# Ko Sanders

# On state spaces for perturbative QFT in curved spacetimes

** Abstract: **

From QFT in curved spacetime it has become clear that the
construction of the extended algebra needed for perturbation
theory (including time-ordered and Wick products), is intimately
related to the set of physical states. For a free scalar field
the physical states are those whose two-point distribution has a
certain given singularity structure, namely the Hadamard states.
In this talk we explain how the Hadamard condition, together with
commutation relations, allows us to estimate the singularity
structure of all higher n-point distributions. (Our result here
actually holds more generally.) From that we conclude that the
full state space of the extended algebra is exactly the space of
Hadamard states. Finally we describe an algebraic abstraction of
this last fact: given any topological *-algebra $A$ with a
suitable state space $S$ we can construct an extended topological
*-algebra $A^{ext}$ for which $S$ is the full set of states. In
fact, $S$ can be recovered entirely from a suitable choice of
topology on $A$.