Quantum field theory on spacetimes with a compactly generated Cauchy horizon.

*(English)*Zbl 0883.53057This work is motivated by S. Hawking’s “chronology protection conjecture”, according to which the laws of physics prevent one from manufacturing a time machine [Phys. Rev. D, III. Ser. 46, 603-611 (1992; MR 93c:83086)]. A quite general class of spacetimes \((M,g)\) containing a time machine (i.e., closed timelike curves) are those with a compactly generated Cauchy horizon. This means that \((M,g)\) is time orientable and contains a partial Cauchy surface \(S\) with \(D^+(S) \not= I^+(S)\). In addition, the existence of a compact set \(K\) is required such that all past directed null generators of the future Cauchy horizon \(H^+(S)\) eventually enter and remain within \(K\). Each such \((M,g)\) with a noncompact \(S\) must violate the weak energy condition [S. Hawking, loc. cit.]. Hence a time machine cannot be constructed with classical matter; quantum effects must be used instead. Therefore the authors pose the question whether a free scalar quantum field theory can be consistently formulated on a spacetime \((M,g)\) with compactly generated Cauchy horizon. However, since the usual framework of quantum field theory, which is well established on globally hyperbolic spacetimes, is useless in the present situation (especially locality has no meaning on \((M,g)\)), it is not clear from the outset how “consistent” has to be interpreted. Therefore the authors use a scheme introduced by the first author in [B. S. Kay, Rev. Math. Phys., Spec. Issue, 167-195 (1992; Zbl 0779.53052)] which assumes the existence of a field algebra satisfying the “F-locality condition”.

In this setup the authors prove the following two theorems: 1. The field algebra, defined in the usual way on the initial globally hyperbolic region \(D(S)\) of \((M,g)\) (which satisfies F-locality) cannot be extended to the whole spacetime. 2. A two-point function which is Hadamard on \(D(S)\) cannot be extended to a distributional bisolution on the whole spacetime which is locally weak Hadamard at each point \(x \in M\).

The proofs of these theorems are based on the propagation of singularity theorems of Duistermaat and Hörmander [L. Hörmander, ‘The analysis of linear partial differential operators IV’ (Grundlehren 275, Springer, Berlin) (1985; Zbl 0612.35001)]. This is an interesting and well written paper and a very useful contribution to the discussion about the possibility of time machines.

In this setup the authors prove the following two theorems: 1. The field algebra, defined in the usual way on the initial globally hyperbolic region \(D(S)\) of \((M,g)\) (which satisfies F-locality) cannot be extended to the whole spacetime. 2. A two-point function which is Hadamard on \(D(S)\) cannot be extended to a distributional bisolution on the whole spacetime which is locally weak Hadamard at each point \(x \in M\).

The proofs of these theorems are based on the propagation of singularity theorems of Duistermaat and Hörmander [L. Hörmander, ‘The analysis of linear partial differential operators IV’ (Grundlehren 275, Springer, Berlin) (1985; Zbl 0612.35001)]. This is an interesting and well written paper and a very useful contribution to the discussion about the possibility of time machines.

Reviewer: M.Keyl (Berlin)

##### MSC:

53Z05 | Applications of differential geometry to physics |

58J45 | Hyperbolic equations on manifolds |

81T20 | Quantum field theory on curved space or space-time backgrounds |

83C47 | Methods of quantum field theory in general relativity and gravitational theory |

81T05 | Axiomatic quantum field theory; operator algebras |