A. Zee, Quantum Field Theory in a Nutshell · Problem VI.1.4
Field Theory Redux
Physics — QFT
Problem
Consider L=γaε∂a−(1/4g2)f2. Calculate the propagator and show that the gauge boson is massive. Some physicists puzzled by fractional statistics have reasoned that since in the presence of the Chern-Simons term the gauge boson is massive and hence short ranged, it can't possibly generate fractional statistics, which is manifestly an infinite ranged interaction. (No matter how far apart the two particles being interchanged are, the wave function still acquires a phase.) The resolution is that the information is in fact propagated over an infinite range by a q=0 pole associated with a gauge degree of freedom. This apparent paradox is intimately connected with the puzzlement many physicists felt when they first heard of the Aharonov-Bohm effect. How can a particle in a region with no magnetic field whatsoever and arbitrarily far from the magnetic flux know about the existence of the magnetic flux?