A reformulation of the Ponzano-Regge quantum gravity model in terms of surfaces

Abstract

We reformulate the Ponzano-Regge quantum gravity model in terms of surfaces on a 3-dimensional simplex lattice. This formulation (1) has a clear relation to the loop representation of the canonical quantum general relativity in 3-dimensions, (2) may have a 4-dimensional analogue, in contrast to the 6-j symbolic formalism of the Ponzano-Regge model, and (3) is purely a theory of surfaces, in the sense that it does not include any field variables; hence it is coordinate-free on the surface and background-free in spacetime. We discuss implications and applications of this formulation.

Figures (5)

• Figure 1: The tetrahedron with edge lengths $l_i\ (i=1,2,\cdots,6)$.
• Figure 2: The 2d lattice (left) and its dual lattice (right) on the faces of a single tetrahedron.
• Figure 3: The gluing of lines at the dual site, corresponding to contractions of spin-${1\over2}$ parallel transports.
• Figure 4: The six elementary surfaces in a tetrahedron. Each one is a portion of a dual face.
• Figure 5: An example of the specific numbers for dual lines, and the 3-j symbols determined by them. All these numbers are determined by a (multiple) surface on a fixed tetrahedral lattice.