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arXiv:2607.26999

Two-state generator extraction: property currents and a two-layer arrow of time in pre- and post-selected quantum dynamics

Seiki Saito

quant-phcond-mat.stat-mechSubmitted Jul 29, 2026• Announced Jul 30, 2026• 0 citations• Revised Aug 15, 2026 (v2)
v1 · Jul 29, 2026v2 · Aug 15, 2026
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Abstract

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 $dA_w/dt=i\langle[H,A]\rangle_w$ 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 $γ_{fwd}=γ_A+γ_S$: $γ_A$, antisymmetric about the midpoint, carries the modes' boundary-condition physics; $γ_S$, symmetric, comes from the differencing scheme; both follow from the same data as $(γ_{fwd}\pmγ_{bwd})/2$. 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, $γ_S$ dominating $γ_A$ at every size and class. The difference is one of degree: $γ_S$ 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 $2^8$ to $2^{20}$ dimensions: $|γ_A|/γ_S=0.09$ to $0.27$ across five classes; self-averaging makes it insensitive to class among those sharing a boundary modulation, removing the $2^{-N/2}$ overlap obstruction for $N$ qubits.

Comments
21 pages, 11 figures, 5 tables

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EPJ Web of Conferences•Jan 1, 2026•0 citations
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.
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.
Complex Hilbert space is usually assumed at the start of quantum mechanics. This paper formulates a reverse construction for the one-particle and operator-algebraic setting. Starting from a real Hilbert space carrying an orthogonal representation, the admissible imaginary units are identified with the skew-adjoint square roots of − Id in the Real commutant of the complexified representation. The novelty is not the isolated use of complexification, spectral orientation, or polar decomposition, but their order of dependence: first the symmetry representation defines an intrinsic operator-valued phase space; then an oriented internal time flow selects a distinguished point of that phase space; only afterward are the usual complex one-particle, observable, probability, and Weyl–CCR structures invoked under their standard hypotheses. In the terminology used here, this oriented internal flow is the reversible or inner-generative side of a dual-time distinction. It is not a second observer clock and not an additional Lorentzian coordinate: mathematically it is the strongly continuous orthogonal group U ( t ) = exp ( t A ) on the real Hilbert space. Its Stone-dual variable is energy/frequency λ ; canonical conjugation pairs λ with − λ ; positive energy orients this pair and selects the canonical phase on the dynamic sector through A = J | A | = | A | J . Thus inner-generative time is the source of phase orientation, while irreversible record time, projection, memory, and decoherence enter only as later reduced-dynamics layers. No spectral gap at zero is required, and zero modes remain visible as residual phase data rather than being hidden by the construction. The selected phase is invariant under time-orientation-preserving symmetries and changes sign under time reversal, giving the usual antiunitary action after complex reconstruction. The paper also makes explicit the monoidal limitation: two selected local phases do not make the raw real tensor product the standard composite system; the standard composite is recovered only after imposing the J -balanced, equivalently complex, tensor product rule. Thus the result is a reconstruction of compatible complex kinematics on the selected one-particle/operator-algebraic sector, not a derivation of all quantum foundations from purely real data.
Understanding how macroscopic irreversible hydrodynamics emerges from the reversible unitary dynamics of isolated quantum many-body systems remains a central challenge. Conventional approaches force spin density dynamics into purely diffusive models, obscuring the interplay of pressure, spin current, and local friction, and leave open the role of the observer's temporal resolution. We introduce a fully data-driven framework based on generalized Extended Dynamic Mode Decomposition (gEDMD) integrated with the Mori-Zwanzig projection. By expanding the observable dictionary to include spin currents, we extract the Navier-Stokes hydrodynamic coefficients from a chaotic XXZ spin chain across temporal coarse-graining scales. Our unconstrained extraction reveals a dichotomy: the mechanical elasticity (c2c^2c2) derives from the exact unitary dynamics, preserving microscopic reversibility, whereas the macroscopic friction (γγγ) and kinematic viscosity (ννν) show zero net dissipation, oscillating rapidly around zero in the exact-derivative limit. Genuine macroscopic transport therefore requires finite temporal coarse-graining: at a finite observation timescale (Δtcg>0Δt_{cg} > 0Δtcg​>0) the system passes through a crossover where these fluctuations average out, establishing an intermediate functional regime with strictly positive friction and viscosity. We further show that the coarse-grained generator is a predictive reduced model requiring only O(N)O(N)O(N) macroscopic observables, and that the sign of the emergent friction is inherited from the causal, forward-in-time direction of the inference: time-symmetric estimators yield zero net dissipation at any coarse-graining scale. Macroscopic friction in isolated quantum systems is thus not an absolute property but an emergent phenomenon dictated by the temporal resolution and causal direction of the observer's description.
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.
Microscopic quantum laws are time-symmetric: nothing in the Schrödinger equation or its relativistic extensions distinguishes future from past. Yet measurements produce irreversible records, an apparently one-way causal flow, and the familiar notion that causes precede effects. Within the Quantum Time-Symmetric Interpretation (QTSI), this asymmetry is not fundamental but emergent. Isolated quantum systems are described by a two-component temporal state containing forward- and backward-propagating amplitudes. Their mixing, governed by a parameter Δ(ϕ)Δ(ϕ)Δ(ϕ), defines a retrocausal coherence time τRC(ϕ)τ_{RC}(ϕ)τRC​(ϕ) beyond which advanced components are suppressed. As the system couples to amplifying environments characterized by a macroscopic parameter ϕϕϕ, Δ(ϕ)Δ(ϕ)Δ(ϕ) decreases and the backward component is dynamically eliminated, giving rise to classical causality and effective collapse. QTSI aligns naturally with time-symmetric approaches from Wheeler--Feynman, Aharonov, and Price, agrees with all weak-measurement and quantum eraser results in their operational regimes, and predicts specific signatures in temporal echoes and chaotic cavities. Detailed formal and experimental developments appear in the Supplementary Addenda.
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.