scholarly journals Intertwining operators for symmetric hyperbolic systems on globally hyperbolic manifolds

2020 ◽  
Author(s):  
Simone Murro ◽  
Daniele Volpe
Keyword(s):  
Hyperbolic Systems ◽  
Hyperbolic Manifolds ◽  
Intertwining Operators ◽  
Geometric Proof ◽  
Hermitian Forms ◽  
Homogeneous Solutions ◽  
Globally Hyperbolic ◽  
Algebraic Quantum ◽  
Suitable Parameter ◽  
Hadamard States

Abstract In this paper, a geometric process to compare solutions of symmetric hyperbolic systems on (possibly different) globally hyperbolic manifolds is realized via a family of intertwining operators. By fixing a suitable parameter, it is shown that the resulting intertwining operator preserves Hermitian forms naturally defined on the space of homogeneous solutions. As an application, we investigate the action of the intertwining operators in the context of algebraic quantum field theory. In particular, we provide a new geometric proof for the existence of the so-called Hadamard states on globally hyperbolic manifolds.

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

1997 ◽  
Vol 09 (05) ◽  
pp. 635-674 ◽  
Author(s):  
Rainer Verch
Keyword(s):  
Symplectic Form ◽  
Scalar Product ◽  
Classical System ◽  
Symplectic Space ◽  
Haag Duality ◽  
Globally Hyperbolic ◽  
Classical Energy ◽  
Hyperbolic Spacetime ◽  
Klein Gordon ◽  
Hadamard States

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 III1-properties). A brief review of this circle of notions, as well as of properties of Hadamard states, forms part of the article.

scholarly journals MICROLOCAL SPECTRUM CONDITION AND HADAMARD FORM FOR VECTOR-VALUED QUANTUM FIELDS IN CURVED SPACETIME

2001 ◽  
Vol 13 (10) ◽  
pp. 1203-1246 ◽  
Author(s):  
HANNO SAHLMANN ◽  
RAINER VERCH
Keyword(s):  
Vector Bundle ◽  
Scalar Fields ◽  
Quantum Fields ◽  
Curved Spacetime ◽  
Globally Hyperbolic ◽  
Commutation Relations ◽  
Hyperbolic Spacetime ◽  
Hadamard Condition ◽  
Spectrum Condition ◽  
Hadamard States

Some years ago, Radzikowski has found a characterization of Hadamard states for scalar quantum fields on a four-dimensional globally hyperbolic spacetime in terms of a specific form of the wavefront set of their two-point functions (termed "wavefront set spectrum condition"), thereby initiating a major progress in the understanding of Hadamard states and the further development of quantum field theory in curved spacetime. In the present work, we extend this important result on the equivalence of the wavefront set spectrum condition with the Hadamard condition from scalar fields to vector fields (sections in a vector bundle) which are subject to a wave-equation and are quantized so as to fulfill the covariant canonical commutation relations, or which obey a Dirac equation and are quantized according to the covariant anti-commutation relations, in any globally hyperbolic spacetime having dimension three or higher. In proving this result, a gap which is present in the published proof for the scalar field case will be removed. Moreover we determine the short-distance saling limits of Hadamard states for vector-bundle valued fields, finding them to coincide with the corresponding flat-space, massless vacuum states.

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