Generally covariant geometric momentum and geometric potential for a Dirac fermion on a two-dimensional hypersurface

Abstract

Geometric momentum is the appropriate momentum for a particle constrained to move on a curved surface, which depends on the extrinsic curvature and leads to observable effects, and curvature-induced quantum potentials appear for a nonrelativistic free particle on the surface. In the context of multi-component quantum states, the geometric momentum should be rewritten as a generally covariant geometric momentum, which contains an additional term defined as the gauge potential. For a Dirac fermion constrained on a two-dimensional hypersurface, we derive the generally covariant geometric momentum, and demonstrate that no curvature-induced geometric potentials arise on a pseudosphere or a helical surface. The dynamical quantization conditions are verified to be effective in dealing with constrained systems on hypersurfaces, enabling the derivation of both the generally convariant geometric momentum and the geometric potential for a spin particle constrained on parametrically defined surfaces.

arXiv comment8 pages, 2 figures

References in corpus (16)

• Quantum mechanics on curved 2D systems with electric and magnetic fields
• Quantum field theory in curved graphene spacetimes, Lobachevsky geometry, Weyl symmetry, Hawking effect, and all that
• The surface states of topological insulators - Dirac fermion in curved two dimensional spaces
• Geometric potential and transport in photonic topological crystals
• Geometry induced charge separation on a helicoidal ribbon
• Geometric Momentum: the Proper Momentum for a Free Particle on a Two-dimensional Sphere
• Monopole Gauge Fields and Quantum Potentials Induced by the Geometry in Simple Dynamical Systems
• Coordinate-Free Quantization of Second-Class Constraints
• Geometric momentum for a particle constrained on a curved hypersurface
• Geometric Potential Resulting from Dirac Quantization
• Electronic states in a quantum Beltrami surface
• Flat bands without twists: periodic holey graphene
• General covariant geometric momentum, gauge potential and a Dirac fermion on a two-dimensional sphere
• No existence of the geometric potential for a Dirac fermion on two-dimensional curved surfaces of revolution
• Quantum particle on the surface of a spherocylindrical capsule
• The curvature-induced gauge potential and the geometric momentum for a particle on a hypersphere