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

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.

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Classical and quantum dynamics of a constrained particle on three-dimensional spaces of constant curvature: An algebraic approach on the superintegrable problem

Najafizade

Panahi

2022

Mod. Phys. Lett. A

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In this study, we discuss a particle which is the constraint on a four-dimensional Euclidean space [Formula: see text]. In this respect and under the framework of Dirac’s approach, we analyze the classical and quantum dynamics of a particle constrained on an embedded surface in supersphere [Formula: see text], which has constant curvature and signature equal to [Formula: see text] and [Formula: see text], respectively. In fact, it is presented that the Cartesian components of momentum and position exhibit a proper description for the motion of the particle embedded in a generalized curved space. Thus, quantization of dynamics for a particle on the hypersurface [Formula: see text] derives an appropriate representation of momentum called geometric momentum as it depends on the mean curvature. The invariance algebra generated by the quantization process is an extension of [Formula: see text] with curvature parameters in both classical and quantum dynamics. Moreover, superintegrability appears in the deformed Hamiltonian [Formula: see text] arising from the Dirac brackets, in which [Formula: see text] of the underlying space behaves as the curvature (deformation) parameter.

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

Cited by 3 publications

References 80 publications

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

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

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

Classical and quantum dynamics of a constrained particle on three-dimensional spaces of constant curvature: An algebraic approach on the superintegrable problem

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