Intertwining operators for symmetric hyperbolic systems on globally hyperbolic manifolds
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Intertwining operators for symmetric hyperbolic systems on globally hyperbolic manifolds Simone Murro1 · Daniele Volpe1 Received: 7 July 2020 / Accepted: 9 September 2020 © The Author(s) 2020
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. Keywords Symmetric hyperbolic systems · Dirac operators · Wave equations · Cauchy problem · Green operators · Intertwining operators · Algebraic quantum field theory Mathematics Subject Classification Primary: 53C50 · 58J45 · Secondary: 53C27 · 81T05
1 Introduction Symmetric hyperbolic systems are an important class of first-order linear differential operators acting on sections of vector bundles on Lorentzian manifolds. The most prominent examples are the classical Dirac operator and the geometric wave operator, which can be understood by reducing a suitable second-order normally hyperbolic differential operator to a first-order differential operator. In the class of Lorentzian manifolds with empty boundary known as globally hyperbolic, the Cauchy problem of a symmetric hyperbolic system is well posed. As a consequence, the existence of advanced and retarded Green operators is guaranteed. These operators are of essential importance in the quantization of a classical field theory: Indeed, they implement the canonical commutation relation for a bosonic field theory or the canonical Simone Murro and Daniele Volpe acknowledge the support of the INFN-TIFPA project “Bell”. * Simone Murro [email protected]; [email protected] Daniele Volpe [email protected] 1
Dipartimento di Matematica, Università di Trento and INFN-TIFPA, Via Sommarive 14, 38123 Povo, Italy
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Vol.:(0123456789)
Annals of Global Analysis and Geometry
anti-commutation relation for fermionic field theory. Moreover, their difference, dubbed causal propagator (or Pauli-Jordan commutator), can be used to construct quantum states. For further details, we recommend the recent reviews [4, 10, 35]. In this paper, we investigate the existence of a geometrical map connecting the space of solutions of different symmetric hyperbolic systems over (possibily different) globally hyperbolic manifolds. A summary of the main result obtained is the following (cf. Theorem 3.4):
Theorem 1.1 Let 𝛼 ∈ {0, 1} and 𝖬𝛼 = (𝖬, g𝛼 ) be globally hyperbolic manifolds admitting the same Cauchy temporal function. Consider the symmetric hyperbolic systems 𝖲𝛼 over 𝖬𝛼 acting on sections of a real (or complex) vector bundle 𝖤𝛼 endowed with a nondegenerate s
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