# How exact is the analogy between statistical mechanics and quantum field theory?

Famously, the path integral of quantum field theory is related to the partition function of statistical mechanics via a Wick rotation and there is therefore a formal analogy between the two. I have a few questions about the relation between the two objects.

1. Loop diagrams in quantum field theory have a nice interpretation in terms of virtual particles. What is the interpretation of loops in statistical mechanics?

2. Does the relation between the two objects imply that for every quantum mechanical phenomenon there's a corresponding statistical mechanics phenomenon and vice versa? If not, where does the analogy fail?

3. If so, what would be the phenomena analogous to the Aharonov-Bohm (AB) effect? It's hard for me to see the analog for any quantum effect that depends on phase interference like the AB effect.

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There are insights to be gained into QFT by looking at the analytic structure of S-matrix elements and such things. I would suspect that those results would be altered due to the Wick rotation -- while insights based on formal reasong using path integrals might carry over more easily. Just my 2 cents of thinking out loud. – Siva Sep 12 '14 at 18:38
I think you two last questions are linked in fact. One has to pay great attention when one wants to deal with thermal field theory in a gauge context. Perhaps you might be interested in this related question: physics.stackexchange.com/q/99683/16689 and the associated answer. – FraSchelle Feb 28 '15 at 22:19

I think it will depend the kind of statistical mechanics. For classical statistical mechanics, there is no time, so it is really hard to imagine a nice physical picture of the propagation of something. But nevertheless we still talk of loops as propagating "particles" (we give the "momenta", for instance, which is conserved, etc.). Interestingly, renormalization (a la Wilson) is easier to understand on a physical ground in statistical physics, where the coarse graining has a very nice interpretation.

On the other hand, in quantum statistical physics, the analogy is bit more direct, though time is still imaginary, so nothing really propagates. But in some sense, we still sum over all the possibilities (in a static sense, though). In this case, AB effect will give the quantization of the flux, or the Quantum Hall Effect.

Anyway, concerning the first question, keep in mind that loops, Feynman diagrams and virtual particles are artifacts of pertubation theory, and therefore have no real physical interpretation.

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I've heard your statement about loops just being some artifact of perturbation theory which shouldn't really be interpreted as virtual particles. What's wrong with the virtual particle interpretation? Do you have any references for this statement, also? – user26866 Nov 21 '13 at 13:39
@user26866, they are a pictorial tool, but they don't capture non-perturbative physics like instantons. How would you represent the Coulomb-type electric field around an electron in terms of virtual particles? You can use coherent states, but these don't have definite particle number. You might have an idea of what a virtual particle is, but it really looks nothing like a real particle, (hence, virtual). – lionelbrits Dec 6 '13 at 1:07
Adding to what lionelbrits said: Virtual particles are only an approximation to the sum over field histories. The sum over field histories is a mathematical representation of the time evolution of the system, but you should be careful ascribing too much physical meaning to it. The summands are not physically meaningful. They are not observable; we have no way of measuring what's going on between measurements. (And if that doesn't convince you, let me point out another problem: the summands are in general not gauge invariant.) – user1504 Dec 6 '13 at 16:17
@user26866: On virtual particles: physics.stackexchange.com/a/22064/7924 – Arnold Neumaier Dec 8 '13 at 15:49

In my naive view, this is merely a mathematical trick that should not be taken too seriously in term of physical interpretation.

After all, a "Wick rotation" applied to the Schrodinger equation yields a diffusion equation. This is helpful for some mathematical problems but the physics it describes is very very different from quantum mechanics; not even mentioning that one is a wave equation while the other is a Fokker-Planck equation.

Now, because quantum field theory and statistical field theory share the same mathematical structure (i.e. a path integral as a generating functional), they also share useful tools like Green functions, Wick theorem, Feynman diagrams and so on but this more a mathematical coincidence than a deep meaningful analogy between the two, in my opinion...

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[By statistical mechanics I mean classical statistical mechanics throughout this answer. If you are curious to think about the complications added with making the statistical side of the story quantum mechanical, that sounds like a very good exercise. For clarification look at Chap. 3 of "Conformal Field Theory" by Di Francesco et al.]

The analogy between "Euclidean quantum field theories" and "equilibrium statistical mechanics near second order phase transitions" is exact, once you identify $\hbar$ (on the quantum side) with $1/\beta$ (on the statistical side). Being careful with the terms Euclidean and equilibrium is important to avoid misguided analogies. The proximity to a second order phase transition guarantees that (the continuum limit of the underlying statistical system approximates it well, and thus) the statistical mechanics can be well approximated by statistical `field theory'.

1) Roughly speaking, in real time quantum field theory, every intermediate stage happens with a probability proportional to $e^{iS/\hbar}$. Often you interpret those intermediate stages as "virtual particles". In Euclidean (or imaginary time) quantum field theory, there is no "intermediate" stage, so the right interpretation is (not in terms of virtual particles, but) that all possible classical configurations contribute to the partition function with a probability proportional to $e^{-S/\hbar}$. Now to connect this Euclidean QFT situation with one in equilibrium statistical mechanics near a 2nd order phase transition, one only needs to specify in what sense "all possible classical configurations contribute to the statistical partition function with a probability proportional to $e^{-\beta S}$". The sense in which the above statement is true in equilibrium statistical mechanics is of course, the Ergodic sense.

In sum, the answer to your first question is that i) the virtual particle interpretation does not apply to Euclidean QFT (which, unlike real time QFT, is analogous to equilibrium statistical mechanics near second order phase transitions), ii) in both Euclidean QFT and equilibrium statistical mechanics, every allowed classical configuration contributes to the partition function; it is just that in Euclidean QFT this has a fundamentally probabilistic interpretation, whereas in equilibrium statistical mechanics it has a statistical interpretation supported by the Ergodic theorem.

2) Yes. In fact, every Euclidean quantum field theory can be regarded as describing an equilibrium statistical physics system near a 2nd order phase transition. The term Statistical Field Theory is applied whenever the field theory is interpreted as describing some statistical system.

3) There is no Aharonov-Bohm effect (in the sense of electrons propagating and interfering with each other) in Euclidean QFT. This is a confusion similar to the one with "virtual particles" which is due to not keeping the word Euclidean in mind; there is no propagation in imaginary time QFT. Also on the equilibrium statistical mechanics side, there is no such a thing. However, if you are looking for manifestations of non-trivial gauge bundles, you can find such manifestations on both sides by looking at Wilson loops circulating around solenoids installed in your quantum or statistical system.

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i will have to disagree with some of the answers posted in this question.

First, this involves a matter of interpretation of the quantum formalism (and a prevailing "interpretation", the Copenhagen one)

Although this interpretation (which i find unsatisfactory and non-physical) may seem prevailing (and indeed it might be), is not because it offers a better or more clear understanding of quantum mechanics (indeed the known R. Feynman quote might be relevant, "no one understands quantum mechanics")

Most physicists just work on a formalism and not enter into any interpretation aspect, although they may find it unsatisfactory.

(sometimes this turns into a "scientific taboo")

So the answers posted that speak of a coincidental resemblnace between statistical mechanics and quantum mechanics, actually talk about interpretation (ie. the Copenhagen interpretation)

So a specific interpretation (which in the best of cases is there as a historical artifact or maybe tradition, but not necessarily science), leads to an associated answer.

Leaving aside all this for a minute (my stance is that the connection between quantum mechanics and statistical mechanics, specifically entropy is very interesting, see for example http://math.stackexchange.com/a/782596/139391),

lets see some other relations between the QM and SM:

1. Planck's constant (and indeed the start of QM) was on a statistical mechanics problem (black-body radiation). Furthermore Planck's constant h, was computed using statistical methods.

2. Wick rotation, has a physical meaning (there is 1-1 correspondence between a "quantum" system and a "statistical" one), The formalism actually reflects this fact

3. There are theories (more or less sponsored) which derive quantum mechanics as an extension of statistical mechanics (or vice-versa). Eg. Stochastic Mechanics (a good attempt), Generalised Thermodynamics (in progress), etc..

4. Quantum mechanics without complex numbers (and Hilbert spaces) is just Statistical mechanics (and Euclidiean spaces). One use of complex numbers is to define a boundary, a closed system, periodic conditions. Since quantum mechanics can represent a SINGLE system (unlike statistical mechanics which represents ENSEMBLES of systems), it all goes back to the double-slit experiment (and the interpretation thereof)

5. There is still the problem of quantum measumerment asymmetry (and possible relations to entropy), which lacks a good explanation/interpretation/re-formulation (the Copenhagen interpretation might be the worst intrepreation in this case)

Thanks

UPDATE: in a light-hearted fashion one can say that QM seems to be the SQUARE-ROOT of SM, or mutatis-mutandis SM is the SQUARE of QM

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the second is more apt, it is all those phases that make the difference which disappear with the complex conjugate square – anna v Sep 27 '15 at 14:21
@annav, thanks anna where do you refer exactly, 2) point about wick rotation, or sth else? – Nikos M. Sep 27 '15 at 15:51
it is on your update, the mutatis part – anna v Sep 27 '15 at 15:59
@annav, ahh yes :) did you read the linked post on math.se which includes a small analysis about this post as well? – Nikos M. Sep 28 '15 at 11:49