2017/09/18 by Q. H. Liu, Xing Yang, Liu, Q. H. +3
Physics and Astronomy · #FOS: Physical sciences #General Physics (physics.gen-ph) #Noncommutative and Quantum Gravity Theories #Quantum Electrodynamics and Casimir Effect #Quantum and Classical Electrodynamics
paper · pdf · doi:10.48550/arxiv.1709.07299
openalex publication_date 2017/09/18 · openalex created_date 2018/04/13 · openalex updated_date 2026/07/28
As a nonrelativistic particle constrained to remain on an N-1 (N≥ 2) dimensional hypersurface embedded in an N dimensional Euclidean space, two different components pi and pj (i,j=1,2,3,...N) of the Cartesian momentum of the particle are not mutually commutative, and explicitly commutation relations [pi,pj]( ≠ 0) depend on products of positions and momenta in uncontrollable ways. The % generalized Dupin indicatrix of the hypersurface, a local analysis technique, is utilized to explore the dependence of the noncommutativity on the curvatures on a local point of the hypersurface. The first finding is that the noncommutativity can be grouped into two categories; one is the product of a sectional curvature and the angular momentum, and another is the product of a principal curvature and the momentum. The second finding is that, for a small circle lying a tangential plane covering the local point, the noncommutativity leads to a rotation operator and the amount of the rotation is an angle anholonomy; and along each of the normal sectional curves centering the given point the noncommutativity leads to a translation plus an additional rotation and the amount of the rotation is one half of the tangential angle change of the arc.