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Measurement of a quantum observable (in an appropriate, old-fashioned sense) necessarily involves coupling to a system with a macroscopically large number of degrees of freedom. Entanglement with this "apparatus" takes care of the decoherence. It is often said (I can provide references upon request) that the remaining problem is the one of "selection", and this is the point where one invariably invokes something philosophically radical, like many-worlds interpretation.

In the above (pretty standard) context, I am trying to make sense of the following observation. Looking at the measuring system from a statistical mechanics point of view, it seems that triggering a particular macroscopic outcome requires spontaneous symmetry breaking via a (thermodynamically) irreversible transition of the "apparatus" from a metastable to a higher entropy final state. My attitude is that "statistical mechanics point of view" is not far from "decoherent large quantum system".

So, the question is:

Is it fair to say that statistical irreversibility ("the second law") and quantum measurement irreversibility (the "wave-function collapse") are necessarily linked? Can this link be made more concrete (e.g., traced in details in a particular model)? Can you give references to approaches to the measurement problem that explore this connection?

Qmechanic
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Slaviks
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2 Answers2

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Quantum statistical irreversibility ("the second law") and quantum measurement irreversibility are almost the same thing. Indeed,the latter is the special case of the former where one assumes a more specific situation in which you consider the statistical mechanics of a small system coupled to a large one. Equilibrium and nonequilibrium statistical mechanics, where one studies a single large system only, are other special case of general quantum statistical irreversibility.

The techniques used (essentially the projection operator formalism and its variants) are essentially the same in every case, but each kind of system has its own specialties.

See, e.g., the arXiv papers cond-mat/0102428 and cond-mat/0203460 for some exactly solvable problems that shed light on your problem.

For more details in the general case see the discussions at http://www.physicsoverflow.org/35537 and https://www.physicsforums.com/threads/collapse-from-unitarity.860627/

A good book from which one can learn the necessary statistical mechanics background necessary to understand how irreversibility arises in real macroscopic systems is the statistical physics book by Linda Reichl. For the classical side, see also Oettinger, Beyond equilibrium thermodynamics.
For the quantum side, see also Calzetta and Hu, Nonequilibrium quantum field theory.

Arnold Neumaier
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  • Fantastic! The references you give is exactly the development of ideas I was looking for. Will accept your answer soon (if nothing emerges). – Slaviks Mar 26 '12 at 13:56
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    @Arnold Neumaier: I was digging through questions on physics.SE related to the measurement problem since I was trying to find out what the actual solution is that everyone one seems to find so obvious. And this answered my question! excellent answer and references: +1 – Wolpertinger Apr 10 '16 at 17:25
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    @Numrok: see also the new references at the end! – Arnold Neumaier Jun 23 '16 at 16:49
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    @Slaviks: Could you please indicate why you unaccepted my answer? If possible with references to papers that made you discontent with my answer! – Arnold Neumaier Jun 23 '16 at 16:50
  • @ArnoldNeumaier thanks! also have some advertising for this great answer. I am currently trying to work towards understanding these papers, have to learn some more statistical physics first. would you happen to have a good textbook recommendation on the topic? I am using Altland and Simons at the moment. – Wolpertinger Jun 26 '16 at 08:35
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    @Numrok: I like the statistical physics book by Linda Reichl. (Different editions vary a great deal; maybe look at several of them!) – Arnold Neumaier Jun 26 '16 at 10:54
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    For the classical side, see also Oettinger, Beyond equilibrium thermodynamics. – Arnold Neumaier Jun 26 '16 at 10:55
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    For the quantum side, see also Calzetta and Hu, Nonequilibrium quantum field theory. – Arnold Neumaier Jun 26 '16 at 11:00
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From Boltzmann's formula, S = k log W , where W is the number of possiblities, we see that entropy, S, increases with multiple possible outcomes. Since the point of a measurement is to find which of multiple possibilities is actually true, then a measurement will necessarily increase entropy. (Unless you are deliberately performing a repeated measurement, where you know the outcome in advance.)

Michael C Price
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