scholarly journals A new proof of Geroch's theorem on temporal splitting of globally hyperbolic spacetime

2021 ◽  
Author(s):  
Ali Bleybel
Keyword(s):  
Causal Sets ◽  
Globally Hyperbolic ◽  
Hyperbolic Spacetime

Abstract In this paper we use our results concerning temporal foliations of causal sets in order to provide a new proof of Geroch's Theorem on temporal foliations in a globally hyperbolic spacetime.

Metric completeness of C(p, q) in a globally hyperbolic spacetime

2014 ◽  
Vol 20 (2) ◽  
pp. 144-147
Author(s):  
B. S. Choudhury ◽  
H. S. Mondal
Keyword(s):  
Globally Hyperbolic ◽  
Hyperbolic Spacetime

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.

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 ASPECTS OF NONCOMMUTATIVE LORENTZIAN GEOMETRY FOR GLOBALLY HYPERBOLIC SPACETIMES

2003 ◽  
Vol 15 (10) ◽  
pp. 1171-1217 ◽  
Author(s):  
VALTER MORETTI
Keyword(s):  
Causal Structure ◽  
Order Relation ◽  
Partial Ordering ◽  
Lorentzian Geometry ◽  
Causal Order ◽  
C Algebra ◽  
Globally Hyperbolic ◽  
C Algebras ◽  
Hyperbolic Spacetime ◽  
Algebra Approach

Connes' functional formula of the Riemannian distance is generalized to the Lorentzian case using the so-called Lorentzian distance, the d'Alembert operator and the causal functions of a globally-hyperbolic spacetime. As a step of the presented machinery, a proof of the almost-everywhere smoothness of the Lorentzian distance considered as a function of one of the two arguments is given. Afterwards, using a C*-algebra approach, the spacetime causal structure and the Lorentzian distance are generalized into noncommutative structures giving rise to a Lorentzian version of part of Connes' noncommutative geometry. The generalized noncommutative spacetime consists of a direct set of Hilbert spaces and a related class of C*-algebras of operators. In each algebra a convex cone made of self-adjoint elements is selected which generalizes the class of causal functions. The generalized events, called loci, are realized as the elements of the inductive limit of the spaces of the algebraic states on the C*-algebras. A partial-ordering relation between pairs of loci generalizes the causal order relation in spacetime. A generalized Lorentz distance of loci is defined by means of a class of densely-defined operators which play the role of a Lorentzian metric. Specializing back the formalism to the usual globally-hyperbolic spacetime, it is found that compactly-supported probability measures give rise to a non-pointwise extension of the concept of events.

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.

scholarly journals The art of the state

2018 ◽  
Vol 27 (11) ◽  
pp. 1843007 ◽  
Author(s):  
Christopher J. Fewster
Keyword(s):  
Infinite Family ◽  
Vacuum State ◽  
Natural Conditions ◽  
Spacetime Geometry ◽  
High Energies ◽  
Hyperbolic Spacetime ◽  
Hadamard Condition ◽  
Free Scalar ◽  
Model Independent ◽  
Hadamard States

Quantum field theory (QFT) on curved spacetimes lacks an obvious distinguished vacuum state. We review a recent no-go theorem that establishes the impossibility of finding a preferred state in each globally hyperbolic spacetime, subject to certain natural conditions. The result applies in particular to the free scalar field, but the proof is model-independent and therefore of wider applicability. In addition, we critically examine the recently proposed “SJ states”, that are determined by the spacetime geometry alone, but which fail to be Hadamard in general. We describe a modified construction that can yield an infinite family of Hadamard states, and also explain recent results that motivate the Hadamard condition without direct reference to ultra-high energies or ultra-short distance structure.

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