The curvature-induced gauge potential and the geometric momentum for a particle on a hypersphere. (English) Zbl 1471.81034
Summary: A particle that is constrained to freely move on a hyperspherical surface in an \(N(\geq 2)\) dimensional flat space experiences a curvature-induced gauge potential, whose form was given long ago [Y. Ohnuki and S. Kitakado, J. Math. Phys. 34, No. 7, 2827–2851 (1993; Zbl 0789.22045)]. We demonstrate that the momentum for the particle on the hypersphere is the geometric one including the gauge potential and its components obey the commutation relations \(\left[p_i , p_j\right]=-i\hbar J_{ij}/r^2\), in which \(\hbar\) is the Planck’s constant, and \(p_i(i,j=1,2,3,\dots N)\) denotes the \(i\)-th component of the geometric momentum, and \(J_{ij}\) specifies the \(ij\)-th component of the generalized angular momentum containing both the orbital part and the coupling of the generators of continuous rotational symmetry group \(SO(N-1)\) and curvature, and \(r\) denotes the radius of the \(N-1\) dimensional hypersphere.
MSC:
| 81Q35 | Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices |
| 81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |
| 81Q60 | Supersymmetry and quantum mechanics |
| 14J70 | Hypersurfaces and algebraic geometry |
| 81S05 | Commutation relations and statistics as related to quantum mechanics (general) |
| 81S08 | Canonical quantization |
| 81R05 | Finite-dimensional groups and algebras motivated by physics and their representations |
| 22E70 | Applications of Lie groups to the sciences; explicit representations |
Citations:
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