Scale and Conformal Transformations in Galilean-Covariant Field Theory

Abstract

It is demonstrated that there exists the possibility of defining scale and conformal transformations in such a way that these constitute exact invariance operations of the Schr\"odinger equation. Unlike the relativistic case there is only a single conformal transformation and the usual eleven-parameter extended Galilei group is consequently enlarged to a thirteen-parameter group. The generalization to the case of fields of arbitrary spin is carried out within the framework of minimal-component theories whose interactions respect scale and conformal invariance. One finds that the bare-internal-energy term can be used to break these additional invariance operations in much the same way as the mass term in special relativity. The generators and conservation laws associated with all space-time symmetries of minimal-component Galilean-invariant field theories are derived, it being shown that, in analogy to the relativistic case, the operators which appear in these equations can be redefined so as to allow the formulation of scale and conformal invariance entirely in terms of those operator densities relevant to the transformations of the Galilei group.