I have seen several claims to that quantum mechanics is required to explain the arrow of time which I take to mean the macroscopic irreversibility of physical systems. This is presumably to resolve Loschmidt's paradox. A recent example is a recent Simons Foundation article Time’s Arrow Traced to Quantum Source. Another example from is A quantum solution to the arrow-of-time dilemma. There is also an older expository article that contains the following rather strongly worded paragraph:

The application of classical mechanics to the explanation of thermodynamics resulted in a complex and disgusting mess that explains nothing, though great efforts were expended along this line. This is not surprising, since thermodynamics depends sensitively on mechanics at an atomic scale, for which classical mechanics fails spectacularly. ... with the introduction of quantum mechanics, it has been possible to understand entropy clearly and completely ...

However, as I understand, coarse grain irreversibility has been been demonstrated for theoretically and by numerical simulations for purely classical idealized systems like "billiards" and "gasses" in 2 and 3 dimensions. I am particularly thinking of Simanyi's results on ergodicity and the numerical investigations of Orban & Bellemans, Levesque & Vertlet and Komatsu & Abe. These would suggest that macroscopic irreversibility comes from the geometrical structure of the vector field on the phase space representing the evolution of the systems. States on entropy decreasing trajectories are unstable and the smallest perturbations result in trajectories that become entropy increasing after time. This means that unless systems can are prepared with infinite precision, then any system prepared is almost certain to be on an entropy increasing trajectory unless it is at equilibrium already. What is required is that the dynamics allow trajectories in phase space to diverge nonlinearly.

This would imply that, although there are deep connections between quantum mechanics and irreversibility, quantum mechanics is not strictly required for irreversibility as some articles seem to be claiming. What is still left to explain why the universe is in a far from equilibrium state, but the articles cited at the beginning do not seem to be addressing this aspect either.

What am I missing?

  • 4
    $\begingroup$ I don't think you're missing anything. You're quite correct (IMHO) in saying Loschmidt's paradox has a perfectly good resolution in classical mechanics, but lots of people are still confused about it and continue to believe it needs a special explanation in terms of quantum mechanics. $\endgroup$
    – N. Virgo
    Apr 21, 2014 at 7:43
  • $\begingroup$ If the article means that they have a quantum mechanical explanation for the special initial conditions that lead to the arrow of time, why not. But anything else is misguided. I also note the "complex and disgusting mess that explains nothing" in your quote. Doesn't really invite to read further. $\endgroup$ Apr 21, 2014 at 7:52
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    $\begingroup$ I don't think quantum mechanics solves the irreversibility problem, since its mechanical equation (Schrodinger equation) is just as reversible as classical equations. Perhaps you can appeal to the Born rule (the rule converting amplitudes -> probabilities) but where does the arrow of time come from there? $\endgroup$
    – Nanite
    Apr 21, 2014 at 10:12
  • $\begingroup$ I think too much is stored in the word "entanglement" . It just means that the solutions of the quantum mechanical equations can be evaluated with the given boundary conditions. In my opinion what they are examining is the arrow of time as defined by classical entropy in a more complicated mathematically manner. All classical behaviors emerge from ensembles of quantum mechanical ones, and the smart thing is to use the classical ones for classical concepts, as the arrow of time. It may be interesting to see the emergence from the density matrix formulation, nothing revolutionary. $\endgroup$
    – anna v
    May 1, 2014 at 7:00

2 Answers 2


The question of irreversibility is inextricably linked with thermodynamics, more precisely, with the lack of information of an observer (and a notion of relevant entropy). Entropy increase needs loss of information. This is true in classical mechanics theory as it is in quantum mechanics theory. Indeed, intrinsically, hamiltonian dynamics (be it quantum or classical) is unitary, deterministic and time-reversible, hence isentropic (e.g. information preserving). Consequently, there is no objective flow of time. In my opinion, the best nowaday answer to the issue of time is the thermal time hypothesis worked out by C. Rovelli, A. Connes and P. Martinetti. For more on this topic, have a look at:

  • Diamonds’s Temperature: Unruh effect for bounded trajectories and thermal time hypothesis Pierre Martinetti, Carlo Rovelli, February 2004 http://arxiv.org/abs/gr-qc/0212074v4

  • Von Neumann algebra automorphisms and time-thermodynamics relation in general covariant quantum theories A. Connes, C. Rovelli, June 1994

  • "Forget time", Essay written for the FQXi contest on the Nature of
    Time, August 2008, C. Rovelli


I feel there is little consensus in the physics community about the origins of the arrow of time. This would suggest that we really know little about the nature of time. So most of them merely display their points of view and interpretations but fall short of providing proofs.

The Wikipedia article on the arrow of time already mentions a couple of different conceptions of the arrow of time, It is not always clear which one is being talked about, and there are at least a couple of different dilemmas, as none of them is perfectly understood.

In this situation, the best I can do is to express my personal views.

So I think Loschmidt's paradox (for the thermodynamic arrow of time) can be resolved by purely statistical arguments. Referring to the improbability of spontaneous entropy reduction, I think the thermodynamic arrow is explained well enough by the fact the low entropy state of the universe (which remains unresolved).

In quantum mechanics, there is a different arrow of time, defined by the collapse of the wave function. But in recent years, some researchers have arrived at a quite good understanding what causes the impression of the collapse of the wave function in a macroscopic world, using quantum thermodynamics. So basically you split your quantum world into the small system you consider and the rest which is the reservoir. Then you analyze the time evolution of the projection of the wave function of the small system to quantum states of the reservoir. And you get the collapse of the wavefunction which happens on an extremely short time scale, related to the size of the small system an the coupling between the two. The effect is called disentanglement. It turns out that it is notoriously hard to avoid disentanglement. This fact is being experienced by the folks trying to build a quantum computer, for which you need to keep your qbits entangled.

So the twist to the story is, that in this particular view of the affairs, Statistical arguments and concepts from classical thermodynamics lead to the explanation of the arrow of time in quantum mechanics. Furthermore it tells us something about the relation between quantum mechanics and thermodynamics.

This, on a first read of the Simons Foundation article you are citing, seems to strongly contradict the findings of Lloyd, who claims, entanglement would emerge in accord with thermal equilibrium. Further he claims that entanglement is what causes equilibration. Yet, the effect of disentanglement described above happen on vastly different time scales and scale completely differently with the number of particles. So Lloyd's view is greatly incompatible with mine.

In contrast, the paper by Maccone seems to highlight a fairly interesting fact about the consequences of quantum mechanics. But I think his view is quite consistent with what others say who are being cited in that same article. The way how Maccone presents the claims of the other papers grossly overexaggerates the importance of his own work. (But I think there is no serious scientist who would not do that.) I do not think Maccone does explain thermodynamics arrow of time, rather he already uses it in the step with the wave function collapse. So it would be evident that he gets it back.

As a result, I would not be concerned too much about those claims. Time remains a mystery for the time being, while Loschmidt's paradox is not a paradox at all if you take into account probabilities, I think.


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