Theorems on the uniqueness and thermal properties of stationary, nonsingular, quasifree states on spacetimes with a bifurcate killing horizon

Bernard S. Kay; Robert M Wald

doi:10.1016/0370-1573(91)90015-E

Theorems on the uniqueness and thermal properties of stationary, nonsingular, quasifree states on spacetimes with a bifurcate killing horizon

Bernard S. Kay, Robert M Wald

Mathematics

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Abstract

This paper is concerned with the study of quasifree states of a linear, scalar quantum field in globally hyperbolic spacetimes possessing a one-parameter group of isometries with a bifurcate Killing horizon. Some results on the uniqueness and thermal properties of such states are well known in the special cases of Minkowski, Schwarzschild, and deSitter spacetimes, and our main aim is to present new stronger results and to generalize them to this wide class of spacetimes. As a preliminary to proving our theorems, we develop some aspects of the theory of globally of hyperbolic spacetimes with a bifurcate Killing horizon, we give some new results on the structure of Bose quasifree states of linear fields (the class which includes all the usual “Fock vacua”), and we clarify and further develop the notion of a “Hadamard state”. We then consider the quasifre e states on these spacetimes which have vanishing one-point function, are invariant under the one-parameter isometry group, and are nonsingular in a neighborhood of the horizon in the sense that their two-point function is of the Hadamard form there. We prove that, on a large subalgebra of observables (which are determined by observables localized in compact regions of the horizon) such states are unique and pure. Furthermore, if the spacetime admits a certain discrete “wedge reflection” isometry (as holds automatically in the analytic case) we prove that this state - if it exists - must be a KMS state at the Hawking temperature T=¿/2p when restricted to those observables in our subalgebra which are localized in one of the (“right” or “left”) wedges of the spacetime where the Killing orbits are timelike near the horizon. Here, ¿ denotes the surface g ravity of the horizon. Under the further assumption that the nonsingularity of the state holds globally and that there are no “zero modes” in the one-partic le Hilbert space belonging to the state, we extend the uniqueness result to all observables localized in the full “domain of determinacy” of the hori zon. However, existence of states satisfying the hypotheses of our theorems does not hold in general and, indeed, we prove the nonexistence of any stationary (not necessarily quasifree) Hadamard state on the Schwarzschild-deSitter and Kerr spacetimes. We remark that nowhere in the analysis do we need to assume any form of Einstein's equations.

Original language	English
Pages (from-to)	49-136
Number of pages	88
Journal	Physics Reports
Volume	207
Issue number	2
DOIs	https://doi.org/10.1016/0370-1573(91)90015-E
Publication status	Published - Aug 1991

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10.1016/0370-1573(91)90015-E

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title = "Theorems on the uniqueness and thermal properties of stationary, nonsingular, quasifree states on spacetimes with a bifurcate killing horizon",

abstract = "This paper is concerned with the study of quasifree states of a linear, scalar quantum field in globally hyperbolic spacetimes possessing a one-parameter group of isometries with a bifurcate Killing horizon. Some results on the uniqueness and thermal properties of such states are well known in the special cases of Minkowski, Schwarzschild, and deSitter spacetimes, and our main aim is to present new stronger results and to generalize them to this wide class of spacetimes. As a preliminary to proving our theorems, we develop some aspects of the theory of globally of hyperbolic spacetimes with a bifurcate Killing horizon, we give some new results on the structure of Bose quasifree states of linear fields (the class which includes all the usual “Fock vacua”), and we clarify and further develop the notion of a “Hadamard state”. We then consider the quasifre e states on these spacetimes which have vanishing one-point function, are invariant under the one-parameter isometry group, and are nonsingular in a neighborhood of the horizon in the sense that their two-point function is of the Hadamard form there. We prove that, on a large subalgebra of observables (which are determined by observables localized in compact regions of the horizon) such states are unique and pure. Furthermore, if the spacetime admits a certain discrete “wedge reflection” isometry (as holds automatically in the analytic case) we prove that this state - if it exists - must be a KMS state at the Hawking temperature T=¿/2p when restricted to those observables in our subalgebra which are localized in one of the (“right” or “left”) wedges of the spacetime where the Killing orbits are timelike near the horizon. Here, ¿ denotes the surface g ravity of the horizon. Under the further assumption that the nonsingularity of the state holds globally and that there are no “zero modes” in the one-partic le Hilbert space belonging to the state, we extend the uniqueness result to all observables localized in the full “domain of determinacy” of the hori zon. However, existence of states satisfying the hypotheses of our theorems does not hold in general and, indeed, we prove the nonexistence of any stationary (not necessarily quasifree) Hadamard state on the Schwarzschild-deSitter and Kerr spacetimes. We remark that nowhere in the analysis do we need to assume any form of Einstein's equations.",

author = "Kay, {Bernard S.} and Wald, {Robert M}",

year = "1991",

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doi = "10.1016/0370-1573(91)90015-E",

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T1 - Theorems on the uniqueness and thermal properties of stationary, nonsingular, quasifree states on spacetimes with a bifurcate killing horizon

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AU - Wald, Robert M

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N2 - This paper is concerned with the study of quasifree states of a linear, scalar quantum field in globally hyperbolic spacetimes possessing a one-parameter group of isometries with a bifurcate Killing horizon. Some results on the uniqueness and thermal properties of such states are well known in the special cases of Minkowski, Schwarzschild, and deSitter spacetimes, and our main aim is to present new stronger results and to generalize them to this wide class of spacetimes. As a preliminary to proving our theorems, we develop some aspects of the theory of globally of hyperbolic spacetimes with a bifurcate Killing horizon, we give some new results on the structure of Bose quasifree states of linear fields (the class which includes all the usual “Fock vacua”), and we clarify and further develop the notion of a “Hadamard state”. We then consider the quasifre e states on these spacetimes which have vanishing one-point function, are invariant under the one-parameter isometry group, and are nonsingular in a neighborhood of the horizon in the sense that their two-point function is of the Hadamard form there. We prove that, on a large subalgebra of observables (which are determined by observables localized in compact regions of the horizon) such states are unique and pure. Furthermore, if the spacetime admits a certain discrete “wedge reflection” isometry (as holds automatically in the analytic case) we prove that this state - if it exists - must be a KMS state at the Hawking temperature T=¿/2p when restricted to those observables in our subalgebra which are localized in one of the (“right” or “left”) wedges of the spacetime where the Killing orbits are timelike near the horizon. Here, ¿ denotes the surface g ravity of the horizon. Under the further assumption that the nonsingularity of the state holds globally and that there are no “zero modes” in the one-partic le Hilbert space belonging to the state, we extend the uniqueness result to all observables localized in the full “domain of determinacy” of the hori zon. However, existence of states satisfying the hypotheses of our theorems does not hold in general and, indeed, we prove the nonexistence of any stationary (not necessarily quasifree) Hadamard state on the Schwarzschild-deSitter and Kerr spacetimes. We remark that nowhere in the analysis do we need to assume any form of Einstein's equations.

AB - This paper is concerned with the study of quasifree states of a linear, scalar quantum field in globally hyperbolic spacetimes possessing a one-parameter group of isometries with a bifurcate Killing horizon. Some results on the uniqueness and thermal properties of such states are well known in the special cases of Minkowski, Schwarzschild, and deSitter spacetimes, and our main aim is to present new stronger results and to generalize them to this wide class of spacetimes. As a preliminary to proving our theorems, we develop some aspects of the theory of globally of hyperbolic spacetimes with a bifurcate Killing horizon, we give some new results on the structure of Bose quasifree states of linear fields (the class which includes all the usual “Fock vacua”), and we clarify and further develop the notion of a “Hadamard state”. We then consider the quasifre e states on these spacetimes which have vanishing one-point function, are invariant under the one-parameter isometry group, and are nonsingular in a neighborhood of the horizon in the sense that their two-point function is of the Hadamard form there. We prove that, on a large subalgebra of observables (which are determined by observables localized in compact regions of the horizon) such states are unique and pure. Furthermore, if the spacetime admits a certain discrete “wedge reflection” isometry (as holds automatically in the analytic case) we prove that this state - if it exists - must be a KMS state at the Hawking temperature T=¿/2p when restricted to those observables in our subalgebra which are localized in one of the (“right” or “left”) wedges of the spacetime where the Killing orbits are timelike near the horizon. Here, ¿ denotes the surface g ravity of the horizon. Under the further assumption that the nonsingularity of the state holds globally and that there are no “zero modes” in the one-partic le Hilbert space belonging to the state, we extend the uniqueness result to all observables localized in the full “domain of determinacy” of the hori zon. However, existence of states satisfying the hypotheses of our theorems does not hold in general and, indeed, we prove the nonexistence of any stationary (not necessarily quasifree) Hadamard state on the Schwarzschild-deSitter and Kerr spacetimes. We remark that nowhere in the analysis do we need to assume any form of Einstein's equations.

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DO - 10.1016/0370-1573(91)90015-E

M3 - Article

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EP - 136

JO - Physics Reports

JF - Physics Reports

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