Assorted topics

3.14 Assorted topics

Hartle [124 , 125 ] has suggested computing the wave functional of the universe in a simplicial approximation, and evaluating the discrete path integral semiclassically near stationary points of the Regge action. He investigated numerically the extrema of the action (12

) on small simplicial manifolds with topologies $4 S$ , $2 ℂ P$ and $2 2 S × S$ [126 ]. The properties of a Hartle–Hawking wave functional for a small complex with an $3 S$ -boundary were studied in [127 ].

Fröhlich [104 ] has advocated the need for a proof of reflection positivity of the Regge path integral, which one may expect to play a role in proving the unitarity of the theory. This can be formulated as a condition on the path-integral measure (including the action) under the gluing of two simplicial four-manifolds along a three-dimensional boundary.

Other authors have suggested associating gauge-theoretic instead of metric variables with the building blocks of a simplicial complex, for the case of the Poincaré group [71 ], the Lorentz group [140 ], and for Ashtekar gravity with gauge group $SU (2,ℂ )$ [131 , 132 ], and reformulating the quantum theory in terms of them.

Hamiltonian 3+1 versions of Regge calculus have been studied classically (see [190 ] for a review), but attempts to quantize them have not progressed very far. One meets problems with the definition of the constraints and the (non-)closure of their Poisson algebra. A recent proposal for constructing a canonical quantum theory is due to Mäkelä [158 ], who constructed a simplicial version of the Wheeler–DeWitt equation, based on the use of area instead of length variables (which however are known to be overcomplete). In a similar vein, Khatsymovsky [141 ] has suggested that the operators measuring spatial areas ought to have a discrete spectrum.