We construct Hadamard states for Klein–Gordon fields in a spacetime
equal to the interior of
the future lightcone
from
a base point
in a globally
hyperbolic spacetime
.
Under some regularity conditions at the future infinity of
,
we identify a boundary symplectic space of functions on
,
which allows us to construct states for Klein–Gordon quantum fields in
from
states on the CCR algebra associated to the boundary symplectic space.
We formulate the natural microlocal condition on the boundary state on
, ensuring that the
bulk state it induces in
satisfies the Hadamard condition.
Using pseudodifferential calculus on the cone
,
we construct a large class of Hadamard 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
.