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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.

edited Aug 16 2010 at 18:25

asked Aug 16 2010 at 15:41

Tim
33●2

	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 2010 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 2010 at 17:37

Andreas Thom · Accepted Answer · 2010-08-16 20:22:29Z UTC

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.

Classification of representations of CCR algebras?

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