# Classification of representations of CCR algebras?

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.

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Good question. I am more familiar with the fermionic case (i.e., CAR as opposed to CCR) where the uniqueness theorem fails basically due to the existence of Bogoljubov transformations relating different inequivalent vacua. A similar story should hold for the CCR case, but since I'm not sure and unable to check at present, I'll leave this as a commment. –  José Figueroa-O'Farrill Aug 16 '10 at 16:14
I think finite dimensional reps are dealt with here en.wikipedia.org/wiki/Stone-von_Neumann_theorem For the more generaly case, I'm not sure. Maybe an operator algebras or C^*-algebras tag should be added to lure someone more suited to answer –  Owen Sizemore Aug 16 '10 at 17:37
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.