On Momentum Operators Given by Killing Vectors Whose Integral Curves Are Geodesics

Abstract

The paper considers momentum operators on intrinsically curved manifolds. Given that momentum operators are Killing vector fields whose integral curves are geodesics, the corresponding manifold is flat or of the compact type with positive constant sectional curvature and dimensions equal to 1, 3, or 7. Explicit representations of momentum operators and the associated Casimir element are discussed for the 3-sphere S3. It is verified that the structural constants of the underlying Lie algebra are proportional to 2 ℏ/R, where R is the curvature radius of S3 and ℏ is the reduced Planck’s constant. This results in a countable energy and momentum spectrum of freely moving particles in S3. The maximal resolution of the possible momenta is given by the de Broglie wave length, λR=πR, which is identical to the diameter of the manifold. The corresponding covariant position operators are defined in terms of geodesic normal coordinates, and the associated commutator relations of position and momentum are established.