Fractional quantum fields with Le'vy paths

Abstract: We develop a path integral approach to quantum field theory that is defined over the paths of the Le'vy flights possessing a fractal dimension $1<d_f<2$. In standard quantum field theory, the fractality of the Brownian trajectories lead to a dispersion relation of quadric form. While the Le'vy paths lead fractional quantum field theory to a fractional dispersion relation. By considering Le'vy paths in time, we calculate density of states for a massless scalar field with box boundary condition. The density of states show behaviors dual to lower dimensional system, and the corresponding black body radiation has an energy spectrum dual to that in lower dimensional black body radiation. We derive the fractional equations of motion for scalar field, vector field and spinor field in zero temperature. Their propagators have been calculated. Based on above derivation, we calculate the one-loop self-energy of electron in Lev'y paths to show how this scheme works equally in renormalization as dimensional regularization does. Nevertheless, We found that gauge symmetry prevent Dirac spinor field from Le'vy paths and demonstrate that these paths in Dirac spinor field leds to unstable electrons, thus Le'vy paths are overcame by ordinary Brownian paths. While Le'vy paths are permitted in electromagnetic field by gauge symmetry. It led to a non-local phase and interaction with electron where electron charge is a composite quantity rather than a fundamental constant which maybe physically observable.