In the fall of August 2001 I was in Gargnano on Garda Lake in Italy to participate at the Conference “Mysteries, Puzzles and Paradoxes in Quantum Mechanics”. This was one of a series of Conferences with the same title organized by Rodolfo Bonifacio, a former full professor at University of Milan and now retired (latest news say that he is taking sun in Brasil). These Conferences were very successful as the participants were generally the most representative in the field of quantum optics and fundamental physics. I have had also the luck to meet interesting people that are still in touch with me like Federico Casagrande, an associate professor at University of Milan currently carrying on relevant research in quantum optics and laser physics. That year there was also Vittorio Giovannetti. Vittorio took a PhD in Physics at University of Camerino with Paolo Tombesi and David Vitali that are behind an international renowned group of quantum optics and gave also to the community a number of high quality researchers. At that time Vittorio was a post-doc at MIT and was working together with Seth Lloyd and another brilliant Italian post-doc Lorenzo Maccone. This collaboration produced a lot of relevant papers, mostly in applications of quantum mechanics, that appeared on Nature, PRL and several other high impact archival journals.

Bonifacio was involved with an original idea about intrinsic decoherence. He got a paper published on Nuovo Cimento B and another, with the collaboration of Camerino’s group, on PRA. After we listened at his talk about this interesting matter I exit the room where talks were taken place and exchanged some words with Vittorio and another person. In a while I averted my attention from Vittorio and the other person and started to mumbling thinking about decoherence. Than, looking at Vittorio I said loudly: “Yes, thermodynamic limit! Classical limit can be obtained from quantum mechanics much in the same way thermodynamics is obtained from statistical mechanics!”. Vittorio stared at me and repeated “Yes, thermodynamic limit.” than kept on talking with the other person. This was the start of a lot of papers I have got published on this matter and some interesting experimental work has also been done. The question is still open. The proceedings of the Conference are here.

Today there is a lot of confusion in physics about classical limit and interpretation of quantum mechanics. Indeed, there is a lot of people accepting without critics many-world interpretation without realizing that are out of the realm of physics in this case. If a theory has no criteria to undergo an experimental check is not a theory and we have to forget about this. I have seen a lot of unprepared people talking about many-worlds without elementary cognitions of physics. This is bad and this is why we are living this times today. Mathematics is not enough to be a physicist.

“If a theory has no criteria to undergo an experimental check is not a theory and we have to forget about this.” This is too strong. We build models of theories, and see how closely some feature in the model matches experimental results. It will not be perfect, but we accept it or not depending on what competing theories and models there are for an experiment. It’s about comparison of models as much as it is about theories. The differences between theories allow hard and fast criteria in simple cases, but in more interesting and delicate cases, particularly when differences between theories are conceptual, pragmatic choices are necessary, which in some cases we may only be able to characterize well decades after the experiment.

The pragmatic reasons for preferring one theory and its models over another theory and its models include such things as engineering effectiveness, particularly how easy it is to build useful models of the theory, and the process is always subject to a failure to construct a good model in one theory while a good model is easier to construct in the other (that might be a good reason to prefer the second theory, but the first theory is only ruled out absolutely if it can be proved that no model of the theory, whatsoever, and of no large or small modification of the theory that we are prepared to countenance, can accommodate the evidence in way that is satisfactory to a large number of Physicists — no ad-hocery, but you know that’s a funny call). Suppose we threw out quantum theory because it happened that no-one was creative enough to construct BCS theory within a year of the experimental question being posed? Particularly in statistical physics, model-building is a demanding past-time, even though the theory is well established.

I’m pleased, incidentally, that you have apparently been careful not to recite a Popperian falsificationism, which has finally started to fade from the lexicon of Physicists.

But as to the intuition, yes!!! Nice story, too. I have recently been taking the following quote from Feynman and Hibbs, pp 22-23, “The usual separation of observer and observed which is now needed in analyzing measurements in quantum mechanics should not really be necessary, or at least should be even more thoroughly analyzed. What seems to be needed is the statistical mechanics of amplifying apparatus” to be a side of what I take is approximately the same idea (I’ve been using it rather obsessively, in fact, so I’m looking for another quote from as reputable a source). Best, Peter.

Hi Peter,

Many thanks for your nice and useful comment. You are right: I do not share Popperian views but anyhow I find unbearable unproven truths to be passed to people as deep truths and I think that it is just a matter of time before we can forget about blind alleys that belong to scientific endeavor.

I have always wanted to know why such a simple mathematical evidence that large number of particles and Planck constant going to zero are equivalent is generally missed. It is just plain mathematics but it does seem really difficult to catch! I have heard at a Conference in Piombino in 2006 a physicist intervening during a talk to protest against the speaker that simply showed this mathematical result. So both agreed that “This is the semiclassical limit but not like the semiclassical limit”!

Best,

Marco

Oh! I see this everywhere, so correct me if you see it as having nothing to do with yours, but my Aha! moment was a dozen years ago: there are correlations at space-like separation in a classical statistical field theory (which I now call a random field, following the mathematicians), and correlations at space-like separation in a quantum field theory, so what’s the difference? It shouldn’t have taken me so long to get there, but now I can tell you that the quantum field vacuum is Lorentz invariant, whereas the thermal equilibrium state of a classical statistical field theory is invariant under the little group of a time-like vector (as a subgroup of the Lorentz group, of course). This is sort of inevitable, insofar as we construct thermal states using the Hamiltonian and phase space, whereas the quantum field vacuum is defined to be Lorentz invariant.

There are, indeed, at least for free quantum fields where it’s more-or-less obvious, two different kinds of fluctuations, quantum and thermal. The amplitude of quantum fluctuations is determined by Planck’s constant in almost exactly the same way as the amplitude of thermal fluctuations is determined by the temperature: more temperature=more thermal fluctuations; bigger Planck’s constant=more quantum fluctuations.

Do I have to go and read your papers? I think I might. I suppose what you say above conflicts with what I say above, however. Taking the number of particles to infinity will give different results depending on precisely how the thermodynamic limit is taken. If we take the limit in a way that leads to a thermal state, that has nothing to do with Planck’s constant, but it has to do with thermal fluctuations; if we take the limit in such a way that it leads to a Lorentz invariant state, then it has to do with Planck’s constant, but not to do with the thermodynamic limit as it is traditionally intended. I think it is useful to keep clear which limit we are taking, whatever we call them.

Is this helpful for you? Peter.

Dear Peter,

Your comment is absolutely interesting and I am aware of the deep analogy between QFT and statistical mechanics and the limitations you point out. But my question does not take me out of the realm of QFT: E.g. take Laughlin and Pines Hamiltonian and apply thermodynamic limit to this “everything theory”. What do you get? I proved that you get Thomas-Fermi approximation, that means semiclassical limit. This is a very simple theorem but difficult to be digested by the community. If you have time read this paper of mine

http://arxiv.org/abs/quant-ph/0611024

Marco

Usually I am not sure what people mean by saying taking the classical limit, roughly speaking they usually mean taking planck’s constant to zero, but I feel uneasy taking a dimensionful quantity to zero!

Dan

Dan,

when people in physics takes some dimensionful constant to a limit, it is always implied that there is some other physical quantity that has the same dimension varying in some way. I mean, when you take the limit of the speed of light going to infinity you are just saying that the ratio of whatever measured velocity to that of light is so small that formally in all the formulas you can take any ratio between a measured velocity and the speed of light to be almost zero and the speed of light going to infinity. In this way you get a mathematical consistent framework.

For the Planck constant is the same thing. You will have a ratio between an action and the Planck constant and the numerator is varying largely with respect to the latter making it very small formally. This is exactly what happens in the thermodynamic limit granting the emerging of the classical limit.

Marco