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REVIEW 4 major objections 5 minor 71 references

The arrow of time, irreversibility, equilibrium and measurement in quantum mechanics

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Pith's one-line read This paper claims that quantum mechanics, in the thermodynamic limit, becomes intrinsically irreversible: isolated systems evolve to microcanonical equilibrium, and measurement outcomes are objectively selected with probabilities given by t

desk verdict A well-built formal restatement of the Prigogine program whose advertised results—irreversibility and collapse—are imposed by boundary condition and conserved initial data, not derived from the dynamics. read the letter →

arxiv 2607.19142 v2 pith:QFFVXH45 submitted 2026-07-21 quant-ph cond-mat.stat-mechmath-phmath.MPphysics.hist-ph

Christopher J. N. Coveney , Peter V. Coveney This is my paper · ORCID
classification quant-phcond-mat.stat-mechmath-phmath.MPphysics.hist-ph
keywords arrowoftimeirreversibilitythermodynamiclimitmicrocanonicalensemblequantummeasurementwavefunctioncollapseBornruleLiouvillianspectrum
topics The Measurement Problem
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper aims to show that quantum mechanics is already complete: once large isolated systems are described in the thermodynamic limit, their Liouville-von Neumann dynamics breaks time-reversal symmetry on its own, without added collapse rules or open-system assumptions. The key mechanism is the absolutely continuous spectrum of the Liouvillian, which turns the unitary evolution group into retarded and advanced semigroups. In the long-time limit, every initial pure state flows to the microcanonical equilibrium state, entropy increases, and coherence is lost. Applied to a spin coupled to a macroscopic measuring device, the same dynamics makes the reduced spin state diagonal with the standard quantum probabilities, and the full system converges to a convex combination of disjoint equilibrium states—which the authors identify with objective wavefunction collapse. If correct, the result unifies the arrow of time, irreversibility, and measurement within unmodified quantum mechanics.

What carries the argument

The central object is the resolvent of the Liouvillian, (z−L)^{-1}, and its analytic continuation to the second Riemann sheet. For systems with purely absolutely continuous Liouvillian spectrum, the resolvent has a branch cut across the real axis; continuation through the cut yields a unique simple pole at z=0—the invariant microcanonical state—plus resonance poles with negative imaginary parts and complex branch-cut background. This structure converts reversible unitary dynamics into a forward-time dissipative semigroup, produces the long-time limit ω^{mce}_{eq}, and determines the degenerate equilibrium manifold in the measurement model. The paper also uses the reduced density matrix's sel

What would settle it

Compute the resolvent of a concrete non-integrable model (for example, the spin-boson Hamiltonian of Eq. 28 beyond the one-particle sector) and check whether the continued Green's function has a unique simple pole at z=0 and only sub-real-axis resonances; if additional real-axis branch points or poles on the real axis beyond z=0 appear, the predicted unique microcanonical attractor and the quantum measurement mixture fail. An experimental observation of long-time recurrences or persistent coherence in an isolated macroscopic superposition would likewise contradict the predicted irreversible ap

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Extended reading notes

Core claim

The paper's central claim is that for isolated, non-integrable quantum systems in the thermodynamic limit—where the Liouvillian's spectrum is absolutely continuous—the resolvent develops a branch cut across the real axis, and the unitary evolution splits into a retarded semigroup for t>0 and an advanced semigroup for t<0. The physical, forward-time semigroup is obtained by analytically continuing the Green's function through the branch cut to a second Riemann sheet, where a unique simple pole at z=0 gives the microcanonical invariant measure and additional resonance poles with negative imaginary parts give exponentially decaying corrections. As a result, lim_{t→∞} ω_t(A)=ω^{mce}_{eq}(A) for

Load-bearing premise

The argument rests on the assumption that, in the thermodynamic limit, the Liouvillian of a non-integrable many-body system has a purely absolutely continuous spectrum and that its Green's function admits a meromorphic continuation to a second Riemann sheet with the required simple pole structure—a spectral and analytic property that the paper asserts but does not prove for realistic Hamiltonians.

Editorial extensions

If this is right

  • If the claim holds, isolated macroscopic quantum systems genuinely reach microcanonical equilibrium and never exhibit Poincaré recurrence, so finite-size recurrences are irrelevant in the thermodynamic-limit description.
  • Quantum measurement requires no separate postulate: the projection postulate and the standard quantum probabilities emerge from the same dynamics that produces equilibrium in the total system; the reduced density matrix's decoherence is derived rather than assumed.
  • Coherence times of quantum devices are set by the complex resonance poles of the analytically continued Liouvillian, giving a first-principles route to computing decoherence rates in large systems.
  • The approach supplies a dynamical derivation of eigenstate thermalization: every energy eigenstate is already microcanonical because the equilibrium state is the attractor of the dynamics.
  • Entropy increase is built into the forward semigroup, so the second law of thermodynamics is compatible with reversible microscopic evolution without ad hoc coarse graining.
  • The analysis shows that to record a stable measurement outcome, the measured observable must be conserved by the dynamics; otherwise the record decays and the system simply relaxes to the microcanonical ensemble.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • An extension the paper leaves implicit: the same resolvent machinery could be used to derive quantum kinetic equations and transport coefficients for realistic many-body systems, with the resonance poles playing the role of relaxation rates.
  • The disjointness of the equilibrium states implies that measurement outcomes are objective and not observer-relative; if this is right, it would dissolve the need for many-worlds or observer-involved interpretations, a consequence the paper does not spell out.
  • A testable implication the authors do not pursue: the spin-boson model's decoherence rate is set by the imaginary part of the leading resonance pole, so engineered quantum simulators could measure this rate and directly test the predicted exponential decay.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper develops an algebraic formulation of quantum dynamics and claims that, when the Liouvillian has absolutely continuous spectrum (as asserted for the thermodynamic limit), a branch cut in the resolvent splits the unitary group into retarded and advanced semigroups, producing irreversibility, loss of coherence, and convergence to the microcanonical equilibrium state. For a spin-boson measurement Hamiltonian, it further claims that the full isolated system converges to a mixture p_up omega_up + p_down omega_down, and that the dynamics selects one of these disjoint extremal states, thereby yielding objective collapse with Born probabilities. The central claims are Eqs. (26), (42), and (51)-(52).

Significance. If the claims were established, this would be a major unification of quantum dynamics, the second law, and the measurement process, with substantial implications for foundations and for practical estimates of coherence times. The algebraic-state framework is appropriate, and the paper correctly emphasizes that the reduced density matrix is only a shadow of the global state. It also identifies a concrete coupling asymmetry needed for pointer distinguishability. However, the central conclusions rest on unsupported spectral-analytic assumptions and on two externally imposed elements: the discarding of the advanced semigroup and the selection of one extremal equilibrium state. The paper contains no machine-checked proofs, and the critical analytic-continuation and spectral assertions are stated rather than demonstrated. I therefore do not regard the main claims as established.

major comments (4)
  1. [Emergence of the arrow of time; Eqs. (17)-(19)] Eq. (19) is not a derivation of irreversibility. For t>0 the retarded term alone gives e^{-iLt} and the advanced term is zero; for t<0 the advanced term gives e^{-iLt}. The unitary group is thus exactly recovered. The advanced sector is then discarded because it 'is in direct contradiction with the second law' and because 'the universe began in a low entropy state.' This is a boundary-condition input, not an emergent property of unitary dynamics. The claimed time-symmetry breaking is therefore imposed rather than derived.
  2. [Emergence of the arrow of time; Eqs. (21)-(26) and App. D] The approach to microcanonical equilibrium rests on two unsupported assumptions: (i) L has purely absolutely continuous spectrum for the spin-boson Hamiltonian (Eq. 28) and for generic non-integrable many-body Hamiltonians; (ii) G_A(z) admits a meromorphic continuation through the branch cut to a second Riemann sheet with a unique simple pole at z=0. The text says this is 'straightforward to realize' and 'Suppose that the continuation reveals a series of simple poles' (App. D). No argument is supplied for realistic many-body systems. Moreover, conserved quantities imply L1=0 and L(a_i^dagger a_i)=0, so zero is protected as an eigenvalue in the algebraic setting; the claim of purely absolutely continuous spectrum needs reconciliation with these constants of motion. If assumptions (i) or (ii) fail, Eqs. (24)-(26) do not follow.
  3. [Quantum measurement; Eqs. (37), (49)-(52)] The asymptotic state actually derived is the convex mixture p_up omega_up^mce + p_down omega_down^mce. Because H=H_up + H_down and [H,a_i^dagger a_i]=0, the sector weights are conserved for all t (Eq. 37); no term in Eq. (28) couples the sectors, so the thermodynamic limit cannot change them. Calling omega_up and omega_down 'disjoint extremal states' yields at most an ergodic decomposition of the equilibrium mixture. The assertion that the intrinsic dynamics 'selects an extremal disjoint equilibrium state' is an additional, unstated postulate. This is the load-bearing step for the claimed explanation of wavefunction collapse, and it is not derived.
  4. [Quantum measurement; Eqs. (42), (46)] The statement that the Born rule is 'directly obtained from the underlying dynamics' overstates the result. The probabilities p_i=|alpha|^2,|beta|^2 are not generated by the time evolution; they enter as conserved initial data rho_ii(t)=rho_ii(0) (Eq. 37). The dynamical content of Eq. (42) is the decay of the off-diagonal terms, i.e. dephasing. A definite measurement outcome additionally requires the selection postulate identified above. Thus the Born-rule probabilities are imported through the initial state and a selection rule, not derived.
minor comments (5)
  1. [Time evolution of observables; Eqs. (5)-(6)] The parameter gamma in Eqs. (5)-(6) is not defined; it should be specified, and the contour orientation should be stated.
  2. [Throughout] There are typos: 'respectfully' should be 'respectively' in the measurement section; 'neccessary' in the spectral section; 'Equilibrium Thermalization Hypothesis' should be 'Eigenstate Thermalization Hypothesis'.
  3. [Introduction and 'Thermodynamic limit' section] The abstract and first section say continuous spectra arise only in the thermodynamic limit, while later the paper correctly notes that scattering theory and quantum fields already have continuous spectra. This should be reconciled.
  4. [Appendices C and D] The notation <A,omega> is used both for the ordinary dual pairing and for distributional pairings on the rigged space; marking this distinction would avoid ambiguity.
  5. [Fig. 1(a)] The figure shows entropy increasing toward both past and future, which is confusing given the text's claim that the advanced sector is unphysical; the figure should be redrawn or explained.

Circularity Check

2 steps flagged · score 6.0 of 10

The arrow of time and the Born-rule collapse are not derived: the second law is used to select the retarded semigroup, and the measurement mixture’s weights are conserved initial data, with the “selection” of one extremal state asserted without a mechanism.

  1. self definitional [Section 'EMERGENCE OF THE ARROW OF TIME AND EQUILIBRIUM STATES FROM THE DYNAMICS OF QUANTUM SYSTEMS', after Eq. (27)]
    "However, the advanced-time direction is in direct contradiction with the second law of thermodynamics which requires the entropy of the universe to increase with time. This is ultimately the consequence of the fact that the universe began in a low entropy state. It means that the advanced solution is in direct contradiction with physical reality and must be discarded."

    Eqs. (17)–(19) show that the branch-cut topology of the resolvent generates both the retarded and advanced semigroups, and Eq. (27) shows that the advanced evolution also approaches the microcanonical equilibrium state. The paper selects the retarded semigroup by invoking the second law and the low-entropy initial condition of the universe. Thus the arrow of time and entropy increase are assumed as boundary conditions in order to pick the physical solution, rather than derived from the thermodynamic-limit dynamics. The claim that time-symmetry breaking emerges from the dynamics is circular: the conclusion (entropy increases) is used to discard the only alternative solution that would violate it.

  2. self definitional [Section 'APPLICATION TO QUANTUM MEASUREMENT AND IRREVERSIBLE WAVEFUNCTION COLLAPSE', Eqs. (36)–(37), (42), (51)–(52), and the paragraph after Eq. (52)]
    "ρ s ii (t) =ρ s,0 ii ... the asymptotic solution is given by lim t→∞ ρ s (t) = |α|2 0 0 |β|2 ... lim t→∞ ω t (A) = p ↑ ω mce ↑ (A) + p ↓ ω mce ↓ (A), with p ↑ = |α| 2 and p ↓ = |β| 2. These probabilities appear in the equilibrium state as they are constants of the motion. ... wavefunction collapse into a definite state is a result of the intrinsic dynamics in the thermodynamic limit that selects an extremal disjoint equilibrium state"

    The only dynamical content derived for the spin subsystem is dephasing of the off-diagonal elements, Eq. (40); the diagonal entries are conserved because [H,a_i†a_i]=0 (Eqs. 36–37). Consequently Eq. (51)/(52) is a convex mixture whose weights are the initial Born coefficients, not quantities generated by the dynamics. The paper then asserts that the intrinsic dynamics 'selects an extremal disjoint equilibrium state', but H=H↑⊕H↓ contains no term that couples or breaks the two sectors, so no mechanism is provided for one outcome to be realized. The Born rule and wavefunction collapse are therefore reinserted as conserved initial data plus the extra postulate that one sector is realized; the claimed prediction reduces to the input by construction.

full rationale

The paper develops substantial original machinery involving the analytic structure of the Liouvillian resolvent, algebraic quantum states, and semigroup splitting on the second Riemann sheet. Several intermediate results are internally derived, including the dephasing of off-diagonal coherences and the convex-mixture form of the long-time state. However, the two headline claims are not as derived as presented. First, the arrow of time: Eqs. (17)–(19) and (27) show that both time directions approach the microcanonical state, and the paper discards the advanced solution by invoking the second law and the universe’s low-entropy initial condition. That is an external input, not an emergent output of the thermodynamic limit. Second, the measurement outcome: the asymptotic state is a convex mixture whose weights are the conserved initial sector probabilities (Eqs. 36–37); the claim that the dynamics 'selects an extremal disjoint equilibrium state' is an additional postulate not supported by any term in H=H↑⊕H↓. The paper also relies on unproved spectral assumptions (absolute continuity, meromorphic continuation, simple pole structure), but these are assumptions rather than circularities. Because the central arrow-of-time and Born-rule collapse results reduce to the inputs used to construct them, a score of 6 is appropriate; the paper is not formally circular throughout, but its core predictions are partly circular by construction.

Assumptions & free parameters 1 free parameters · 7 assumptions · 2 invented entities

The paper fits no numbers and makes no quantitative predictions; the ledger is dominated by structural assumptions rather than fitted parameters. The one hand-chosen structure doing load-bearing work is the distinguishing coupling g_↑ − g_↓ that makes the two measurement outcomes macroscopically distinct by construction. The central results rest on unproven spectral/analytic-continuation assumptions, an imposed low-entropy boundary condition selecting the retarded branch, the chosen limit order, and standard ergodic/statistical facts. The two invented objects (second-sheet eigenfunctionals and disjoint 'macroscopic reality' states) carry the interpretive weight of objective collapse but have no independent falsifiable handle.

free parameters (1)
  • Pointer-state coupling asymmetry g_↑(k) − g_↓(k)
    Hand-chosen in the measurement Hamiltonian (Eq. 28) and converted into the pointer observable M via Δg(k) (Eqs. 57–58) so that ω_↑(M) ≠ ω_↓(M) 'by construction'. Outcome distinguishability is therefore an input, not a derived property.
assumptions (7)
  • domain assumption Purely absolutely continuous Liouvillian spectrum for the models (thermodynamic limit, non-integrable, mixing)
    Case 3 of the paper's spectral classification; asserted for Eq. 28 ('similar to the Friedrich's model') without proof. The entire approach to equilibrium and the branch-cut splitting rest on it; point or singular-continuous components would leave non-decaying oscillations.
  • domain assumption Meromorphic continuation of G_A(z) through the branch cut with the stated pole structure (simple poles, Im z_J < 0, unique z_0 = 0)
    Appendix D: 'it is straightforward to realize ... admits a meromorphic continuation'; 'Suppose that the continuation reveals a series of simple poles'. The equilibrium state and the measurement mixture are read off this continuation; no proof is supplied for the many-body Hamiltonians treated.
  • ad hoc to paper The advanced (past-pointing) semigroup is discarded because the universe began in a low-entropy state
    Main text after Eq. 27: the advanced direction 'is in direct contradiction with the second law ... must be discarded. This is ultimately the consequence of the fact that the universe began in a low entropy state.' The arrow of time is thus imposed as a boundary condition, making the second law an input to the derivation of the second law.
  • domain assumption Only the total energy (plus the conserved spin projections) is conserved; the systems are non-integrable
    Needed for the uniqueness of the z = 0 pole and for identifying the invariant state with the microcanonical ensemble (Eqs. 22–26). The two-body interaction v(p) among apparatus modes is included so the bath is mixing, but its spectral properties and sufficiency for mixing are not demonstrated.
  • standard math Classical ergodic facts: absolute continuity ⇒ mixing; mixing ⇒ decay of correlations to the invariant measure (Riemann–Lebesgue)
    Borrowed from classical ergodic theory (refs 7, 9, 44–53); the quantum analogue is assumed to hold in the same form on the dual Banach space of algebraic states.
  • standard math Microcanonical ensemble is the unique equilibrium state compatible with the conserved quantities
    Standard statistical mechanics (refs 19, 23, 24); used to identify the z = 0 residue state ω~0 with ρ_mce in Eq. 25.
  • domain assumption Order of limits: lim_{t→∞} is taken after the thermodynamic limit (Eq. 8), asserted to be 'the physically correct limit'
    This limit-order choice is what converts an almost-periodic finite-system dynamics into irreversibility; it is a modeling assumption about what 'isolated macroscopic system' means, justified heuristically by analogy with phase transitions, not proved.
invented entities (2)
  • Generalized eigenfunctionals of the continued Liouvillian on the second Riemann sheet (Ω̃_J, resonances z_J)
    purpose: Carry the irreversible decay terms (Ruelle–Pollicott resonances) in Eqs. 22–23 and D2–D5; the machinery behind equilibrium and measurement in this paper.
    Inherited from the cited Antoniou–Petrosky–Prigogine literature and redeployed; the paper attaches no falsifiable handle (no predicted rates, no observable signatures) to them.
  • Disjoint extremal equilibrium states / 'macroscopic realities' ω_↑, ω_↓
    purpose: Explains why a single measurement outcome occurs: the equilibrium mixture is claimed to reduce to one extremal disjoint thermodynamic state (Eqs. 51–56 and following text).
    No experimental signature distinguishes this from standard decoherence plus the Born rule; the single-outcome step is asserted, not derived, and the paper's own asymptotic state is the convex combination.

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Cite this review

Pith. "Pith review of The arrow of time, irreversibility, equilibrium and measurement in quantum mechanics." pith.science (2026). https://pith.science/paper/QFFVXH45

@misc{pith2026260719142,
  author       = {Pith},
  title        = {Pith review of: The arrow of time, irreversibility, equilibrium and measurement in quantum mechanics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QFFVXH45}},
  note         = {Machine review of arXiv:2607.19142}
}
read the original abstract

Quantum mechanics is widely recognised as being incomplete. It is not consistent with the second law of thermodynamics and does not provide a scientifically credible physical account of the measurement process, the means by which coherence is broken and classically observable states are recorded. This has led to many ad hoc assumptions being used to account for various properties of quantum systems, among which is the coherence time of quantum devices that determines their ability to perform computations. Here, we show that all these properties can be accommodated naturally and consistently in the context of quantum systems which exhibit continuous spectra, as arises in the thermodynamic limit of large systems. In particular, for isolated systems we show that the time-reversal symmetry associated with unitary time evolution of the quantum state gives rise to time-symmetry breaking and a semi-group evolution which attains thermodynamic equilibrium at long times. Moreover, the emergence of this non-unitary time-asymmetry leads to microcanonical equilibrium states in which all quantum coherence is lost and is accompanied by the transformation of pure states into mixtures, leading in turn to an increase in entropy. Inclusion of a macroscopic measurement apparatus shows how the outcome of a measurement corresponds to the von Neumann projection postulate, arising with probabilities in conformance with the Born rule. The mathematical structure of the theory which applies to quantum systems with continuous spectra is closely analogous to the classical ergodic theory of dynamical systems and the conditions under which they attain equilibrium states.

Figures

Figures reproduced from arXiv: 2607.19142 by the authors.

Figure 1
Figure 1. FIG. 1: (a) Asymptotic approach to equilibrium due to the dissipative dynamics contained in mixing systems that [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Different algebraic structures of the resolvent of the Liouvillian due to the nature of its spectrum. The left [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗

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Pith tools

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