http://rev-inv-ope.univ-paris1.fr/IMG/pdf_35214-09.pdf, “Hilbert’s Sixth Problem: Descriptive statistics as new foundations for probability” which, in common, with von Plato, reverses the usual approach and uses descriptive statistics of actual data as the foundations for probability theory, instead of using probability theory as the foundations for statistics as in Cramer and practically everybody else. ]]>

Regarding a summer placement, I am not sure whether you mean within India or internationally. Within India there are a few professors who work on quantum foundations, but I am not too familiar with them or with what programs exist for summer placements in India. Ravi Kunjwal, a grad student from IMS Chennai, recently visited us and he is working on quantum foundations, so you might want to contact him for advice on where to go in India. His website is http://www.imsc.res.in/~rkunj/

Internationally, I would say that it is rather rare for an institution to have funding for an undergraduate to visit from elsewhere. We have programs for undergraduates to do research placements, but they are mostly intended for undergrads at our own institutions. Even at the graduate student level, short research visits are not all that common, and it would normally be expected for the student to have research experience and to have published relevant papers. The exception, of course, is if you can obtain your own money to fund the visit. For example, I know that the American Physical Society has funds explicitly for graduate students from India to visit the USA, but that is only for graduate students. Nevertheless, you might want to look at the websites of professional physics societies and of research funding agencies to see if there are any schemes that might apply to you. Professors are likely to look more kindly upon a request to visit if you can fund it yourself in this way.

The other option for you would be to look into doing your masters degree abroad after you finish your undergrad. At Perimeter Institute, we have the Perimeter Scholars International course, and that is fully funded for successful applicants. I know you have looked at Rob Spekkens’ lectures, but the courses vary from year to year and it would also provide an opportunity to do a research project with a faculty member. Similar courses exist in other places, but usually not funded so you would need to obtain an separate scholarship. Maths part III at Cambridge and the Theoretical Physics masters at Imperial College both have a quantum component and would give you the opportunity interact with faculty interested in quantum foundations. Similar opportunities probably exist in other countries, but I don’t know about them.

Since you say you are interested in Everett and decoherence, it would be remiss of me not to mention the philosophy of physics group at Oxford, because this is basically Everett central. They have a masters degree program in philosophy of physics, which would be a good way of getting into that. On the physics side, you might try contacting one of the decoherence people like Wojczech Zurek to see if he knows of any opportunities for summer placement, but, as I said, I think such opportunities are rare at the undergrad level.

]]>The thing is, I have no one to guide me in this respect in my institute (which apparently is one of the leading institutions of India).. I’ve got a summer coming up, and I think it would be nice if I work under an experienced professor’s guidance in that time towards the questions of the foundations of quantum physics.. Could you suggest programs, summer schools or professors who could help me in this respect? – let me take up a project maybe?

]]>Yes, because the conditional states formalism and Bohmian mechanics both seem very nice and natural to me, so I was wondering if it’s possible to eat my cake and have it! Thanks for the lengthy discussion! I think I understand the issues much better now.

I think I have only one last question for the moment. Does the conditional states formalism apply to infinite dimensional Hilbert spaces?

]]>Yes it does. One just puts an ordinary classical probability distribution over the particle configurations and derives the analog of the Liouville equation from the guidance equation. More generally, you could put a probability measure on the space of wavefunctions as well, but I’ve never seen anyone do that.

“If that’s possible, could one say that the conditional states formalism also applies to Bohmian nonequlilibrium, since the conditional states formalism contains standard probability?”

You really seem to want to stretch things so that Bohmian mechanics looks compatible with the conditional states formalism. Yes, in principle you could do this, but it is not a very natural thing to do from either perspective. In the conditional states formalism, classical probability applies if all the quantum states and conditional states involved are diagonal in a common product basis. Therefore, you would have to imagine that there were two quantum states associated with a quantum system. The first is the usual Bohmian wavefunction, but then you also need a state that is diagonal in a product basis to represent the classical probabilities over the particle configurations. Even in equilibrium you would still have two quantum states.

Now, one of the virtues of the Bohmian interpretation is that it does not require a modification of classical probability theory. The probabilities within it are just ordinary classical probabilities and can be interpreted in any of the standard ways (e.g. Bayesian, frequentist, etc.). Therefore, for a Bohmian it is very bizarre to introduce an exotic type of probability where none is needed. On the other hand, from the conditional states point of view, quantum states should be thought of as generalized probabilities. In your reconciliation, you have one quantum state that isn’t to be interpreted in that way, i.e. the usual Bohmian wavefunction that guides the particle trajectories, and another that is interpreted that way, but however is always in the classical special form. Thus, the interpretation of a quantum state as a generalized probability is playing absolutely no role.

Ultimately, if you have two viable accounts of quantum theory then it will always be possible to munge them together in some way in order to make them look consistent because, at the end of the day, both accounts must make the exact same predictions. However, if the two accounts are based on opposite ideas about the ontology underlying quantum theory then the union will likely be a kludge and I fail to see the point in pursuing it.

]]>If that’s possible, could one say that the conditional states formalism also applies to Bohmian nonequlilibrium, since the conditional states formalism contains standard probability?

]]>Yes, that’s what I mean. In Newtonian mechanics you can use whatever probability distribution on phase space you like and compute its evolution via the Liouville equation. It does not have to be the equilibrium case.

Also, the emergence of classicality and the passage to thermal equilibrium are somewhat independent of each other. Although it is the case that many examples of decohering enviroments also thermalize the system and vice versa, this is not always be true. It depends on the relative strengths of the system Hamiltonian and the system-environment interaction, and on whether the environment is itself in a thermal state. Therefore, you can have systems that behave classically without being in thermal equilibrium.

]]>When you say that standard probability applies to non-equilibrium situations, do you include the completely deterministic case of Newtonian mechanics for a single particle in a potential? (I’m trying to understand whether you mean that in classical mechanics, standard probability always works, and never has to get turned on.)

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