Edwards field theory for glasses and granular matter

A minimal description of the inherent states of amorphous solids is presented. Using field theory, applicable when a system is probed at long length scales, it is shown that athermal amorphous solids have long-range correlations in their stresses, as recently observed in supercooled liquids, colloids, and granular matter. Explicit predictions for the correlators are presented, in both 2D and 3D, in excellent agreement with simulation data on supercooled liquids. It is shown that when applied to solids with strictly repulsive interactions, the simplest, na\"ive theory leads to a paradox. This paradox is resolved, and it is shown that a nontrivial, non-Gaussian theory is necessary for such materials. Modifications to the correlators are shown, at the saddle-point level. In all cases, `equations of state' relating fluctuations to imposed stresses are derived, as well as field equations that fix the spatial structure of stresses in arbitrary geometries. A new holographic quantity in 3D amorphous systems is identified.