November 28, 2005

November 28

An interesting paper, somewhat off the main stream, is Differential Structures – the Geometrization of Quantum Mechanics by Torsten Asselmeyer-Maluga and Helge Rose. They ask what is the geometry of  quantum mechanics. Finding the answer to this question  would bring us to a unified theory of physics in terms of one entity alone, the four dimensional space-time manifold. Matter is characterized in this theory by the topology of three dimensional particles. The differential structure gives us two pictures of matter: geometrical and topological. The former is related to the change in the non-local,wave-like four dimensional connection and the latter to the local three dimensional singularities.

I try to give some idea of the first few pages of the paper. Manifolds are described by maps h from its subsets W to the linear space R^n with subsets U. A coordinate transformation between two charts is a map between subsets of linear spaces. The overlapping origin of two W's is mapped into two images U_ij = h_i(W_ij) and U_ji = h_j(W_ij). A coordinate transformation between two charts is a map between subsets of linear spaces, h_ij: U_ij -> U_ji.  Two charts are compatible if U_ij, U_ji are open  and the coordinate transformations h_ij, h_ji are diffeomorphisms. A family of pairwise compatible charts that covers the whole manifold is an atlas. Two atlases are equivalent if their union is an atlas.  A differential structure consists of the equivalence classes of the atlases of a manifold. The number of differential structures from dimension 1 up to 11 are 1, 1, 1, infinity, 1, 1, 28, 2, 8, 6, and 992, resp. In dimension four there is an uncountable number differential structures for most non-compact four-manifolds (countable for compact).

The main hypothesis of the authors is that the structures of quantum mechanics are induced by the differential structure of the space-time manifold. To support the idea they prove a theorem in which they end up in Temperley-Lieb algebra. This algebra represents the set of operators of transitions of the differential structure of space-time.  They end the paper with a comparison of their model with Loop quantum gravity, knots and links appearing in both. More papers are promised by the authors.