What are the potential connections between Quantum Mechanics and Model Theory?

[nomadreid]

Although a physicist probably would just shake his head and point to the usual Hilbert space framework, it is nonetheless tempting for anyone dabbling in Mathematical Logic to see if any of the existing structures out of Model Theory could be appropriate for quantum physics concepts. The closest I can find is some sort of possible world semantics, such as Kripke Frames, but I hit some snags very quickly. If we have a collection of possible worlds which are connected by an accessibility relation, then how is a superposition of different worlds represented? Are indeterminate states then part of the accessibility relation, or is a representation by truth values in a complex lattice sufficient? How are entanglement and the uncertainty relations handled? Perhaps I am not searching correctly, but I find no decent answers on the Internet. Can anyone point me in the right direction?

 

[chiro]

Hey nomadreid.

Given that you have some linguistic framework for representing "stuff", is the quantum extension to that to have things as a kind of "random variable" (with any kind of extra conditional attributes or complex probability relationships) where you model probabilities and collapse based on those of evolution operators (like the Hamiltonian) in that system?

 

[nomadreid]

chiro, obviously whatever framework one will work in for quantum mechanics, one will end up with probabilities and Hamiltonians in there somewhere, although sometimes in the guise of possible world selections, and so forth. However, I was looking for something more precise out of Model Theory, since most searches for "quantum logic" simply refer to the standard Hilbert space formalism, leaving out all the concepts which have grown out of the field of Model Theory. One of the more interesting attempts I found was arxiv.org/gr-qc/9910005, although I don't think it fully answers the issues I raised in my original post. As far as modeling collapse, this "measurement problem" is still an ongoing debate even in the usual framework.

 

[chiro]

I'm wondering if you can use a power set approach: In a quantum system of n-q bits, you have a state space of 2^n different states with the standard superposition of states.

Maybe the use of substituting power sets where appropriate and taking a new look at the theory with this approach.

If this was going to get formal, you may have to result to theoretical results in measure theory and sigma algebra's and stochastic processes.

The thinking is that you develop a logic in terms of the power set in an abstract way, then you can look at the logic in terms of attributes of the power set and make deductions on the logic on the attributes of the power set rather than on the individual characteristics themselves.

The sigma algebra's would be useful to because they include some of the properties a power set has.

 

[nomadreid]

chiro, thanks for the suggestion. Yes, sigma algebras, or filters/ideals might be handy, even perhaps ultrafilters/principal ideals (although there is the issue of completeness that is tricky for a quantum theory which allows virtual particles). Appropriate equivalence relations might work for modeling entanglement, but set intersection does not model superposition very well. So the way forward is still not very clear.

 

[Stephen Tashi]

What properties make an axiomatic system a "logic" as opposed to a physical theory?

As I understand the axiomatics of Quantum Mechanics, there is nothing unusual about the logic used in the mathematics. It's the same logic you would use in doing proofs in other fields. In the sense that the conclusions of Quantum Mechanics can contradict commonly held intuitions, it has an unusual "logic".

 

[nomadreid]

Thanks for the reply, Stephen Tashi.

What properties make an axiomatic system a "logic" as opposed to a physical theory?

A logic only has to be self-consistent; a physical theory has to be both internally and externally consistent; i.e., it has to agree with experimental data. In other terms, a logic is purely syntactical, and is valid if there exists any model for it, whereas a physical theory includes both the syntax and semantics, and is valid only if it is fulfilled by a particular model. However, that is splitting hairs, so I have no problem in accepting the term "quantum logic" to mean either one.

As I understand the axiomatics of Quantum Mechanics, there is nothing unusual about the logic used in the mathematics. It's the same logic you would use in doing proofs in other fields.

You are, of course, absolutely correct. The problem lies in what I was trying to express when I threw in the bit about Model Theory, Kripke semantics, etc. To be more explicit, I am going to go back to splitting hairs. The mathematics to which you are referring would be, in Model Theory, the syntax (the "theory" in the sense of Model Theory). The "reality" which it is trying to describe, the semantics (the "model" in the sense of Model Theory). The link between them is the "interpretation function" (assigning elements form the model to the symbols of the theory in order to give the theory meaning). This is the crux, and is at the heart of debates, old and new: some of them are: "It from Bit?" (Wheeler), "is the wave function real?", the Everett many-worlds interpretation versus the Copenhagen interpretation, Philosophical Idealism versus Philosophical Materialism, Platonism versus Formalism, etc. Some attempts have been made to make a Model-Interpretation-Theory structure for each of these disputes, starting with making a semantics for the standard mathematics of quantum theory for the Theory, but each one that I have looked seems to have its limitations. However, perhaps I am not looking hard enough.

In the sense that the conclusions of Quantum Mechanics can contradict commonly held intuitions, it has an unusual "logic".

Ah, counter-intuitive notions in Model Theory rival those in Quantum mechanics! It is part of the folklore among logicians that one mathematician(sorry, I would have to search for the name) threw up his hands in despair when hearing from his logician friend (again, I would have to search) that you could build a countable model to satisfy the statement that there are uncountable sets (Skolem's paradox). The large cardinals get even weirder. A mathematician is not bound by reality, so in a game of "who's weirder" between mathematician and physicist, I would put my money on the mathematician. :-)