 96448 Verch R.
 Continuity of symplectically adjoint maps and the algebraic
structure of Hadamard vacuum representations for quantum fields
on curved spacetime
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Sep 27, 96

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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
oneparametric 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 KleinGordon 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 energynorm, and an application of the above
result yields that its ``purification'' induces on the oneparticle space
of the quantized system a topology which coincides with that given by the
twopoint 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
KleinGordon field in an arbitrary globally hyperbolic spacetime, such as
local definiteness, local primarity and Haagduality (and also split and
type III_1properties). A brief review of this circle of notions, as well as
of properties of Hadamard states, forms part of the article.
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