Verch R.
Continuity of symplectically adjoint maps and the algebraic
structure of Hadamard vacuum representations for quantum fields
on curved spacetime
(134K, LaTex)

ABSTRACT.  We derive for a pair of operators on a symplectic space which are adjoints
of each other with respect to the symplectic form (that is, they are
symplectically adjoint) that, if they are bounded for some scalar product
dominating the symplectic form, then they are bounded with respect to a 
one-parametric family of scalar products canoncially associated with the
initially given one, among them being its ``purification''. As a typical
example we consider a scalar field on a globally hyperbolic spacetime 
governed by the Klein-Gordon equation; the classical system is described by
a symplectic space and the temporal evolution by symplectomorphisms (which
are symplectically adjoint to their inverses). A natural scalar product is
that inducing the classical energy-norm, and an application of the above
result yields that its ``purification'' induces on the one-particle space
of the quantized system a topology which coincides with that given by the
two-point functions of quasifree Hadamard states. These findings will be 
shown to lead to new results concerning the structure of the local (von
Neumann) algebras in representations of quasifree Hadamard states of the 
Klein-Gordon field in an arbitrary globally hyperbolic spacetime, such as
local definiteness, local primarity and Haag-duality (and also split- and
type III_1-properties). A brief review of this circle of notions, as well as
of properties of Hadamard states, forms part of the article.