My question is: can we say that QM can be expressed as a given mathematical structure?(adsbygoogle = window.adsbygoogle || []).push({});

More precisely, GR can be understood easily as "something to do" with differential geometry (even though some authors, like Weinberg, prefer not even to talk about DG when talking about GR). You have your Riemann tensor, ... and some PDE relating DG-concepts.

Thermodynamics is also like GR, where you have a clear mathematical structure, and you only need a "dictionary" between the mathematics and the physics.

So, a mathematician could understand GR as DG + some additional axioms that encapsulate the physics.

Can something like this be done with QM, and even better, QFT? I have studied some QM books, and most of them start with QM postulates. But they do not look like the GR postulates. I think that for a mathematician it would be more difficult to disentangle what are the mathematical structures, and what are the additional, physical postulates.

Of course, in QM there are mathematical structures: Hilbert spaces, self-adjoint operators, distributions, ... but, at least to my taste, they are not equivalent to what DG represents to GR.

I have heard (but not studied) something called C*-algebras, that maybe do what I want, but I am not sure.

Is there a book, or article, that does what I want? That is, state a standard mathematical structure, add a few physical axioms, and then work out the (physical) theory as just postulates, theorems, ... where all physical concepts have their correspondent language in the mathematical structure.

In fact, to me Quantum Mechanics' mathematical structures look like much more like a probability theory: you have correlation functions, the effective lagrangian, ... as a clear, mathematical meaning (in probability theory). Instead, the usual, accepted “quantum language” is functional analysis. Of course, you can understand correlation functions is functional analysis’ terms: it is simply a “matrix element”. But “matrix elements” do not constitute something so important in functional analysis as correlation functions are in a probability theory.

I have read (but not studied) that there have been some trials in this direction (Nelson), but they have not fully worked out. But Kac theorem bridges the functional analysis with the probability in quantum theory, so probability is at least part of the solution.

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# QM: mathematical structure?

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