# How should quantum mechanics be formulated on disconnected spacetime?

I am interested in spacetimes with nontrivial topologies. One such example are those that that transition from spatially connected to disconnected, for example:

Usually, we would have some initial data set on a Cauchy surface ($\sigma'$ in this case), and we time evolve it in the usual manner to a data set on a future Cauchy surface $\sigma ''$. However, this process requires a smooth foliation between $\sigma '$ and $\sigma ''$. In the above spacetime, there is no such smooth family of interpolating Cauchy surfaces between $\sigma '$ and $A$, for example. Hartle proposes some generalizations to quantum theory that may accommodate such spatially disconnected spacetimes here (viz. section VI).

I am especially interested in the more general instance of a disconnected region forming in spacetime (not only spatially, i.e. what is sometimes referred to as a 'baby universe'). Is there a similar generalized theory for such spacetimes?

I would be interested in this result, though I am prepared to accept that such dynamics are prohibited for whatever reason. I am familiar with results that say there may be quantum correlations between classically forbidden zones, though I am not sure that the horizons considered there would apply to these disconnected topologies (and further, such correlations do not transmit information).

If there are no such theories that allow for this tunneling between disconnected regions, what are the theoretical issues inherent in one?

EDIT: I've been asked to clarify this post.

I am familiar with theorems that state topology change in classical general relativity are prohibited if you assume an everywhere non-singular Lorentzian manifold. However, I am also aware that if this condition is tossed, you can recover some interesting results (specifically, I am familiar with, though do not claim to understand in their entirety, these lecture notes from G. Gibbons).

I am primarily motivated from an interested in this paper from Stephen Hsu that proposes a spacetime topology change as a resolution of the black hole information paradox. I borrow his assumptions regarding QG, namely that:

• Gravitational collapse leads to a region of Planckian curvature where QG effects are large.
• QG tunneling in this high curvature region causes a topology change and a new disconnected region of spacetime.

To be more precise, I am interested in what the relation is, if any, between the quantum states in the two disconnected regions. The paper mentioned above regarding correlations across clasically forbidden horizons has prompted me to wonder if such correlations (and possibly more) can exist even between disconnected regions. I realize that with the absence of a theory of quantum gravity this question may not be know or have a definitive answer, but I am curious to see proposed ideas and am looking for literature recommendations.

• I do not understand well. Here $A$ is not a Cauchy surface for the left portion of spacetime, nor $B$ is for the right one. There are maximally extended causal curves contained in the left tube which do not meet $A$. This spacetime is globally hyperbolic and Cauchy surfaces are all homeomorphic to $\sigma'$. – Valter Moretti Feb 12 '18 at 21:44
• Disconnected spacelike $3$ surfaces like $A$ and$B$ in the figure can be constructed in Minkowski spacetime, too. – Valter Moretti Feb 12 '18 at 21:46
• $A$ and $B$ have no particular physical meaning here. Their union is not a Cauchy surface of the whole spacetime so that you cannot conclude that the spatial topology has changed. – Valter Moretti Feb 12 '18 at 21:57