In a multiple particle Hilbert space (any space of any multi-particle system), is it sufficient to define creation and annihilation operators by their action (e.g. mapping an n-particle state to an n+1-particle state) or does one have to do anything else, like “proving existence”. Intuitively, one can apriori say it exists if you can write it down, but I donno.
If the action is given, the existence is obvious (unless the definition is faulty). But one would usually want to verify that all creation operators commute,
all annihilation operators commute, and the commutator of a creation operator and an annihilation operator is a c-number.
Moreover, for a field theory, one would usually want to verify that this c-number transforms covariantly under Poincare transformations, and vanishes at spacelike separation. This applies for free field theories. For interactive field theories, there are no intrinsic c/a operators, as their properties are destroyed by renormalization. But one can associate (according to Haag–Ruelle theory) to each bound state a family of creation and annihilation operators parameterized by momentum, describing the free asymptotic motion in a scattering process.