3.14 Assorted topics

Hartle [124125] 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 (12View Equation) 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 ) [131132], 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.

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