Wednesday, August 08, 2007
The reason for gauge invariance
Why is there gauge invariance? Despite the opinion of many physicists it is not because God likes it. Rather it is the form Poincaré transformations take for massless objects and are possible for these only. This has been discussed in depth in the book Massless Representations of the Poincaré Group (see booklist). It can be explained trivially. Consider an electron and photon with momenta parallel and spins along the momenta (so parallel). There are transformations that leave the momenta unchanged, changing the spin direction of the electron, but cannot change that of the photon. EM waves are transverse. (This is required by the Poincaré group, not God). Thus there are transformations acting on the electron but not on the photon, which is impossible. What are these transformations? Obviously gauge transformations. And that is exactly what the Poincaré group gives; all their properties follow. They are not possible for massive objects but are a required property of massless ones.