[1][2] They are less studied in physics because, unlike the rotations and translations of Poincaré symmetry, an object cannot be physically transformed by the inversion symmetry.
In 1831 the mathematician Ludwig Immanuel Magnus began to publish on transformations of the plane generated by inversion in a circle of radius R. His work initiated a large body of publications, now called inversive geometry.
The most prominently named mathematician became August Ferdinand Möbius once he reduced the planar transformations to complex number arithmetic.
In the company of physicists employing the inversion transformation early on was Lord Kelvin, and the association with him leads it to be called the Kelvin transform.
) so that space-time is Euclidean and the equations are simpler.
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.
The propagator for this theory for a string starting at the endpoints
A string field in endpoint-string theory is a function over the endpoints.
Although it is natural to generalize the Poincaré transformations in order to find hidden symmetries in physics and thus narrow down the number of possible theories of high-energy physics, it is difficult to experimentally examine this symmetry as it is not possible to transform an object under this symmetry.
The indirect evidence of this symmetry is given by how accurately fundamental theories of physics that are invariant under this symmetry make predictions.
Other indirect evidence is whether theories that are invariant under this symmetry lead to contradictions such as giving probabilities greater than 1.
So far there has been no direct evidence that the fundamental constituents of the Universe are strings.