QUANTUM-FIELD THEORY ON SPACETIMES WITH A COMPACTLY GENERATED CAUCHY HORIZON

We prove two theorems which concern difficulties in the formulation of the quantum theory of a linear scalar field on a spacetime, (M,g(ab)) , with a compactly generated Cauchy horizon. These theorems demonstrat e the breakdown of the theory at certain base points of the Cauchy hor izon, which are defined as 'past terminal accumulation points' of the horizon generators, Thus, the theorems may be interpreted as giving su pport to Hawking's 'Chronology Protection Conjecture', according to wh ich the laws of physics prevent one from manufacturing a 'time machine '. Specifically, we prove: Theorem 1. There is no extension to (M, g(a b)) of the usual field algebra on the initial globally hyperbolic regi on which satisfies the condition off-locality at any base point, In ot her words, arty extension of the field algebra must, in any globally h yperbolic neighbourhood of any base point, differ from the algebra one would define on that neighbourhood according to the rules for globall y 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 distributio n and a local Hadamard distribution cannot be given by a bounded funct ion in any neighbourhood (in M x M) of(x, x). In consequence of Theore m 2, quantities such as the renormalized expectation value of Phi(2) O r. Of the stress-energy tensor are necessarily ill-defined or singular at any base point. The proof of these theorems relies on the 'Propaga tion of Singularities' theorems of Duistermaat and Hormander.