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Measuring the Fold

Laurence A Jacobs, Alejandro Frank

Published Sep 30, 2026• 0 citations
Publisher

Abstract

In the folded-histories reformulation of the path integral, the two-state-vector formalism acquires a coupling constant: forward- and backward-evolving states exchange amplitude at rate $\Delta$. This paper answers the question that result raises ---how would one measure $\Delta$? and four results are new. First, a parity selection rule: the signed metric intertwines the generators exactly, $GH(\Delta)G=H(-\Delta)$, so with absorber-defined boundary data every sense-blind observable is an even function of $\Delta$; all intensity-based signatures begin at second order, and first-order access requires sense-superposed boundary data or sense-odd probes. Second, the information is nonetheless there: the quantum Fisher information about $\Delta$ is finite at $\Delta=0$ even for physical preparation -it resides in the amplitude, and we exhibit the hierarchy exactly: at $\Delta=0.02$ (in units of the turn rate) the intensity pointer protocol captures $0.155\%$ of the QFI, a bare sense-odd projective measurement $36\%$, and the locally optimal basis all of it. Third, an exact weak-value-amplification verdict for this system: because the measured observable has eigenvalues $\pm1$ the pointer protocol is closed form, and sweeping the post-selection overlap at linearity-saturating coupling shows no fundamental gain -total Fisher information per trial dips and plateaus- but seventy-fold concentration of information per detected event, a node at which the click rate is momentarily blind to $\Delta$, and a revival beyond it where click statistics carry a comparable share. Fourth, a design equation: the population-inverted absorber, the one proposed measurement requiring no new instrument, is Fourier-limited with detectability threshold $\Delta_{\min}T_w=2\hbar|\sin(ET_w/\hbar)|\le2\hbar$, independent of the particle energy on the envelope -integration time alone sets the reachable coupling.

DOI
10.5281/zenodo.23058159

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In the folded-histories reformulation of the path integral, the two-state-vector formalism acquires a coupling constant: forward- and backward-evolving states exchange amplitude at rate Δ\DeltaΔ. This paper answers the question that result raises ---how would one measure Δ\DeltaΔ? and four results are new. First, a parity selection rule: the signed metric intertwines the generators exactly, GH(Δ)G=H(−Δ)GH(\Delta)G=H(-\Delta)GH(Δ)G=H(−Δ), so with absorber-defined boundary data every sense-blind observable is an even function of Δ\DeltaΔ; all intensity-based signatures begin at second order, and first-order access requires sense-superposed boundary data or sense-odd probes. Second, the information is nonetheless there: the quantum Fisher information about Δ\DeltaΔ is finite at Δ=0\Delta=0Δ=0 even for physical preparation -it resides in the amplitude, and we exhibit the hierarchy exactly: at Δ=0.02\Delta=0.02Δ=0.02 (in units of the turn rate) the intensity pointer protocol captures 0.155%0.155\%0.155% of the QFI, a bare sense-odd projective measurement 36%36\%36%, and the locally optimal basis all of it. Third, an exact weak-value-amplification verdict for this system: because the measured observable has eigenvalues ±1\pm1±1 the pointer protocol is closed form, and sweeping the post-selection overlap at linearity-saturating coupling shows no fundamental gain -total Fisher information per trial dips and plateaus- but seventy-fold concentration of information per detected event, a node at which the click rate is momentarily blind to Δ\DeltaΔ, and a revival beyond it where click statistics carry a comparable share. Fourth, a design equation: the population-inverted absorber, the one proposed measurement requiring no new instrument, is Fourier-limited with detectability threshold Δmin⁡Tw=2ℏ∣sin⁡(ETw/ℏ)∣≤2ℏ\Delta_{\min}T_w=2\hbar|\sin(ET_w/\hbar)|\le2\hbarΔmin​Tw​=2ℏ∣sin(ETw​/ℏ)∣≤2ℏ, independent of the particle energy on the envelope -integration time alone sets the reachable coupling.
Version 3 (2026-08). One correction, chair-directed after an independent text-only rebuild: the setting's conventions dictionary is regenerated from the code of record. The pointer packet is an ENERGY-basis Gaussian launched at tick zero (v1/v2 described it as a tick-space packet of tick-width sigma_n; the temporal width is in fact the inverse, NC/(4 pi sigma_n) ticks); the tick-state sign convention, the second geometry's sigma_n = 3, the twin/control definitions, and the post-selection probabilities are now printed; a standing text-only rebuild gate (fresh implementation from the setting paragraph alone, matching the exact conditioned state at 3.5e-14) guards the dictionary. Every published number stands (the independent rebuild reproduced the NC = 24 physics cold, the lane's first exterior verification, recorded in the data-and-code paragraph); the dictionary was broken. The methods law is minted program-wide: a geometry prints its dictionary, generated from the code, never transcribed. Version 2 (2026-08). Changes from v1: the clock-width profile is completed end to end: v1's spike-plus-plateau is the interior of a measured disturbance VALLEY with an optimal clock width (minimum at sigma_n* = 2.5; the broad side rises 93x by sigma_n = 24, tracked by the value law against exact evolution at ratios 0.985 and 0.995), so the least-disturbing clock is neither time-sharp nor energy-sharp; a pre-registered identification of the plateau with the sharp-clock limit was killed by a factor of 160 and is ledgered; the open problems are re-posed at the valley's optimum. When a finite quantum clock gates a register's interactions and the run is conditioned on the clock's final reading, the register's conditioned dynamics is derived at two orders and machine-verified against exact evolution. First order: the drive is the complex Aharonov-Albert-Vaidman weak value of the gating window projector (real part = Hermitian envelope, imaginary part = post-selection reweighting); a twin evolved under this envelope removes the entire first-order mismatch a mean-envelope control retains (state-metric deficits g^1.1 vs g^2.1, factor 56 at g = 0.1). Second order: the deficit is governed by the connected two-time window correlator under post-selection, resolved along the interaction picture; the resulting parameter-free value law matches measured deficits at ratios 0.92-1.05 across the clock-width grid and at a second geometry (1.03-1.07). The when/what trade-off of a participating clock is promoted from confounded excess to derived law: the disturbance is the connected fluctuation of the clock's timing answer, its clock-width profile is a predicted spike-plus-plateau (not a power law), and two earlier exponent claims are superseded with the reconciliation recorded. Two honest derivation failures (equal-time invisibility resurrected by the interaction picture; the kernel belonging inside the integral) are ledgered as the steps that located the law.
A wave packet localized within a binding potential feeds an outgoing probability current as it tunnels through a finite-width potential barrier, imprinting its localization-related information on the outgoing probability-current waveform. The product of the wave number and the gradient of the outgoing probability current provides a readout for this information, connecting to the wave-packet localization, Fisher information, and related information entropy production, thus revealing the information-flow underpinning of tunneling dynamics.
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.
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.