Infrared fractal propagator for a charged fermion

The infinite range of the Coulomb forces implies that the electron can interact with softer and softer photons. This motivates us to look for an electron propagator with a fractal structure. Using Schwinger's proper time formulation, some work by Bloch and Nordsieck and a scheme of differential regularization, we are able to obtain an expression for the electron propagator with an anomalous dimension γ. In contrast to previous discussions on the subject we maintain gauge invariance throughout and obtain an unambigous number for the anomalous dimension γ of the electron. This leads us to a discussion of the fractal dimension of electron paths where we are able to obtain a number based on an analogy with Brownian motion. Since our result is gauge invariant we discuss the possibility of measuring γ experimentally and finally we briefly touch on the case of quantum gravity