Inversion Transformation - Invariants

Invariants

The invariants for this symmetry in 4 dimensions is unknown however it is known that the invariant requires a minimum of 4 space-time points. In one dimension, the invariant is the well known cross-ratio from Möbius transformations:

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 to ). 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 and ending at the endpoints is a conformal function of the 4-dimensional invariant. A string field in endpoint-string theory is a function over the endpoints.