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

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 had 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.

• The paper shows that an absolutely continuous Liouvillian spectrum in the thermodynamic limit separates unitary evolution into retarded and advanced semigroups, producing irreversible relaxation to the microcanonical ensemble and a monotonic entropy increase.
• The paper derives equilibrium as a dynamical attractor through resolvent branch cuts, Ruelle–Pollicott resonances, and a zero-frequency conservation pole, offering a quantum explanation of thermalization for non-integrable, mixing systems.
• The paper models measurement with a spin–boson apparatus and finds that coherences decay as approximately t⁻² for an ohmic bath, while conserved Born weights define disjoint macroscopic outcome sectors, although single-outcome selection remains unresolved.

Overview and central claim

This paper by C. J. N. Coveney and P. V. Coveney argues that the two most prominent deficiencies of quantum mechanics—its incompatibility with the second law of thermodynamics and its lack of an internally consistent account of measurement—are resolved within standard quantum theory itself once systems are treated in the thermodynamic limit. The key technical condition is that the Liouvillian $\mathcal{L}$ possess an absolutely continuous spectrum, which occurs for macroscopic quantum systems with $N, V \to \infty$ at fixed density. Under this condition, the authors show that time-reversal symmetry is broken, unitary evolution splits into retarded and advanced semi-groups, isolated systems relax to the microcanonical ensemble, pure states evolve into mixtures with monotonically increasing entropy, and the von Neumann projection postulate together with the Born rule emerges from the intrinsic dynamics of a measurement apparatus coupled to a subsystem. No modification of quantum mechanics is invoked; the formalism is built on algebraic quantum states governed by the Liouville–von Neumann (LvN) equation.

Framework: algebraic states and the resolvent

The dynamics is formulated on the Banach algebra of observables $\mathcal{A}$ and its dual space of states $\mathcal{A}^*$, rather than on Hilbert space vectors. This choice is not cosmetic: in the thermodynamic limit the identity operator is not trace-class ($\text{Tr}\{\mathbbm{1}\}$ diverges), so Hilbert–Schmidt formulations cannot represent the microcanonical distribution. The authors explicitly note that earlier work by Petrosky and Prigogine, based on Hilbert–Schmidt Liouville space, fails on this point; the algebraic formulation repairs it.

Time evolution is solved via the Laplace transform of the LvN equation, so that all dynamical information resides in the analytic structure of the resolvent $R(z) = (z\mathbbm{1} - \mathcal{L})^{-1}$:

Spectral type	Resolvent structure	Long-time behavior
Pure point (finite systems)	Meromorphic, discrete poles	Periodic; strong Poincaré recurrence
Point + continuous above thresholds	Poles plus branch cuts	Quasi-periodic; partial dissipation
Absolutely continuous (mixing)	Branch cut across entire real axis	Irreversible relaxation to equilibrium

For finite systems the inverse Laplace transform can be closed into a single contour for all $t$, recovering the full unitary group. For absolutely continuous spectra, the resolvent develops a discontinuity across the real axis—the spectral function—and belongs to operator-valued Hardy spaces supported only in one half-plane. Consequently the unitary group decomposes as $\hat{U}(t) = \hat{U}_+(t) + \hat{U}_-(t)$, where each sector carries a Heaviside step function and forms a semi-group. A notable methodological point is that these causal step functions arise intrinsically from the resolvent topology, whereas in much of quantum field theory they are inserted by hand in the definition of projected Green's functions.

Time-symmetry breaking and the microcanonical attractor

The physical limit is the non-commuting order $\lim_{t\to\infty}\lim_{N,V\to\infty}$, analogous to how spontaneous symmetry breaking requires taking the thermodynamic limit before removing an external field. Analytic continuation of the Green's function through the branch cut onto the second Riemann sheet yields Ruelle–Pollicott resonances at complex eigenvalues $z_J$ with $N, V \to \infty$0, plus a unique simple pole at $N, V \to \infty$1 guaranteed by probability and energy conservation for non-integrable systems. All resonant and continuum background contributions decay (exponentially or via power-law tails), leaving

$N, V \to \infty$2

where $N, V \to \infty$3 is the microcanonical ensemble—the quantum analogue of the classical Sinai–Ruelle–Bowen measure. Because this state is a distributional eigenstate outside Hilbert space, it lives on the second Riemann sheet, which is why rigged Hilbert space (Gel'fand triple) machinery is required.

Two features deserve emphasis. First, symmetry-broken solutions come in pairs: backward-time evolution also converges to the same invariant measure, but this advanced solution contradicts the second law given the low-entropy initial state of the universe and is discarded, in the same way boundary conditions select physical solutions of differential equations. Second, the result provides a dynamical derivation of the Equilibrium Thermalization Hypothesis (ETH): the microcanonical ensemble is shown to be the actual attractor of non-integrable mixing dynamics, rather than a property asserted of individual energy eigenstates. The authors also construct a Lyapunov functional from the biorthogonal resonance expansion that decreases strictly until equilibrium, making the entropic arrow explicit.

Application to quantum measurement

The measurement analysis uses a spin-boson model in which a two-level system couples to a bosonic field representing a macroscopic apparatus, with conserved spin projections $N, V \to \infty$4 and distinct couplings $N, V \to \infty$5 so that the apparatus can distinguish outcomes. The exact reduced density matrix obeys a Volterra equation closed by a self-energy whose off-diagonal components acquire nonzero imaginary parts inherited from the branch cut of the full resolvent. The exact cumulant solution gives

$N, V \to \infty$6

with $N, V \to \infty$7 producing long-time power-law decay of coherences. Under an ohmic spectral density the off-diagonal elements decay as $N, V \to \infty$8, i.e., $N, V \to \infty$9 asymptotically, and the subsystem equilibrates to the diagonal mixture with weights $\mathcal{A}$0 and $\mathcal{A}$1—the Born probabilities appearing directly as constants of motion.

The stronger claim concerns the full isolated system. Because the total Hamiltonian has an absolutely continuous spectrum, the global state also relaxes irreversibly to a mixed equilibrium state that decomposes into disjoint extremal sectors,

$\mathcal{A}$2

with the degeneracy of the $\mathcal{A}$3 pole set by the conserved spin projections. The disjoint extremal states correspond to mutually exclusive macroscopic pointer realities, analogous to thermodynamic phases; collapse into a definite outcome is then attributed to selection of one such extremal state, with probability equal to its conserved spectral weight. The authors argue this is strictly stronger than decoherence theory, which addresses only the reduced density matrix and does not establish irreversibility of the full isolated dynamics nor explain how a single outcome is realized. An appendix analyzing a competing Hamiltonian without conserved projections shows that when the measured observable is not approximately conserved, the system relaxes fully to the microcanonical ensemble and no stable measurement record survives—clarifying why conservation of the measured quantity is essential for a pure measurement.

Limitations and open questions

The argument rests on several assumptions stated or implicit in the paper. The approach to equilibrium requires genuine mixing, i.e., an absolutely continuous Liouvillian spectrum with energy as the only conserved quantity; integrable or finite systems are excluded, and the paper concedes that oscillatory behavior persists there indefinitely. The analytic continuation assumes simple poles without multiplicity, and the treatment of singular continuous spectral contributions is set aside as irrelevant. The claim that collapse selects one disjoint extremal state identifies the mechanism qualitatively but does not derive a dynamical equation for when or how the selection occurs beyond the probabilistic weighting—the inherently random single outcome remains asserted rather than computed. The measurement model is idealized (von Neumann-type coupling, two-state subsystem, specific spectral density assumptions), and extending the framework to realistic apparatuses, quantum kinetic equations, and transport phenomena is identified by the authors as subsequent work rather than completed here.

Conclusion

The paper presents a unified account in which irreversibility, equilibration, entropy increase, wavefunction collapse, and the Born rule all follow from the analytic structure of the Liouvillian resolvent in the thermodynamic limit, using algebraic quantum theory and rigged Hilbert space methods without altering quantum mechanics. Its principal contributions are the rigorous construction of the microcanonical ensemble as a second-Riemann-sheet attractor state and the demonstration that measurement outcomes correspond to disjoint extremal equilibrium sectors weighted by conserved Born coefficients. Whether the selection among disjoint macroscopic realities can be given a more explicit dynamical characterization, and whether the framework extends quantitatively to realistic measurement devices, remain open questions raised by the analysis.