Continuity of Symplectically Adjoint Maps and the Algebraic Structure of Hadamard Vacuum Representations for Quantum Fields on Curved Spacetime

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 sympletically adjoint) that, if they are bounded for some scalar product on the symplectic space dominating the symplectic form, then they are bounded with respect to a one-parametric family of scalar products canonically 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) observable-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 1properties). A brief review of this circle of notions, as well as of properties of Hadamard states, forms part of the article. * Supported by a Von Neumann Fellowship of the Operator Algebras Network, EC Human Capital and Mobility Programme. 1 in a globally hyperbolic spacetime with respect to a certain topology on the Cauchy-data space. (Here, ∇ is the covariant derivative of the metric g on the spacetime, and r an arbitrary realvalued, smooth function.) The Cauchy-data space is a symplectic space on which the said temporal evolution is realized by symplectomorphisms. It turns out that the classical "energy-norm" of solutions of (1.1), which is given by a scalar product µ 0 on the Cauchy-data space, and the topology relevant for the required continuity statement (the "Hadamard oneparticle space norm"), induced by a scalar product µ 1 on the Cauchy-data space, are precisely in the relation for which our result on relative µ 0 − µ 1 continuity of symplectically adjoint maps applies. Since the continuity of the Cauchy-data evolution in the classical energy norm, i.e. µ 0 , is well-known, the desired continuity in the µ 1 -topology follows.The argument just described may be viewed as the prime example of application of the relative continuity result. In fact, the relation between µ 0 and µ 1

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