We investigate a particle motion quantum-mechanically in the neighbourhood of n-dimensional curved manifold Mn embedded in a p-dimensional Euclidean space Rp. The effective Hamiltonian on Mn has a gauge structure determined by geometry of Mn, generally non-Abelian. This gives a generalization of the geometry-induced Aharonov-Bohm effect in the case M1 ⊂ R3.
We investigate a particle motion quantum-mechanically in the neighbourhood of n-dimensional curved manifold Mn embedded in a p-dimensional Euclidean space Rp. The effective Hamiltonian on Mn has a gauge structure determined by geometry of Mn, generally non-Abelian. This gives a generalization of the geometry-induced Aharonov-Bohm effect in the case M1 ⊂ R3.
Investigating a particle, a charged particle in an electromagnetic field and a particle with spin-orbit interaction confined to a space curve in square and circular cases, respectively, we first find that their effective Hamiltonian can be defined in terms of the ground states of their confined dimensions. And we obtain interesting results: the torsion-induced geometric potential appears in the square case, while in the circular case it vanishes; the geometrically induced gauge potential and Zeeman coupling appear in the circular case; and the geometric gauge potential, curvature and torsion simultaneously emerge in the spin-orbit interaction in the circular case. In the example, a helix nanowire, the previous results are all demonstrated. These results provide a way to generate an artificial gauge field and to manipulate spin transport by designing the geometry of curved nanodevice.
The Aharonov–Bohm effect in general gauge theories, for particles in gauge-curvature-free regions, is studied using the quantum mechanical propagator in the form of a Feynman sum over paths. Following Schulman [L. S. Schulman, Techniques and Applications of Path Integration (Wiley, New York, 1981)], such paths are divided into their homotopy equivalence classes, and the contributions from each class of paths of the Feynman sum are identified with propagators of a wave equation in the universal covering manifold of M, resulting in a simple form for the propagator on M. A group homomorphism from ℋ, the fundamental homotopy group of M, to the gauge group G is shown to characterize possible Aharonov–Bohm effects, which can be divided into two types, Abelian and non-Abelian, according to whether ℋ*, the image of this homomorphism, is Abelian or non-Abelian. For a non-Abelian Aharonov–Bohm effect, it is necessary that both ℋ and G be non-Abelian. Simple examples illustrate the theory.
A realistic analysis shows that constraining a quantomechanical system produces the effective dynamics to be coupled with abelian/nonabelian gauge fields and quantum potentials induced by the intrinsic and extrinsic geometrical properties of the constraint’s surface. This phenomenon is observable in the effective rotational motion of some simple polyatomic molecules. By considering specific examples it is shown that the effective Hamiltonians for the nuclear rotation of linear and symmetric top molecules are equivalent to that of a charged system moving in a background magnetic-monopole field. For spherical top molecules an explicit analytical expression of a non-abelian monopole-like field is found. Quantum potentials are also relevant for One of the first questions addressed by researchers who first studied the properties of surfaces embedded in the three-dimensional euclidean space, at
The motion of a quantum particle constrained to a two-dimensional non-compact Riemannian manifold with non-trivial metric can be described by a flat-space Schroedinger-type equation at the cost of introducing local mass and metric and geometry-induced effective potential with no classical counterpart. For a metric tensor periodically modulated along one dimension, the formation of bands is demonstrated and transport-related quantities are derived. Using S-matrix approach, the quantum conductance along the manifold is calculated and contrasted with conventional quantum transport methods in flat spaces. The topology, e.g. whether the manifold is simply connected, compact or non-compact shows up in global, non-local properties such as the Aharonov-Bohm phase. The results vividly demonstrate emergent phenomena due to the interplay of reduced-dimensionality, particles quantum nature, geometry, and topology.
In this paper, we apply covariant affine integral quantization to study motion in the 2D punctured plane. The associated four-dimensional phase space is identified with the similitude group SIM(2), representing transformations of the plane composed of translations, rotations, and dilations. Near the punctured point, we observe that the topology influencing quantum fluctuations gives rise to an affine vector potential, which can be interpreted as the Aharonov-Bohm (AB) gauge potential generated by an infinite coil. This finding suggests that the AB effect originates from the topology enforced by the impenetrable coil, rather than from a classical gauge potential. Our results offer a novel perspective on the AB effect, emphasizing the fundamental role of topology in quantum mechanics.
We investigate the effective dynamics of a particle confined near a space curve. In the strict thin-layer reduction of the nondegenerate transverse ground state, torsion does not enter the local effective Hamiltonian, which contains only the curvature-induced scalar geometric potential. In contrast, for a thin guide with finite transverse width, a leading-order adiabatic projection onto the twofold-degenerate first-excited transverse band renders the rotation of the Frenet normal frame dynamically relevant and generates a matrix-valued Abelian gauge potential. Using a projection-based derivation in a co-rotating Frenet-frame basis, we show that this effective gauge potential is directly determined by the local torsion of the curve. The resulting effective Hamiltonian takes a gauge-covariant form and produces two transverse-mode branches whose parabolic dispersions are shifted in opposite directions in momentum space. For closed curves, the associated holonomy is controlled by the integrated torsion and leads to geometric interference. These results provide a direct realization of a Wilczek--Zee-type connection induced purely by spatial geometry in curved quantum waveguides. We further construct a classical-wave analogue using the degenerate bending modes of an isotropic elastic rod, demonstrating that the same torsion-induced gauge structure appears in continuum wave physics.