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Folded histories: Time-Symmetric Quantum Mechanics from Feynman's Sum

Laurence A Jacobs, Alejandro Frank

Published Sep 30, 2026• 0 citations
Publisher

Abstract

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 $\Delta$ 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 $SU(1,1)$ mixing, so the coupling enters the dispersion as $E^2=c^2p^2+\hbar^2a^2-\Delta^2$, reducing mass and forcing second quantization, under which the fold is a two-mode squeezing with conjugate-amplitude ratio $R=\tanh(\tfrac12\,\mathrm{artanh}\,\Delta/E)$. 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 $\Delta$, 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.

DOI
10.5281/zenodo.23057605

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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.
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.
Conditioning on both past and future assigns intermediate-time properties a causal observer does not; these time-symmetric assignments obey exact symmetry theorems and are measurable from trajectories. We use two-state generator extended dynamic mode decomposition (gEDMD): because weak values obey dAw/dt=i⟨[H,A]⟩wdA_w/dt=i\langle[H,A]\rangle_wdAw​/dt=i⟨[H,A]⟩w​ exactly, generator extraction, with an exact-derivative baseline, applies unchanged to them. First, a reflection involution on the pre-/post-selected ensemble splits every window-fitted friction uniquely as γfwd=γA+γSγ_{fwd}=γ_A+γ_Sγfwd​=γA​+γS​: γAγ_AγA​, antisymmetric about the midpoint, carries the modes' boundary-condition physics; γSγ_SγS​, symmetric, comes from the differencing scheme; both follow from the same data as (γfwd±γbwd)/2(γ_{fwd}\pmγ_{bwd})/2(γfwd​±γbwd​)/2. At a fixed inference resolution the arrow of time has two layers: the coherent-mode arrow reverses at the midpoint, the fluctuation-level one does not, γSγ_SγS​ dominating γAγ_AγA​ at every size and class. The difference is one of degree: γSγ_SγS​ is 34 times larger there than at the mode layer, and with the exact derivative both layers reverse: immunity belongs to the inference, not the ensemble. Second, in a lattice interferometer conditioned only at its ports, the quantum Cheshire-cat structure emerges unimposed: particle and polarization obey separate continuity equations, and a local field in the polarization-carrying arm rotates that phase alone, at exactly twice the field strength, entering the generator as a rigid imaginary shift, while the particle's weak density stays invariant to machine precision. We verify the sample-level identity and the two layers from 282^828 to 2202^{20}220 dimensions: ∣γA∣/γS=0.09|γ_A|/γ_S=0.09∣γA​∣/γS​=0.09 to 0.270.270.27 across five classes; self-averaging makes it insensitive to class among those sharing a boundary modulation, removing the 2−N/22^{-N/2}2−N/2 overlap obstruction for NNN qubits.
The fundamental equations of quantum mechanics are time-symmetric, yet measurements generate irreversible records and a one-way causal structure at macroscopic scales. The Quantum Time-Symmetric Interpretation (QTSI) explains this contrast as emergent rather than fundamental. Isolated quantum systems are described by a two-component temporal bispinor containing both forwardand backward-evolving amplitudes, coupled through a pseudo-Hermitian SU(1,1) Hamiltonian. Their coupling is governed by a mixing parameter Δ that depends on a macroscopic amplification fraction φ = N m /N t , where N t is the total number of relevant microscopic events and N m is the number that trigger irreversible macroscopic records. The associated retrocausal coherence time τ_RC = ħ/|Δ(φ)| is the characteristic timescale on which ψ+ and ψ− mix. As φ increases, τ_RC grows, temporal interference requires progressively longer times to form, and in the limit φ → 1 it becomes physically inaccessible. Δ vanishes, the two temporal sectors decouple dynamically, time symmetry is broken, and the system enters the irreversible classical regime. Standard quantum mechanics is recovered in this limit. The Born rule emerges naturally as ψ− → 0, since |ψ+ + ψ−|² reduces to |ψ+|² without any additional postulate. QTSI is a locally causal, ontologically realist, no-signaling interpretation: Bell correlations arise from time-symmetric global boundary conditions rather than nonlocal influences. Observable deviations from standard quantum mechanics are predicted in weakly monitored interferometric regimes, including the visibility formula V(φ) = V₀[1−exp(−T_av(1−φ)Δ₀/ħ)] — which decreases with monitoring fraction φ and reduces to V ≈ V₀(1−φ) for T_avΔ₀/ħ ≫ 1 — and a new three-slit signature in which the Fourier spectrum of the intensity pattern contains phase-coherent sum-phase peaks absent in standard quantum mechanics, whose Fourier amplitude scales as (1−φ) with the monitoring fraction.
Measurements distributed in time provide information about a system at more than one time; they cannot be described in terms of the conventional language of a system quantum state evolving in time. This paper, the second in a series, explores connections among various ways of formulating a quantum-mechanical description of time-distributed measurements. The natural formulation, involving a ``sum over histories,'' arises directly from Feynman's rules for combining probability amplitudes. One equivalent formulation uses a standard measurement model, in which the system is coupled to a set of ``measuring apparatuses.'' A second equivalent formulation uses the language of ``effects'' and ``operations.'' Still a third formulation attempts to create a new language of multiple-time states and multiple-time eigenstates.