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The proper-time gauge appears to be the simplest one consistent with the invariance properties of the gravitational action. It also permits one to implement in a direct manner the requirement of causality in the quantum theory of gravitation. In this paper the measure for the path integral over gravitational fields in the proper-time gauge is explicitly worked out. The corresponding propagation amplitude results after the following steps: (i) evaluating the amplitude for a transition between two three-geometries for which one specifies the relative pointwise proper-time separation and relative spatial coordinate system, (ii) integrating over all positive proper-time separations with a logarithmic measure, (iii) averaging over all possible choices of the relative coordinate system. (An explicit expression for the measure over the diffeomorphism group in terms of a set of ghost fields emerges from the path integral.)
Claudio Teitelboim (Fri,) studied this question.