The Universal Arrow of Time II (O. Kupervasser, arXiv:1106.6160; published in Electronic Journal of Theoretical Physics 10(29), 39, 2013) argues that in closed, finite-volume quantum systems the thermodynamic time arrows of weakly interacting macroscopic chaotic subsystems — including observers — align, that recoherence (the revival of suppressed quantum superpositions at a Poincaré return) is unobservable not merely 'for all practical purposes' but in principle, and that this makes the Copenhagen and Many-Worlds interpretations experimentally indistinguishable for macroscopic observers. Every load-bearing element of that argument is qualitative: the criterion separating macroscopic chaotic systems from small strongly fluctuating ones, the entropy and isolation cost of any protocol that could render recoherence observable, the scaling of residual off-diagonal corrections and recurrence times, the memory dynamics of an observer during a Poincaré return, and the regime of validity beyond closed weakly interacting chaotic systems. The surviving quantitative open core is: (i) a general quantitative regime criterion built on the saturation of the mutual-entropy bound by classical mutual information; (ii) explicit minimal models bounding the isolation and entropy cost of unperturbative recoherence-observation protocols; (iii) rigorous scaling of residual corrections, recurrence times, and observer-memory models that would turn the in-principle unobservability claim into a theorem or refute it; (iv) delimitation of validity for long-range, non-chaotic, and critical systems; and (v) quantitative experimental lower bounds for the microscopic-to-macroscopic transition. No later work is known to have developed or refuted these claims, and adjacent literatures — macroscopicity measures, quantum-recurrence bounds and experimental blueprints, recoherence and records in isolated quantum systems, heat-flow reversal in correlated qubits — address neighboring questions without closing any strand.

Background

Microscopic dynamics is, to high accuracy, invariant under time reversal, yet macroscopic behavior is not: entropy in the thermodynamic sense increases and records of the past accumulate. Two classical objections make this tension sharp. Loschmidt's paradox notes that every allowed trajectory has a time-reversed counterpart, so entropy-decreasing evolutions exist as solutions. Poincaré recurrence states that a closed system confined to a bounded region of phase space (or a quantum system with discrete energy spectrum and finite volume) eventually returns arbitrarily close to its initial state, so that any apparent equilibration must be undone at some later time. Standard resolutions rest on the improbability of entropy-decreasing states together with special low-entropy initial conditions.

Kupervasser's program ('The Universal Arrow of Time I', Foundations of Physics 2012; 'The Universal Arrow of Time II', the source) proposes instead a dynamical mechanism: when several chaotic macroscopic systems interact weakly, their thermodynamic time arrows (the direction in which each subsystem's coarse-grained entropy, or macroentropy, increases) align, because processes running against the common arrow are unstable against the small interaction noise. In the quantum case the source adds two observations. First, decoherence — the entanglement-driven suppression of off-diagonal terms of a subsystem's reduced density matrix, which selects robust pointer states — has the same effect as weak classical interaction and aligns the arrows of the interacting subsystems. Second, closed finite-volume quantum systems are periodic or almost periodic, so decoherence is eventually undone by recoherence at a Poincaré return; yet the source claims this is unobservable not merely 'for all practical purposes' (FAPP) but in principle, because at introspection (the observer included inside the closed system) the return restores the observer together with the system and thereby erases the memory needed to compare initial and final states, and because for an external observer the weak interaction aligns the arrows so that recoherence exists only in the coordinate time of an idealized dynamics and never with respect to the observer's thermodynamic arrow. From this the source claims a resolution of the Schrödinger-cat and Wigner's-friend paradoxes (Wigner's friend being the scenario in which one observer treats another observer's measurement as a coherent superposition rather than a definite outcome), an in-principle (its 'Gödel theorem of impossibility' for quantum mechanics) experimental indistinguishability of the Copenhagen and Many-Worlds interpretations for macroscopic observers, and an explanation of deviations from exponential decay. Notably, the mainstream decoherence literature treats recoherence and measurement reversal as FAPP-unobservable rather than provably undetectable; the in-principle claims are specific to this program and have been neither adopted nor refuted elsewhere.

Two formal devices carry the argument. The first is the 1962 Daneri–Loinger–Prosperi (DLP) theorem: for a macroscopic measuring device whose phase distribution is chaotic, the unitarily evolved statistical operator cannot be distinguished from the reduced (collapsed) one by any measurement performable with that device. The source reads DLP as a complete, not merely FAPP, resolution of the reduction paradox, because the residual off-diagonal corrections could only resurface at a Poincaré return whose observation the arrow argument excludes. The second is Maccone's arrow-of-time result (Physical Review Letters 103, 080401 (2009)): for a device A correlated with a system C, any local entropy decrease must erase the correlation the observer holds. Maccone related the ensemble mutual entropy S(A:C)=S(ρA)+S(ρC)−S(ρAC)S(A:C)=S(\rho_A)+S(\rho_C)-S(\rho_{AC}) — where ρA\rho_A and ρC\rho_C are the reduced density matrices of A and C, ρAC\rho_{AC} is their joint state, and S(ρ)S(\rho) is the entropy of the quantum state ρ\rho — to the classical mutual information I(A:C)I(A:C), the information about C extractable from A maximized over all local measurements, described mathematically by POVMs (positive-operator-valued measures), through the inequality S(A:C)≥I(A:C)S(A:C)\ge I(A:C), the bound on locally accessible classical information known as the Holevo bound. Jennings and Rudolph objected (Physical Review Letters 104, 148901 (2010)) with a three-qubit example in which mutual entropy decreases while classical mutual information increases, so the inference fails for small systems; they then developed the entanglement-arrow connection for small strongly fluctuating systems (Physical Review E 81, 061130 (2010)). The source concedes the objection is physically trivial — no thermodynamics applies to three qubits — but exploits its mathematical content: in the DLP measurement setup the equality I(A:C)=S(A:C)I(A:C)=S(A:C) holds before and after measurement, and the source proposes this saturation as a measure of macroscopicity of chaotic quantum systems, with the gap S(A:C)−I(A:C)S(A:C)-I(A:C) as a criterion of fluctuation strength.

What the source never supplies is the quantitative content of its own claims: the regime in which the saturation criterion applies is not delimited (no theorem on when the equality holds, no finite-size or coupling scaling, no operational threshold); the assertion that the entropy costs of organizing an unperturbative observation (an observation protocol that does not disturb the observed state) 'will exceed considerably' the observed entropy decrease, and that the system 'needs to be isolated very strongly from environment noise', is made without any model; the scaling of residual off-diagonal corrections, their recurrence times, and the observer's memory dynamics during a Poincaré return are asserted, not derived; the analysis is restricted to closed, weakly interacting, chaotic finite systems; and no quantitative experimental bounds (size, temperature, decoherence rates) are given for where the microscopic regime, in which interpretational differences or recoherence effects might be visible, crosses over to the macroscopic thermodynamic regime the argument covers. No later work is known to have developed or refuted these claims.

Problem Statement

Within the source's physical setting — closed quantum systems of finite volume with finitely many degrees of freedom, macroscopic subsystems that are chaotic (random energy-level statistics) and interact weakly, and observers either included in the system (introspection) or weakly coupled — place the source's in-principle claims on a quantitative, checkable footing, or exhibit their failure. The common objective is to convert the qualitative in-principle unobservability of recoherence and the alignment of thermodynamic time arrows into explicit quantitative statements with a delimited regime of validity. A solution must jointly deliver:

  1. Regime criterion. Turn the proposed saturation measure — the equality of classical mutual information and mutual entropy, I(A:C)=S(A:C)I(A:C)=S(A:C), between a macroscopic device A and the measured system C — into a general quantitative criterion for the regime in which thermodynamic-arrow reasoning, pointer-state selection (the selection, under decoherence, of robust preferred measurement states), and arrow alignment apply: state under what spectral, size, and coupling conditions the equality holds, with explicit finite-size corrections, and how the gap S(A:C)−I(A:C)S(A:C)-I(A:C) scales with system size and fluctuation strength, so that 'macroscopic chaotic' versus 'small strongly fluctuating' becomes an operational distinction rather than an illustrative one.

  2. Recoherence-observability bounds. Construct explicit minimal models that quantify (i) the degree of isolation from environmental noise and (ii) the entropy (accounting) cost of the experimental organization, for any unperturbative observation protocol (one that does not disturb the observed state) that would render recoherence — the Poincaré-return revival of suppressed macroscopic superpositions — observable; the deliverable is a bound on experimentally achievable recoherence within the regime fixed by item 1, or a physically admissible protocol that defeats such a bound.

  3. Rigor of the in-principle unobservability claim. Derive the scaling of the residual off-diagonal corrections left by decoherence, their recurrence times, and their detectability, including an explicit model of the observer's record-keeping (memory) subsystem during a Poincaré return — at introspection the observer returns together with the system, so what must be shown is that no admissible memory dynamics allows the initial and returned states to be compared — such that the claimed in-principle (not merely 'for all practical purposes', FAPP) unobservability of recoherence, and the associated in-principle experimental indistinguishability of the Copenhagen and Many-Worlds interpretations for macroscopic observers, becomes a theorem with explicit hypotheses, or fails.

  4. Regime of validity. Determine where the arrow-alignment argument breaks down — long-range interactions, non-chaotic macroscopic systems, and systems near criticality, where pointer-state structure and locality assumptions may change — at minimum by locating the phase boundaries at which the hypotheses of items 1–3 cease to hold.

  5. Experimental delimitation. Derive quantitative lower bounds (system size, temperature, decoherence rates) delimiting the microscopic regime, in which interpretational differences or recoherence effects might in practice be visible, from the macroscopic thermodynamic regime the source's argument covers.

Items 1–3 are the load-bearing core: the unobservability statement is only as strong as the regime within which its hypotheses are guaranteed (item 1) and the cost-and-memory analysis inside that regime (items 2–3); items 4 and 5 delimit the theorem's domain and its empirical contact and are judged by consistency with the core. A resolution that supplies a quantitative regime criterion plus a proven unobservability bound for that regime — or an admissible counterexample protocol within it — resolves the central objective; items 4–5 may then be satisfied by locating where the criterion and bound cease to apply.

References

  1. Oleg Kupervasser, The Universal Arrow of Time II: Quantum mechanics case, arXiv:1106.6160 (2011; published as Electronic Journal of Theoretical Physics 10(29), 39, 2013), DOI 10.48550/arXiv.1106.6160, https://arxiv.org/abs/1106.6160
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