Hilbert Bundles and the Hydrodynamic Approach to Quantum Gravity

Several papers from the mid to late 1990s suggest that Einstein’s equations should be thought of as the hydrodynamic equations of a special class of quantum systems. A classical solution defines subsystems by dividing space-time up into causal diamonds and Einstein’s equations are the hydrodynamics of a system that assigns density matrices to each diamond with the property ⟨K⋄⟩= ⟨(K⋄−⟨K⋄⟩)2⟩=A⋄. These define 4GN the empty diamond state, the analog of the quantum field theory vacuum, in the background geometry. The assignment of density matrices to each diamond enables one to define the analog of half sided modular flow along geodesics in the background manifold, as a unitary embedding of the Hilbert space of a given diamond into the next one in a nesting with Planck scale time steps. We conjecture that this can be enhanced to a full set of compatible unitary evolutions on a Hilbert bundle over the space of time-like geodesics, using a Quantum Principle of Relativity defined in the text. The compatibility of this formalism with the experimental success of quantum field theory (QFT) is discussed, as well as the theoretical limits in which QFT emerges.