scholarly journals Complex Structures for Klein-Gordon theory on globally hyperbolic spacetimes

2021 ◽  
Author(s):  
Albert Much ◽  
Robert Oeckl
Keyword(s):  
Differential Equation ◽  
Complex Structure ◽  
Complex Structures ◽  
Main Ingredient ◽  
Globally Hyperbolic ◽  
Klein Gordon ◽  
Globally Hyperbolic Spacetimes ◽  
Hyperbolic Spacetimes ◽  
Friedmann Robertson Walker ◽  
Rigorous Method

Abstract We develop a rigorous method to parametrize complex structures for Klein-Gordon theory in globally hyperbolic spacetimes that satisfy a completeness condition. The complex structures are conserved under time-evolution and implement unitary quantizations. They can be interpreted as corresponding to global choices of vacuum. The main ingredient in our construction is a system of operator differential equations. We provide a number of theorems ensuring that all ingredients and steps in the construction are well-defined. We apply the method to exhibit natural quantizations for certain classes of globally hyperbolic spacetimes. In particular, we consider static, expanding and Friedmann-Robertson-Walker spacetimes. Moreover, for a huge class of spacetimes we prove that the differential equation for the complex structure is given by the Gelfand-Dikki equation.

scholarly journals Quantum Linear Scalar Fields with Time Dependent Potentials: Overview and Applications to Cosmology

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 115 ◽  
Author(s):  
Jerónimo Cortez ◽  
Guillermo A. Mena Marugán ◽  
José Velhinho
Keyword(s):  
Scalar Fields ◽  
Complex Structures ◽  
De Sitter ◽  
Quantum Theories ◽  
Bianchi I ◽  
Uniqueness Results ◽  
Klein Gordon ◽  
Globally Hyperbolic Spacetimes ◽  
Hyperbolic Spacetimes ◽  
Linear Scalar

In this work, we present an overview of uniqueness results derived in recent years for the quantization of Gowdy cosmological models and for (test) Klein-Gordon fields minimally coupled to Friedmann-Lemaître-Robertson-Walker, de Sitter, and Bianchi I spacetimes. These results are attained by imposing the criteria of symmetry invariance and of unitary implementability of the dynamics. This powerful combination of criteria allows not only to address the ambiguity in the representation of the canonical commutation relations, but also to single out a preferred set of fundamental variables. For the sake of clarity and completeness in the presentation (essentially as a background and complementary material), we first review the classical and quantum theories of a scalar field in globally hyperbolic spacetimes. Special emphasis is made on complex structures and the unitary implementability of symplectic transformations.

scholarly journals Self-Adjointness in Klein-Gordon Theory on Globally Hyperbolic Spacetimes

2021 ◽  
Vol 24 (1) ◽  
Author(s):  
Albert Much ◽  
Robert Oeckl
Keyword(s):  
Large Class ◽  
Hilbert Spaces ◽  
Hyperbolic Manifolds ◽  
External Potential ◽  
Globally Hyperbolic ◽  
Spatial Part ◽  
Klein Gordon ◽  
Globally Hyperbolic Spacetimes ◽  
Hyperbolic Spacetimes

AbstractWe prove essential self-adjointness of the spatial part of the linear Klein-Gordon operator with external potential for a large class of globally hyperbolic manifolds. The proof is conducted by a fusion of new results concerning globally hyperbolic manifolds, the theory of weighted Hilbert spaces and related functional analytic advances.

scholarly journals Global wave parametrices on globally hyperbolic spacetimes

2020 ◽  
Vol 490 (2) ◽  
pp. 124316
Author(s):  
Matteo Capoferri ◽  
Claudio Dappiaggi ◽  
Nicolò Drago
Keyword(s):  
Globally Hyperbolic ◽  
Globally Hyperbolic Spacetimes ◽  
Hyperbolic Spacetimes

scholarly journals Abelian Duality on Globally Hyperbolic Spacetimes

2016 ◽  
Vol 349 (1) ◽  
pp. 361-392 ◽  
Author(s):  
Christian Becker ◽  
Marco Benini ◽  
Alexander Schenkel ◽  
Richard J. Szabo
Keyword(s):  
Globally Hyperbolic ◽  
Globally Hyperbolic Spacetimes ◽  
Hyperbolic Spacetimes

THE PRINCIPLE OF LOCALITY AND QUANTUM FIELD THEORY ON (NON GLOBALLY HYPERBOLIC) CURVED SPACETIMES

1992 ◽  
Vol 04 (spec01) ◽  
pp. 167-195 ◽  
Author(s):  
BERNARD S. KAY
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Quantum Field ◽  
Field Algebra ◽  
Massless Field ◽  
Globally Hyperbolic ◽  
Local Algebras ◽  
Hyperbolic Spacetime ◽  
Globally Hyperbolic Spacetimes ◽  
Hyperbolic Spacetimes

In the context of a linear model (the covariant Klein Gordon equation) we review the mathematical and conceptual framework of quantum field theory on globally hyperbolic spacetimes, and address the question of what it might mean to quantize a field on a non globally hyperbolic spacetime. Our discussion centres on the notion of F-locality which we introduce and which asserts there is a net of local algebras such that every neighbourhood of every point contains a globally hyperbolic subneighbourhood of that point for which the field algebra coincides with the algebra one would obtain were one to regard the subneighbourhood as a spacetime in its own right and quantize — with some choice of time-orientation — according to the standard rules for quantum field theory on globally hyperbolic spacetimes. We show that F-locality is a property of the standard field algebra construction for globally hyperbolic spacetimes, and argue that it (or something similar) should be imposed as a condition on any field algebra construction for non globally hyperbolic spacetimes. We call a spacetime for which there exists a field algebra satisfying F-locality F-quantum compatible and argue that a spacetime which did not satisfy something similar to this condition could not arise as an approximate classical description of a state of quantum gravity and would hence be ruled out physically. We show that all F-quantum compatible spacetimes are time orientable. We also raise the issue of whether chronology violating spacetimes can be F-quantum compatible, giving a special model — a massless field theory on the “four dimensional spacelike cylinder” — which is F-quantum compatible, and a (two dimensional) model — a massless field theory on Misner space — which is not. We discuss the possible relevance of this latter result to Hawking’s recent Chronology Protection Conjecture.

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