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I cannot find reference, but I read that in curved spacetime, there exists representation that is not Fock satisfying CCR and unitarily inequivalent to a Fock representation.

In usual understanding of quantum field theory in flat spacetime, CCR is, or can be, written between annihilation and creation operators as well, so any representation satisfying CCR is automatically Fock representation.

So is the "satisfying CCR" part referring to CCR not between annihilation and creator operators, and we cannot arrive at creation and annihilation operator or something like that?

OR is this referring to the fact that once we pick one Fock representation, other Fock representations are unitarily inequivalent to this Fock representation and are considered not to be Fock representation?

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Independently of the background spacetime, be it curved or flat, there are uncountably many inequivalent irreducible representations of the algebra of canonical commutation and anticommutation relations.

Many of them are Fock representations, corresponding to fields of the same spin (and charge eventually) but with different masses. There are, however, also interacting representations that are inequivalent to the free Fock representations and that are not of Fock type. Some explicit examples are known in flat spacetimes with $1+1$ and $2+1$ dimensions.

In general, Haag's theorem guarantees that there are inequivalent irrepresentations of any C*-algebra of quantum observables that comes with a representation of some group $G$ (e.g., the Poincaré group), as long as there are at least two different $G$-abelian pure states (the $G$-abelian states are $G$-invariant states that satisfy additional properties).

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  • $\begingroup$ But we consider non-Fock representation as unphysical when doing usual scattering amplitude computations (LSZ reduction), right? (3+1 dimensional spacetime) $\endgroup$ – Krudak Krudak Mar 23 '18 at 17:26
  • $\begingroup$ No, actually this is not true. The non-Fock representations (or at least the inequivalent, w.r.t. the free ones, interacting representations) are unavoidable, because of Haag's theorem. One should therefore take into account that the interacting theory is inequivalent to the free, asymptotic one in scattering theory. The workaround to make scattering theory and LSZ formulas etc. work is called "Haag-Ruelle scattering theory". You can find it described in some (not many) details, and references, in the third volume of Reed-Simon's series of books "Methods of modern mathematical physics". $\endgroup$ – yuggib Mar 23 '18 at 17:47

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