Curvature-induced noncommutativity of two different components of momentum for a particle on a hypersurface

Abstract

As a nonrelativistic particle constrained to remain on an (N − 1)-dimensional ((N ≥ 2)) hypersurface embedded in an N-dimensional Euclidean space, two different components pi and p j (i, j = 1, 2, 3,… N) of the Cartesian momentum of the particle are not mutually commutative, and explicitly commutation relations [ p ˆ i , p ˆ j ] ≠ 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 around 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.

The curvature-induced gauge potential and the geometric momentum for a particle on a hypersphere

“…In quantum mechanics, a constrained dynamical system is usually associated with a gauge structure, and it is quite well-understood in, for instance, gravitational field [1,2], condensed matter physics [3,4], quantum fields [5] and particle physics [6]. We are recently interested in the constrained motion of which a particle remains and freely move on a hypersurface [7][8][9][10][11][12][13][14][15][16][17][18], and there are also a lot of papers paying attention to the curvature-induced gauge structure on it [19][20][21][22][23][24][25][26][27][28][29][30][31]. In present article, we demonstrate that the gauge potential arisen from the fundamental algebra, i.e., a generalized angular momentum algebra, on the hypersphere [19] can be a part of the geometric momentum, and the resulting momentum obeys the fundamental quantization conditions.…”

“…where p = Π+ A and Π is the so-called geometric momentum for a spinless particle on the hypersurface is, [7][8][9][10][11][12][13][14][15][16][17][18]…”

Generally covariant geometric momentum, gauge potential and a Dirac fermion on a two-dimensional sphere

“…The hypersurface Σ N −1 can be described by a constraint in the configurational space as f (x) = 0 and the equation of the surface is chosen as |∇f (x)| = 1, such that the normal vector is n ≡ ∇f (x) = e i n i . As a consequence, only the unit normal vector and/or its derivatives enter the physics equation regardless of the surface equation [11,12]. Within the above quantum conditions, there are many forms of the quantum momentum p because of the operatorordering problem in O {(n i n k , j −n j n k , i ) p k } Hermition , leading to the fact that even the proper form of the momentum and the Hamiltonian cannot be determined unless more conditions are presented [13][14][15][16].…”