Hilbert Bundles and Holographic Space-time: the Hydrodynamic Approach to Gravity

Abstract: Results of Jacobson, Carlip and Solodukhin, from the 1990s, suggest a hydrodynamic approach to quantum gravity in which a classical solution of Einstein's equations determines the density matrices of subsystems associated with causal diamonds in the "empty diamond" state of a corresponding quantum system. The subsystem operator algebras are finite dimensional and correspond to a UV cutoff $1 + 1$ dimensional field theory of fermions living on a "stretched horizon" near each diamond's holographic screen. The fields can be thought of as fluctuations of solutions of the screen's Dirac operator around that of the background geometry, expanded up to a maximal Dirac eigenvalue determined by the Carlip-Solodukhin relation between area and central charge. This cutoff renders the screen geometry "fuzzy". Quantum dynamics is defined in a Hilbert bundle over the space of time-like geodesics on the background geometry. A nesting of diamonds along a given geodesic defines a series of unitary embedding maps of diamond Hilbert spaces into each other, analogous to half sided modular flows in quantum field theory. These can be extended into a consistent set of unitary maps of each fiber Hilbert space into itself by a quantum version of the principle of relativity. According to the QPR, the largest diamond in the overlap between any two diamonds is identified with a tensor factor in each individual diamond Hilbert space, and must have the a density matrix with the same entanglement spectrum no matter which fiber dynamics is used to compute it. This brief review summarizes how these ideas play out in a variety of contexts in different dimensions.