Position Operators, Gauge Transformations, and the Conformal Group

Abstract

The connection between the Lie algebra of the conformal group and the algebra of quantum mechanics is analyzed, applying a method similar to one considered by Segal. The contraction of the 4-parameter special conformal group yields the position operators, and the contraction of the dilatations yields a phase transformation of the states considered. Quantum mechanics appears in this way as a broken symmetry. Thus one gets a relationship between geometrical gauge transformations and phase transformations, which sheds new light on Weyl's conjecture that geometrical gauge transformations and charge conservation are related to each other. Arguments are given as to why the usual interpretation of the special conformal group as a system of transformations connecting frames of constant relative accelerations hardly can be the right one. The main point is that the physically essential group velocity of the wave packets formed by the eigen-functions of the special conformal group has the same form as in the case of the plane waves, whereas the physically irrelevant phase velocity has the hyperbolic structure usually discussed.