The Topology of Branching Universes

The purpose of this paper is to survey the possible topologies of branching space-times, and, in particular, to refute the popular notion in the literature that a branching space-time requires a non-Hausdorff topology. 1 Branching MWI space-times Two basic types of branching universe have been suggested in modern mathematical physics: universes which branch in a style befitting the ‘many-worlds’ interpretation (MWI) of quantum theory; and universes which bifurcate to implement a topology-change cobordism. This section, concerned with the former type of branching universe, is not intended as a defence or analysis of the many-worlds interpretation. It is not even the intention to argue that the many-worlds interpretation genuinely entails such a branching model of space-time. Rather, the intention is to analyse the topological implications of such a branching model of space-time, and to question some of the assumptions normally found in the literature. The many-worlds interpretation holds that if a system is prepared into a state which is a ‘superposition’ Ψ = ∑ i aivi with respect to some quantity A whose eigenvectors provide a basis {vi} of the quantum state space, then when that system undergoes a measurement-like interaction, the universe splits into multiple branches, one for each basis vector vi. Each different branch records a definite value for the quantity A, corresponding to the eigenvector vi. This branching can be thought of as either a local or global process. If the branching is considered to be a global process, then the branches correspond to multiple regions of four-dimensional space-time which emanate from a common three-dimensional hypersurface, (see Figure 1). Each such branching hypersurface has a non-unique future. The notion that the entire universe branches in this style every time there is a measurement-like interaction, renders such branching a highly non-local process, and tacitly supposes that there is a unique global time coordinate for the universe. Treating a measurement-like interaction as a point event in spacetime, there will be many spacelike hypersurfaces which pass through that point; the selection of only one of these as the branching hypersurface requires one to