Abstract

A formulation of gravitation theory originally proposed by Mandelstam is re-examined. The idea is to avoid the use of coordinates while staying in the continuum. This is accomplished by regarding a point as the end of a path. The theory is then formulated in the space of all paths. The analysis relies on the properties of path deformations. These deformations play the role of gauge transfor-mations in path space. Their algebra is established. It closes if and only if the defining conditions of a riemannian geometry hold (Bianchi identity and vanishing of the antisymmetric part of the Riemann tensor in three of its indices). Two problems faced by Mandelstam are solved: (i) An explicit formula is given which establishes when two neighboring paths end at the same point, (ii) An action principle is given, in terms of a functional integral over path space. It is also indicated how to reconstruct the metric from the curvature through gauge fixing in path space. Brief com-ments are offered on the possibility of developing an invariant description of loops regarded as boundaries of two-dimensional surfaces.