Prospects and Puzzles
The program outlined here is both ambitious and highly speculative. We find it attractive as a way of deriving most of the familiar structure of the world from a minimal set of truly quantum ingredients.
In particular, we derive rather than postulating such notions as space, fields, and particles. The fact that our Hilbert space is locally finitedimensional suggests an escape from the famous problems of ultraviolet divergences in quantum field theory, and the emergence of spacetime geometry from quantum entanglement is an interesting angle on the perennial problems of quantum gravity.Numerous questions remain; we will highlight just two. One is the emergence of local Lorentz-invariant dynamics. There are no unitary representations of the Lorentz group on a finite-dimensional factor of Hilbert space. This might seem to imply that Lorentz symmetry would be at best approximate, a possibility that is experimentally intriguing but already highly constrained. It would be interesting to understand how numerically large any deviations from perfect Lorentz invariance would have to be in this framework, and indeed if they have to exist at all.
The other issue is the emergence of an effective field theory in curved spacetime that could describe matter fields in our geometric background. We have posited that a field theory might be identified with infrared degrees of freedom while the geometry is determined by ultraviolet degrees of freedom, but there is much to be done to make this suggestion more concrete. A promising idea is to invoke the idea of a quantum error-correcting code [11, 21]. Such codes imagine identifying a “code subspace” within the larger physical Hilbert space, such that the quantum information in the code can be redundantly stored in the physical Hilbert space. There is a natural way to associate the code subspace with the infrared degrees of freedom of the matter fields, with the rest of the physical Hilbert space providing the ultraviolet entanglement that defines the emergent geometry. Once again, this is a highly speculative but a promising line of investigation.
We are optimistic that this minimal approach to the ontology of quantum mechanics is sufficient, given an appropriate Hamiltonian and quantum state, to recover all of the richness of the world as we know it. It would be a profound realization to ultimately conclude that what is fundamental does not directly involve spacetime or propagating quantum fields, but simply a vector moving smoothly through a very large-dimensional Hilbert space. Further investigation will be needed to determine whether such optimism is warranted, or whether we have just gone mad.
Acknowledgements We are thankful to ChunJun (Charles) Cao for helpful conversations. This research is funded in part by the Walter Burke Institute for Theoretical Physics at Caltech and by DOE grant DE-SC0011632.