Density matrix of the Universe

The Hartle-Hawking wave function for the Universe gives a pure-state density matrix that may be expressed as a path integral over compact Euclidean four-geometries and matter fields which have two disjoint pieces, one of which is bounded by the first argument (a three-geometry with its matter configuration) of the density matrix and the other of which is bounded by the second. Here an alternate proposal is suggested, that the density matrix of the Universe is given by a path integral over all compact Euclidean four-geometries bounded by both three-geometries. Connected four-geometries joining the two three-geometries would be a new contribution to the density matrix which would not factorize, so the entire density matrix proposed here would represent an impure mixed state of the Universe.