at second order, while a reader of the backward strand a gain medium for a matter wave, a post-selection for the electron -displays, where ordinary quantum mechanics records nothing, a faint pattern of near-unit contrast: the backward-in-time branch interfering with itself. And the inverse-scattering algorithms of grating engineering transcribe verbatim into fold-profile tomography. Around these results stands a dictionary: the mathematics of folded histories has run in laboratories since 1914 under five names- the Airy sum of a Fabry--P'erot cavity, the Bragg grating with its net power flux as the signed crossing count, the Manley--Rowe invariant of a parametric amplifier, the Bogoliubov normalization ∣u∣2−∣v∣2=1|u|^2-|v|^2=1∣u∣2−∣v∣2=1 of a quantum gas. Nothing in it changes the classical answers; it changes the questions, defining an observable the classical theories could not write down: how many times did the light turn around? -whose mean is a weak value, complex and anomalous.
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.
Using the most basic mathematical tools, I present the full analysis of the experiment decribed in [A. Danan, D. Farfurnik, S. Bar-Ad, and L. Vaidman, {\em Phys. Rev. Lett.} {\bf 111}, 240402 (2013)]. First, I confirm that the data presented therein are in full agreement with the standard quantum mechanics. I then show other symptoms of presence of photons at all mirrors in the setup. I then analytically explain both the absence of peaks a Readers of [A. Danan, D. Farfurnik, S. Bar-Ad, and L. Vaidman, {\em Phys. Rev. Lett.} {\bf 111}, 240402 (2013)] are made to expect and presence of those not discussed in the Reference.
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.
Sorkin defined a quantum measure as a set function on histories that is nonnegative, permits two-way interference, and satisfies the sum rule that forbids three-way interference. We show that the two-boundary measure of the folded-histories theory (the theory obtained from Feynman's sum by allowing a history to turn in time with an amplitude per unit time Delta) is a quantum measure in this sense for every value of Delta: it arises from a Hermitian, biadditive, strongly positive decoherence functional, and its measure of the whole history space exceeds unity by exactly the squared norm of the backward-in-time strand, which the second-quantized theory identifies as the flux of created pairs. The measure is Sorkin's double sum over histories with one added condition: histories interfere only if they end on the same strand, and only the forward strand is counted. The signed crossing count, which the fold conserves, also satisfies the sum rule but is not positive; Sorkin's positivity axiom excludes it; among the positive pairings that remain, the absorber selects the one restricted to the strand it reads. In unitary quantum measure theory a serial event, one defined by a sequence of projectors, cannot exceed unity; in the folded theory the whole history space, a serial event specified by identity projectors, does whenever the backward strand is populated. A serial event whose measure exceeds unity is therefore the signature of evolution that is not unitary; for the free electron, the only such mechanism in the theory is the turn. The fold introduces two-way interference between strand-resolved classes, while preserving the absence of three-way interference at every real Delta. We prove the measure identities, outline an optical test, and report a finite-system numerical audit.
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.