Covariance and the use of the Schrodinger equation in quantum field theory
The Schrodinger equation is not covariant. Nevertheless, quantum field theory is often formulated using the Schrodinger equation to describe the time evolution of the system, which is equivalent to using Feynman path integrals. It is well known that scattering theory gives covariant results provided that the interaction vanishes in the asymptotic limit of $t \rightarrow \pm \infty$, but there are other situations of interest that do not satisfy those conditions. As an example, the Aharonov-Bohm effect is derived here using second-quantized field theory to describe all the electrons as well as the electromagnetic field, which allows the effects of retarded vector potentials to be calculated in a straightforward way using the Feynman propagator. The results are in agreement with the usual expression for the Aharonov-Bohm effect when the retardation of the electromagnetic field is negligible, but they predict a fractional phase shift, a lack of covariance, and violations of causality when retardation effects are significant. One of the assumptions inherent in the usual proof of causality is shown to be invalid under these conditions. These results suggest that quantum field theory based on the Schrodinger equation or Feynman path integrals is incomplete in the sense that it cannot give a correct description of all observable phenomena.