Hi,


I'm wondering if there is a some classification of representations of CCR algebras (<http://en.wikipedia.org/wiki/CCR_algebra>), where say the underlying vector space is a separable Hilbert space. 


My naive understanding is that for a QFT, one wants a representation of a CCR algebra satisfying certain properties.



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](http://arxiv.org/abs/0705.1988) using the notion of resolvent algebra. This settles the problem very nicely from a mathematical and physical perspective.


