Construction of Hadamard states by characteristic Cauchy problem

We construct Hadamard states for Klein-Gordon fields in a spacetime $M_{0}$ equal to the interior of the future lightcone $C$ from a base point $p$ in a globally hyperbolic spacetime $(M, g)$. Under some regularity conditions at future infinity of $C$, we identify a boundary symplectic space of functions on $C$, which allows to construct states for Klein-Gordon quantum fields in $M_{0}$ from states on the CCR algebra associated to the boundary symplectic space. We formulate the natural microlocal condition on the boundary state on $C$ ensuring that the bulk state it induces in $M_{0}$ satisfies the Hadamard condition. Using pseudodifferential calculus on the cone $C$ we construct a large class of Hadamard boundary states on the boundary with pseudodifferential covariances, and characterize the pure states among them. We then show that these pure boundary states induce pure Hadamard states in $M_{0}$.