Recently I have been learning a lot about what kind of axioms and mathematical formulations there are for non-relativistic quantum mechanics. Unfortunately, I am a little confused, because at first I thought that Von Neumann axioms are nice enough, but then I saw some posts about saying that Von Neumann measurement axiom has to be replaced by POVM (positive operator valued measure) for measurement process to be more realistic. Then I saw some arguments about how Von Neumann axioms are not good enough because they deal only with closed systems. When I considered mathematics books like L. Takhtajan "Quantum Mechanics for Mathematicians" then there axiom about measurement process is not given at all. I also saw that, for example, there are ideas that "wave function collapse" can be reinterpreted by taking unitary evolution of system + apparatus by using some partial trace argument.
In any case, I would like to train myself for research in mathematical quantum optics, so I am interested in what kind of axioms of quantum mechanics are accepted nowadays. Of course, I understand that there are equivalent axioms that make structures isomorphic to each other. I also am aware that physics community does not have one opinion about axioms, so I would like to receive an answer about the most popular approach (if it is possible to estimate which one it is). I assume that there is an answer because I do not think that there could be anything theoretical investigation in, say, quantum optics if there were no axioms on which models are based upon. If there is not (if there are no sets of axioms on which majority of physicists agree upon) then how is mathematical analysis in quantum optics usually conducted?
I would appreciate all kind of opinions and comments about this problem!