Potential in the Canonical Formalism of Gravity

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Abstract
We attempt to derive the gravitational potential from the canonical formalism viewpoint of the gravitational field coupled with matter. The two-body potential in an arbitrary coordinate frame (i.e., gauge) is obtained by solving constraint equations. It is shown, by making a transformation of the time coordinate, that the potential coincides with Hiida's obtained with the aid of the “diagrammatic” quantization. A simple equation to get the static potential to any order is presented. Explicit forms of the static potential are given up to the five-body potential. From the form of potential it is immediately seen that there exists a relation between two-body and many-body static potentials.

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