  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 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 (M,g_{ab}) 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 MXM) of (x,x). Theorem 2
implies quantities such as the renormalized expectation value of \phi^2 or of
the stress-energy tensor are necessarily ill-defined or singular at any base
point. The proofs rely on the `Propagation of Singularities' theorems of
Duistermaat and H\"ormander.
