# Creation and annihilation operators for any operator which has discrete and countable eigenvalues?

I have studied basic quantum mechanics and have come across the creation and annihilation operators in the context of the quantum harmonic oscillator, and of the angular momentum operator. It made me think whether for any observable which has discrete and countable eigenvalues (may be infinite or not) there exist analogous operators that do the same job as creation and annihilation operators? (I suppose even for a continuous spectrum of eigenvalues, there are often operators which can evolve the state such that the eigenvalue of the new state is incremented by some infinitesimal amount, such as the shift and time translation operators).

On the Wikipedia article for creation ad annihilation operators, there is a section on 'Generalised creation and annihilation operators'. I think if I et the jist of it, that it seems to imply that your can create such operators in general, however I am not sure. I am certainly not familiar with the mathematical concepts referred to in this section. I would appreciate if someone was able to summarise this in a less-technical way, if it is applicable to my question.

As an aside, I was also confused by the fact that the Wikipedia article mentions that "Creation/annihilation operators are different for bosons (integer spin) and fermions (half-integer spin). This is because their wavefunctions have different symmetry properties." In my classes and in all of the books I have reffered to, I have seen only one form of the creation and annihilation operators for the quantum harmonic oscillator and the angular momentum, and I think this is the form that Wikipedia says is for the 'bosonic case'. If there are in fact different creation and annihilation operators for fermions, why is it that this is not mentioned in so many books and online? Plus, I see no stage in the working that shows the creation operator increasing the energy eigenstate or angular momentum eigenstate by 1 (or annihilating for the annihilation operator) that depends on the state on which it acts to represent a boson. Granted, I have only come across the use of these operators for a single particle. Perhaps I am confusing 'ladder operators' and 'creation and annihilation operators'. These terms were used synonymously in my lectures, but it might make sense if 'ladder operators' is actually the correct term for an operator that increments the state of a single particle system by 1, whereas 'creation and annihilation' operators refer to operators acting on multi-particle states, that happen to have the same form as the ladder operators in the bosonic case, but not if our many-particle system contains fermions. (In that case, what about a mixture of bosons and fermions!)

• I have put more than one question in this post- I thought they were sufficiently related, but am happy to split them into different posts if that is more suitable. – 21joanna12 Apr 20 '18 at 14:54
• FWIW, note that creation and annihilation operators are never diagonalizable, cf. e.g. this Phys.SE post. – Qmechanic Apr 20 '18 at 15:17