Topology changing processes have played a central role in recent developments in quantum gravity, and have been argued to lead to an abelian algebra of superselected observables. I will discuss one-dimensional theories of quantum gravity in this context. This invites comparison with a worldline formalism of QFT, with its non-abelian algebra of quantum fields. This apparent tension is resolved by considering the different choices to be made when constructing Hilbert spaces for quantum gravity and QFT.