rt.representation theory - Classification of representations of CCR algebras?

The question depends very much on the regularity that you demand. You have to decide before asking the question which operators are supposed to be self-adjoint or merely symmetric as unbounded operators etc. Weyl has solved the problem by exponentiating everything and looking at the resulting relations. This however gives rise to some unphysical representations.

Buchholz and Grundling give a new $C^star$-algebraic approach to the problem in 0705.1988 using the notion of resolvent algebra. This settles the problem very nicely from a mathematical and physical perspective.