# Non-Fock representation of quantum field theory

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?

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).