Because the only invariants under this symmetry involve a minimum of 4 points, this symmetry cannot be a symmetry of point particle theory. Point particle theory relies on knowing the lengths of paths of particles through space-time (e.g., from x to y ). The symmetry can be a symmetry of a string theory in which the strings are uniquely determined by their endpoints. The propagator for this theory for a string starting at the endpoints ( x , X ) and ending at the endpoints ( y , Y ) is a conformal function of the 4-dimensional invariant. A string field in endpoint-string theory is a function over the endpoints.