Factors of the Fock functional

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Abstract
Can the fields φ and π of a representation of the CCR’s be written as φ=1/{φ1×1+1⊗φ2} and similarly for π, such that φi and πi satisfy the CCR’s? What are the possible φi’s and πi’s? This is equivalent to a factorization of the corresponding generating functionals (scaled by 1/). Generalizing this question somewhat we show a noncommutative analog of Cramér’s theorem of probability theory. If φ and π are Fock fields then so are φi, πi, i=1,2; similarly for quasifree representations of the CCR’s. As an application we show that the fields of a representation of the CCR’s whose generating functional differs from a Fock functional by a phase factor only are just shifted Fock fields.

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