Ko Sanders

On state spaces for perturbative QFT in curved spacetimes


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 $A$.