Gravitation Theory in Path Space

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 transformations in path space. Their algebra is established. Itcloses if and only if the defining conditions of a riemannian geometry hold (Bianchi identity and vanishing of the antisymmetric part of the Rieman 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 comments are offered on the possibility of developping an invariant description of loops regarded as boundaries of two-dimensional surfaces.