Geometrodynamics of a 2D Curved Surface due to a Constrained Quantum Particle via its Gravitational Dual: ${\mathcal{S}}^2$ Analytical Model Calculations

Abstract

We provide a unique and novel extension of da Costa's calculation of a quantum mechanically constrained particle. This is achieved by analyzing the perturbative back reaction of the quantum confined particle's eigenstates and spectra upon the geometry of the curved surface itself, thereby addressing the problem of shape optimization in this model. We do this by first formulating a two-dimensional action principle of the quantum constrained particle, which upon variation of the wave function reproduces Schrödinger's equation including da Costa's surface curvature-induced potentials. We further demonstrate that our derived action principle is dual to a two-dimensional dilation gravity theory and we vary its functional with respect to the embedded two-dimensional inverse-metric to obtain the respective geometrodynamical Einstein equation. We solve this resulting Einstein equation perturbatively by first solving the da Costa's Schrödinger equation to obtain an initial eigensystem, which is used as initial-input data for a perturbed metric inserted into the derived Einstein equation. As a proof of concept, we perform this calculation on a two-sphere and show its first iterative perturbed shape evolution. We also turn on external electromagnetic fields and formulate the full field theoretic field equations for future investigation. The external fields manifest themselves via a surface induced, pulled-back $U(1)$ coupling in our two-dimensional dual gravity theory, thereby revealing interesting and rich new surface physics in this specific paradigm.

23 pages, 9 figures, substantially revised and expanded for clarity based upon referee and editorial suggestions. Version accepted as an invited contribution to IOP's Physica Scripta

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
• Electronic materials with nanoscale curved geometries
• Tuning electronic properties of corrugated graphene: confinement, curvature and band gap opening
• Proper Dirac Quantization of Free Particle on D-Dimensional Sphere
• Quantum mechanics of a constrained particle and the problem of prescribed geometry-induced potential
• A Near Horizon CFT Dual for Kerr-Newman-$AdS$
• Geometric effects on the electronic structure of curved nanotubes and curved graphene: the case of the helix, catenary, helicoid, and catenoid
• Electronic properties of a nanotorus
• Curvature control of the superconducting proximity effect in diffusive ferromagnetic nanowires
• Generalized Ellis-Bronnikov graphene wormhole
• Quantum Mechanics of Particle on a torus knot: Curvature and Torsion Effects
• Quantum Scattering of Spinless Particles in Riemannian Manifolds
• Two dimensional electron gas in a non-Euclidean space
• Gravitational wave emission from quadrupole Josephson junction device
• Exploration of The Duality Between Generalized Geometry and Extraordinary Magnetoresistance