Charged Scalar Field in an external magnetic Field: Renormalisation and Universal Diamagnetism

Abstract

The physical and mathematical mechanism behind diamagnetism of N (finite) spinless bosons (relativistic or non-relativistic) is well known. The mathematical signature of this diamagnetism follows from Kato's inequality while its physical way of understanding goes back to Van Leewen. One can guess that it might be true in the field theoretic case also. While the work on systems with a finite number of degrees of freedom suggests that the same result is true in a field theory, it does not by any means prove it. In the field theoretic context one has to develop a suitable regularisation scheme to renormalise the free energy. We show that charged scalar fields in (2+1) and (3+1) dimensions are always diamagnetic, even in the presence of interactions and at finite temperatures. This generalises earlier work on the diamagnetism of charged spinless bosons to the case of infinite degrees of freedom. We also discuss possible applications of the theory.