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
[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.
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Discrete Approaches to Quantum Gravity in Four Dimensions
Renate Loll http://www.livingreviews.org/lrr-1998-13 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |