Can Quantum Mechanics Be Expressed as a Specific Mathematical Structure?

[jordi]

My question is: can we say that QM can be expressed as a given mathematical structure?

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.

 

[jordi]

When I say "But “matrix elements” do not constitute something so important in functional analysis as correlation functions are in a probability theory" is not 100% correct: matrix elements define fully an operator.

What I mean is that for a probability mathematician, to define a probability theory is quite clear: give the dependence of the variables (via correlation functions, copulas, or whatever) and all the properties of the theory can be fully worked out "easily". This have a biunivocal correspondence with the qft physicist (scattering matrices can be calculated using correlation functions).

But for a functional analyst, this language is not true: usually, a functional analyst does not understand its work as finding a set of operators and find a clever basis that diagonalizes its operators. Maybe this can be part of his/her work, but functional analysis is broader. A functional analyst may be interested in properties of operators that have nothing to do with diagonalising them. In other words, a "functional-analysis theory" is not diagonalizing operators (but a "probabilistic theory" is finding correlation functions). This has not a biunivocal correspondence with the qft physicist.

 

[jordi]

Nobody has any comments?

 

[Talisman]

How much QM reading have you done? I ask this because, although I have very little GR background, my first introduction to QM was in terms of the Hilbert space formalism -- where it becomes clear very quickly why certain observables are called "noncommuting," what "causes" EPR correlations, the double-slit experiment interference, etc.

Check out this book perhaps?

https://www.amazon.com/dp/0674843924/?tag=pfamazon01-20

"A formidable and intelligent account of the (partial) Hilbert-space formalization of quantum mechanics and the inevitable philosophical ambiguities that result."

 

[jordi]

Talisman, thank you for your reply. I have a degree in physics, and a MSc in Theoretical Physics. Currently I work in finance, and studying physics is only my hobby.

My question is about the presentation of the theory: GR is usually explained very easily to a mathematician: you understand differential geometry (a theory that existed much before than GR) and then GR is only differential geometry + some additional postulates. It is like applying a known mathematical theory to a particular case.

Instead, QM is presented in a different way. Of course, you need to known functional analysis, probability, ... but it is not the particular case of a given theory.

What I would like is, for example, to understand QM as a particular probabilistic theory: QM is a probability theory where the sigma-algebra is such-and-such (or a given one for the oscillator, another one for the free particle, ...). And then, all physical properties can be understood/calculated just following a standard probability book (for example, correlation functions in probability are correlation functions / S-matrix elements in physics; ...).

 

[jordi]

For example, another thing that bothers me in the presentation of QM is that there are pseudo-explanations of the maths involved. All QM books talk about Hilbert spaces, basis, ...

But when you go to the physics itself, you have the books using freely concepts like the time derivative of an operator ... but I have never seen how a book defines the time derivative of an operator.

Or statements like: "for a free particle, the eigenstates are vectors not lying in the vector space" (meaning that they are delta Diracs, distributions). But they never give a explicit representation of this basis in this vector space. And they do not explain if it is valid or not.

I have the feeling that this kind of statements have a precise, standard meaning, but you need to know much more functional analysis than the functional analysis explained in the first chapter (or the appendix) of a QM book. So, in a sense, they are cheating.

What I would like is:

1. presentation of ALL the functional analysis I will ever need to understand rigorously QM

2. application of this material