(some of this is adapted from a comment I posted at Theories ‘n Things)
How should we think about spacetime in Everettian quantum mechanics? If we adopt the Saunders-Wallace multiple-utterance proposal as a way of accounting for uncertainty (and hence probability) in EQM, we still have a choice. Either we treat spacetime the same way as we treat material objects (in which case spacetime doesn’t literally fission, and there is exactly one object in each spatio-temporal location even counting by identity) or we treat it as a common background to all the branches (in which case it literally fissions, and material objects are literally collocated on a grand scale). This choice is exactly the choice of whether spacetimes diverge or branch.
If we go the latter way, we need to count by world-indexed identity if we want to say that there is exactly one object in a particular spatio-temporal region. But we don’t need to count by world-indexed identity to say (for example) that there is only one cat on the mat, or only one table in this room: the cat, mat, table and room are all branch-bound objects. So then the semantics of ‘there is one table in this room’ and ‘there is one table in this spatio-temporal region’ would be structurally different. This seems good reason to consider the alternative route.
If we go the former way, what remains of the notion that reality is branching? The answer is simply that branches branch. But this can all get very confusing. I’ll try to clear this up a bit by reiterating how everything’s supposed to fit together. This is my current preferred take on things, which presupposes supersubstantivalism, for reasons I’ll discuss below.
1) At the fundamental level, there is just the universal state.
2) Decoherence picks out an approximate basis to decompose the universal state, approximately defining an emergent branching structure.
3) Pick out big-bang-to-heat-death histories from this emergent branching structure and call them branches.
4) Pick out parts of branches and call them branch segments. Branch segments are common to multiple branches.
5) Ordered pairs http://segment,branch are identified with spacetime regions. A special case of this is when the branch segment is the whole branch; the spacetime region http://x, just is the spacetime of branch x.
6) Via supersubstantivalism, subregions of a spacetime are identified with objects and agents.
So on this account spacetime and ordinary objects do not branch, but the underlying emergent branching structure does. This is my preferred set-up at the moment; but I imagine many people will baulk at the supersubstantivalism. Why is it needed? Consider another way of setting it up which appeals to a more orthodox view of spacetime:
5) Ordered pairs http://segment,branch are identified with spacetime regions. A special case of this is when the branch segment in question is the whole branch; the region http://x, is the spacetime of branch x.
6) For some spacetime regions there is a material object occupying that region.
The problem I have with this is that material objects only come in at the final stage. What we have in stages 2-5) is a new kind of entity being extracted from an already-accepted layer of ontology. But in stage 6 a new kind of entity is introduced in a different way; material objects are just stipulated to occupy particular regions of spacetime. But the hypothesis made in 1) is that the quantum state is the only fundamental existent, so these material objects must be emergent from it somehow. But the non-supersubstantivalist story doesn’t give us any picture of how this works.
We could switch things around as follows:
5) Ordered pairs http://segment,branch are identified with material objects. A special case of this is when the branch segment in question is the whole branch; the material object http://x, is the mereological sum of all objects realized by branch x.
6) For every material object there is a spacetime region which that material object occupies.
But this looks even less good. Not only do spacetime regions now seem to float free from the underlying ontology just as material objects did in the previous version, this version doesn’t even seem to get us unoccupied spacetime regions.
Perhaps all this isn’t so much of an argument as a restatement of the intuition motivating supersubstantivalism. Accepting two different kinds of entity (objects and regions) related by a primitive ‘occupation’ relation doesn’t add any extra explanatory power to supersubstantivalism, rather it makes things more mysterious. We are faced either with the problem of explaining where material objects come from, given a spacetime, or the problem of explaining where spacetime comes from, given material objects. Just calling the relation between the two ‘occupation’ won’t cut any ice in a naturalistic picture.