Intertwining operators for symmetric hyperbolic systems on globally hyperbolic manifolds

Murro, Simone; Volpe, Daniele

doi:10.1007/s10455-020-09739-0

Cited by 9 publications

(9 citation statements)

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“…Amongst numerous other nice properties, they ensure that quantum fluctuations of observables are bounded and allow for an extension of the algebra of fields to encompass Wick polynomials [32][33][34][35][36][37][38][39]. Over the years, the notion of Hadamard states has proved successful in a wide range of different settings, see, e.g., [40][41][42][43][44][45][46][47][48][49][50][51][52][53], to name a few.…”

Section: Definition 4 (Quasifree State) a Statementioning

confidence: 99%

Partial Differential Equations and Quantum States in Curved Spacetimes

Avetisyan

Capoferri

2021

Mathematics

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In this review paper, we discuss the relation between recent advances in the theory of partial differential equations and their applications to quantum field theory on curved spacetimes. In particular, we focus on hyperbolic propagators and the role they play in the construction of physically admissible quantum states—the so-called Hadamard states—on globally hyperbolic spacetimes. We will review the notion of a propagator and discuss how it can be constructed in an explicit and invariant fashion, first on a Riemannian manifold and then on a Lorentzian spacetime. Finally, we will recall the notion of Hadamard state and relate the latter to hyperbolic propagators via the wavefront set, a subset of the cotangent bundle capturing the information about the singularities of a distribution.

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Section: Definition 4 (Quasifree State) a Statementioning

confidence: 99%

Partial Differential Equations and Quantum States in Curved Spacetimes

Avetisyan

Capoferri

2021

Mathematics

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“…By setting suitably such a function, we shall show that the resulting Møller operator is actually a unitary map between the spaces of initial data endowed with a naturally defined positive scalar product. Our goal is achieved with the help of [52,66].…”

Section: Møller Operators For Symmetric Weakly-hyperbolic Systemsmentioning

confidence: 99%

“…In [66] a geometric process was realized to compare solutions of symmetric hyperbolic systems on different globally hyperbolic manifolds M 0 := (M, g 0 ) and M := (M, g 1 ) with empty boundary, provided that M 0 and M 1 admit the same Cauchy temporal function and g 1 ≤ g 0 , namely the set of timelike vectors for g 1 is contained in the one for g 0 . This was achieved via the construction of a family of so-called Møller operators [22,35,56].…”

Section: Møller Operators On Manifolds With Timelike Boundarymentioning

confidence: 99%

“…In this section we shall compare the quantization of the Dirac field on two different (yet related) globally hyperbolic spacetimes with timelike boundary. To this end, we shall benefit from [22,38,66], where a class of Møller operator was introduced in order to construct unitary equivalent quantum field theories, together of the results of the previous Sections 2.5-2.6.…”

Section: Conservation Of Positive Definite Hermitian Scalar Productsmentioning

confidence: 99%

“…In [28,38,66], the quantization of a free field theory is realized as a two-step procedure. On the one hand, the physical system classically described by Sol sc,mit (D) is quantized by introducing a unital * -algebra A, whose elements are interpreted as observables for the system under investigation.…”

Section: The Algebra Of Dirac Fields With Mit Boundary Conditionmentioning

confidence: 99%

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Møller operators and Hadamard states for Dirac fields with MIT boundary conditions

Drago¹,

Ginoux²,

Murro³

2021

Preprint

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The aim of this paper is to prove the existence of Hadamard states for Dirac fields coupled with MIT boundary conditions on any globally hyperbolic manifold with timelike boundary. This is achieved by introducing a geometric Møller operator which implements a unitary isomorphism between the spaces of L 2 -initial data of particular symmetric systems we call weakly-hyperbolic and which are coupled with admissible boundary conditions. In particular, we show that for Dirac fields with MIT boundary conditions, this isomorphism can be lifted to a * -isomorphism between the algebras of Dirac fields and that any Hadamard state can be pulled back along this * -isomorphism preserving the singular structure of its two-point distribution.

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The Quantization of Proca Fields on Globally Hyperbolic Spacetimes: Hadamard States and Møller Operators

Moretti

Murro

Volpe

2023

Ann. Henri Poincaré

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This paper deals with several issues concerning the algebraic quantization of the real Proca field in a globally hyperbolic spacetime and the definition and existence of Hadamard states for that field. In particular, extending previous work, we construct the so-called Møller $$*$$ ∗ -isomorphism between the algebras of Proca observables on paracausally related spacetimes, proving that the pullback of these isomorphisms preserves the Hadamard property of corresponding quasifree states defined on the two spacetimes. Then, we pull back a natural Hadamard state constructed on ultrastatic spacetimes of bounded geometry, along this $$*$$ ∗ -isomorphism, to obtain an Hadamard state on a general globally hyperbolic spacetime. We conclude the paper, by comparing the definition of an Hadamard state, here given in terms of wavefront set, with the one proposed by Fewster and Pfenning, which makes use of a supplementary Klein–Gordon Hadamard form. We establish an (almost) complete equivalence of the two definitions.

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Intertwining operators for symmetric hyperbolic systems on globally hyperbolic manifolds

Cited by 9 publications

References 47 publications

Partial Differential Equations and Quantum States in Curved Spacetimes

Partial Differential Equations and Quantum States in Curved Spacetimes

Møller operators and Hadamard states for Dirac fields with MIT boundary conditions

The Quantization of Proca Fields on Globally Hyperbolic Spacetimes: Hadamard States and Møller Operators

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