Are math only questions relevant when the math is that of quantum mechanics?

Question

In my brief study of quantum mechanics I have found that math done using operators typically breaks from traditional mathematics. Almost certainly between linear algebra and probably something else there is plenty of mathematics to accurately sum up operators. That being said, does it make sense to ask questions about the math of operators on math stackexchange when the math is describing entirely physical systems?

For example if you have the operators A and B that both correspond to observables and you have a question on a bit of math involving them, doesn't it make more sense here where you can simply say that you are doing quantum and one immediately understands how to work with the operators? If you asked in math you would have to 1) be able to appropriately tag your question under the relevant fields of math (who knows what they are) and 2) have to explain very well the background to where the math was coming from.

Furthermore, sometimes things such as the canonical commutation relation throw a wrench into the works in a way that mathematicians might not pick up. For example when helping someone from a physics standpoint you would know that problems involving x and p might lead to having an i hbar d/dt pop out somewhere. Whereas in the math forums you would have to explicately remind people that this is something that exists that they could potentially work with.

It just seems like most math questions that stem from quantum mechanics are much more relevant on the physics forum than on the math forum.

Answer 1

There are plenty of mathematicians who know physics; in fact mathematical physics is considered (in most countries) a branch of mathematics.

In math.SE there is, as in phys.SE, the tag "mathematical-physics" to address questions like the ones you are suggesting. However, to ask a question about operators on math.SE you have to know the mathematical language of operator theory (and functional analysis). Operators are studied in mathematics in many branches: e.g. analysis of PDEs, functional analysis, theory of operator algebras. And the terminology, once you get used to it, is not so far from the one used in physics.

That said, I think that physically relevant pure math questions concerning QM are ok in phys.SE. In addition, I have seen more people from mathematical physics active on phys.SE than math.SE (so you would probably get an answer that has a more physical "taste", nevertheless being mathematically accurate). On the contrary, a pure mathematical question that is only vaguely related to QM because it talks about operators is more suitable for math.SE.

To give two concrete examples, I would say that a math question like this is ok for phys.SE:

What are the sufficient conditions on the classical phase space to uniquely define (up to *-isomorphisms) the algebra of canonical commutation relations? And to have then a unique irreducible representation (up to unitary equivalence)?

While a question like this is not:

Does there exist an m-sectorial operator with compact resolvent whose real part has not compact resolvent?

Answer 2

For example if you have the operators A and B that both correspond to observables...If you asked in math you would have to 1) be able to appropriately tag your question under the relevant fields of math (who knows what they are) and 2) have to explain very well the background to where the math was coming from...

As long as you correctly state the question in terms of mathematical objects, mathematicians will understand and will immediately refer to the corresponding areas. In the case at hand I do not see any particular difficulty to address the theory of operators as suggested tag and I do not want to believe that you have to explain where the mathematics comes from; that those are observables in quantum mechanics or mere self-adjoint operators, it makes no difference (that is exactly the entire purpose of using the formalism in physics). For example, the below:

Question: let A, B be two self-adjoint operators such that [A,B] \subset i1 on their domains of definition. Prove that given any L^2 function on the circle then...

perfectly addresses any physics question related to the uncertainty principle and similars that can be asked on the mathematics platform. This said, I am nevertheless a strong supporter of keeping the mathematics questions on the physics area because:

1. despite any personal opinion, mathematics does play a prominent role in modern physics, especially in QFT, string theory, quantum gravity and related areas. Most of nowadays research in the aforementioned is mainly carried exploiting mathematics rather than physical ideas (for instance I do not see any physics in the renormalisation procedures in QFT, to say one, but that could as well just be me...)
2. physicists are, according to me, far better than mathematicians in the area of mathematical physics.
3. (warning, very irrelevant!) those questions are the only ones I answer and understand ;-).