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  • Springer  (6)
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  • 1
    Electronic Resource
    Electronic Resource
    Springer
    General relativity and gravitation 14 (1982), S. 1197-1199 
    ISSN: 1572-9532
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Type of Medium: Electronic Resource
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  • 2
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract We study the options for boundary conditions at the conical singularity for quantum mechanics on a two-dimensional cone with deficit angle ≦ 2π and for classical and quantum scalar fields propagating with a translationally invariant dynamics in the 1+3 dimensional spacetime around an idealized straight infinitely long, infinitesimally thin cosmic string. The key to our analysis is the observation that minus-the-Laplacian on a cone possesses a one-parameter family of selfadjoint extensions. These may be labeled by a parameterR with the dimensions of length—taking values in [0, ∞). ForR=0, the extension is positive. WhenR≠0 there is a bound state. Each of our problems has a range of possible dynamical evolutions corresponding to a range of allowedR-values. They correspond to either finite, forR=0, or logarithmically divergent, forR≠0, boundary conditions at zero radius. Non-zeroR-values are a satisfactory replacement for the (mathematically ill-defined) notion of δ-function potentials at the cone's apex. We discuss the relevance of the various idealized dynamics to quantum mechanics on a cone with a rounded-off centre and field theory around a “true” string of finite thickness. Provided one is interested in effects at sufficiently large length scales, the “true” dynamics will depend on the details of the interaction of the wave function with the cone's centre (/field with the string etc.) only through a single parameterR (its “scattering length”) and will be well-approximated by the dynamics for the corresponding idealized problem with the sameR-value. This turns out to be zero if the interaction with the centre is purely gravitational and minimally coupled, but non-zero values can be important to model nongravitational (or non-minimally coupled) interactions. Especially, we point out the relevance of non-zeroR-values to electromagnetic waves around superconducting strings. We also briefly speculate on the relevance of theR-parameter in the application of quantum mechanics on cones to 1+2 dimensional quantum gravity with massive scalars.
    Type of Medium: Electronic Resource
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  • 3
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract: We prove two theorems which concern difficulties in the formulation of the quantum theory of a linear scalar field on a spacetime, , with a compactly generated Cauchy horizon. These theorems demonstrate the breakdown of the theory at certain base points of the Cauchy horizon, which are defined as ‘past terminal accumulation points’ of the horizon generators. Thus, the theorems may be interpreted as giving support to Hawking's ‘Chronology Protection Conjecture’, according to which the laws of physics prevent one from manufacturing a ’time machine‘. Specifically, we prove: Theorem 1. There is no extension to of the usual field algebra on the initial globally hyperbolic region which satisfies the condition of F-locality at any base point. In other words, any extension of the field algebra must, in any globally hyperbolic neighbourhood of any base point, differ from the algebra one would define on that neighbourhood according to the rules for globally hyperbolic spacetimes. Theorem 2. The two-point distribution for any Hadamard state defined on the initial globally hyperbolic region must (when extended to a distributional bisolution of the covariant Klein-Gordon equation on the full spacetime) be singular at every base point x in the sense that the difference between this two point distribution and a local Hadamard distribution cannot be given by a bounded function in any neighbourhood (in M × M) of (x,x). In consequence of Theorem 2, quantities such as the renormalized expectation value of φ2 or of the stress-energy tensor are necessarily ill-defined or singular at any base point. The proof of these theorems relies on the ‘Propagation of Singularities’ theorems of Duistermaat and Hörmander.
    Type of Medium: Electronic Resource
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  • 4
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract We consider the Klein-Gordon equation (m≧0) on the double Schwarzschild wedge of the Kruskal spacetime, and construct the Hartle-Hawking stateω H as a thermal state relative to the Boulware quantization. We prove that, on the double wedge,ω H is a pure state, and in the corresponding representation, the left- and right-wedgeC* algebras each have the Reeh-Schlieder property, while the corresponding von-Neumann algebras are typeIII 1 factors which are dual to (i.e. commutants of) each other. We discuss the extent to which these properties may generalize to non-quasi-free field theories. Pursuing the Rindler-Fulling-Unruh analogy with the Klein-Gordon equation (m〉0) in (d-dimensional) flat spacetime, we establish an explicit formula for the Minkowski vacuum on a spacelike double wedge as a thermal state relative to the Fulling quantization. We also treat the cased=2,m=0 of this formula since this is essential input for a paper with Dimock on scattering theory for the quantum Klein-Gordon equation on the Schwarzschild metric.
    Type of Medium: Electronic Resource
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  • 5
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract The quantum theory of both linear, and interacting fields on curved space-times is discussed. It is argued that generic curved space-time situations force the adoption of the algebraic approach to quantum field theory: and a suitable formalism is presented for handling an arbitrary quasi-free state in an arbitrary globally hyperbolic space-time. For the interacting case, these quasi-free states are taken as suitable starting points, in terms of which expectation values of field operator products may be calculated to arbitrary order in perturbation theory. The formal treatment of interacting fields in perturbation theory is reduced to a treatment of “free” quantum fields interacting with external sources. Central to the approach is the so-called two-current operator, which characterises the effect of external sources in terms of purely algebraic (i.e. representation free) properties of the source-free theory. The paper ends with a set of “Feynman rules” which seems particularly appropriate to curved space-times in that it takes care of those aspects of the problem which are specific to curved space-times (and independent of interaction). Heuristically, the scheme calculates “in-in” rather than “in-out” matrix elements. Renormalization problems are discussed but not treated.
    Type of Medium: Electronic Resource
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  • 6
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract We give mathematically rigorous results on the quantization of the covariant Klein Gordon field with an external stationary scalar interaction in a stationary curved space-time. We show how, following Segal, Weinless etc., the problem reduces to finding a “one particle structure” for the corresponding classical system. Our main result is an existence theorem for such a one-particle structure for a precisely specified class of stationary space-times. Byproducts of our approach are: 1) A discussion of when a given “equal-time” surface in a given stationary space-time is Cauchy. 2) A modification and extension of the methods of Chernoff [3] for proving the essential self-adjointness of certain partial differential operators.
    Type of Medium: Electronic Resource
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