| Current quantum theories require a Hilbert space of states defined on spacelike surfaces. These theories will not be sufficient to describe quantum systems on spacetimes which cannot be smoothly foliated by spacelike surfaces. A generalized framework is needed to study quantum mechanics on these spacetimes. Using Hartle's generalized quantum mechanics, we will investigate quantum systems and their information on examples of non-foliable spacetimes.;First, we examine a simple two-particle scattering experiment in which there is a bounded region of closed timelike curves (CTCs) in the experiment's future. The transitional probability is shown to be acausal: it depends on the existence and distribution of these future CTCs. Using a CTC spacetime developed by Politzer we show that for certain initial data, the total cross-section of a scattering experiment deviates from the standard value (the value predicted if no CTCs existed). However, this deviation may be negligible for many likely distributions of CTCs. It is also shown that if the CTCs pervade all of space, then the deviation is zero due to a cancellation effect.;Next, we examine quantum systems on a spacetime with the trousers topology. This flat 1+1 dimensional spacetime exhibits a change in spatial topology: S1 ∣→ S1 ⊕ S 1. Using path integrals, amplitudes of transitions across the topology change are calculated for non-relativistic quantum mechanics and scalar field theory. The evolution through the transition is shown to be unitary for the former theory, but non-unitary for latter. Using these transition amplitudes, the Heisenberg-picture formulation of Hartle's generalized quantum mechanics is extended to non-foliable spacetimes. This formulation is used to study spacetime information in the case of non-relativistic quantum mechanics on the trousers spacetime. It is shown that if distinct alternatives are restricted to one leg, then there is an infinite amount of missing information, while if our alternatives are projection operators on the product space of the two legs, we find that complete information is available after the topology change. These results are discussed in connection with the black hole information loss problem: the results support the claim that information is not actually lost if one considers information four-dimensionally. |
