Communications in Mathematical Physics
, Volume 183, Issue 3, pp 533556
First online:
Quantum Field Theory on Spacetimes with a Compactly Generated Cauchy Horizon
 Bernard S. KayAffiliated withDepartment of Mathematics, University of York, Heslington, York YO1 5DD, UK
 , Marek J. RadzikowskiAffiliated withDepartment of Mathematics, University of York, Heslington, York YO1 5DD, UK
 , Robert M. WaldAffiliated withEnrico Fermi Institute, University of Chicago, Chicago, Illinois 60637, USA
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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 Flocality 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 twopoint distribution for any Hadamard state defined on the initial globally hyperbolic region must (when extended to a distributional bisolution of the covariant KleinGordon 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 φ^{2} or of the stressenergy tensor are necessarily illdefined or singular at any base point.
The proof of these theorems relies on the ‘Propagation of Singularities’ theorems of Duistermaat and Hörmander.
 Title
 Quantum Field Theory on Spacetimes with a Compactly Generated Cauchy Horizon
 Journal

Communications in Mathematical Physics
Volume 183, Issue 3 , pp 533556
 Cover Date
 199702
 DOI
 10.1007/s002200050042
 Print ISSN
 00103616
 Online ISSN
 14320916
 Publisher
 SpringerVerlag
 Additional Links
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 Authors

 Bernard S. Kay ^{(A1)}
 Marek J. Radzikowski ^{(A1)}
 Robert M. Wald ^{(A2)}
 Author Affiliations

 A1. Department of Mathematics, University of York, Heslington, York YO1 5DD, UK, UK
 A2. Enrico Fermi Institute, University of Chicago, Chicago, Illinois 60637, USA, US