Dual variables and a connection picture for the Euclidean Barrett–Crane model

Abstract

The partition function of the SO(4)- or Spin(4)-symmetric Euclidean Barrett–Crane model can be understood as the sum over all quantized geometries of a given triangulation of a four-manifold. In the original formulation, the variables of the model are balanced representations of SO(4) which describe the quantized areas of the triangles. We present an exact duality transformation for the full quantum theory and reformulate the model in terms of new variables which can be understood as variables conjugate to the quantized areas. The new variables are pairs of S3-values associated with the tetrahedra. These S3-variables parametrize the hyperplanes spanned by the tetrahedra (locally embedded in ℝ4), and the fact that there is a pair of variables for each tetrahedron can be viewed as a consequence of an SO(4)-valued parallel transport along the edges dual to the tetrahedra. We reconstruct the parallel transport of which only the action of SO(4) on S3 is physically relevant and rewrite the Barrett–Crane model as an SO(4) lattice BF-theory living on the 2-complex dual to the triangulation subject to suitable constraints whose form we derive at the quantum level. Our reformulation of the Barrett–Crane model in terms of continuous variables is suitable for the application of various analytical and numerical techniques familiar from statistical mechanics.