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Holomorphic Quantization in Constant Curvature Backgrounds

Dmitri Bykov, Viacheslav Krivorol

Abstract

We present a holomorphic quantization scheme for free point particles on two-dimensional constant curvature Riemannian backgrounds. The procedure is based on a Lagrangian embedding of the particle configuration space into a product of coadjoint orbits of the background isometry group. Examples are provided by particles on the plane, torus, sphere, and hyperbolic plane, with or without a monopole field. We elaborate the method by recovering the Hamiltonian spectrum and the wave functions on such spaces. As a by-product, we obtain a geometric and physical interpretation of Repka's result on the decomposition of tensor products of $\mathbf{SL}(2,\mathbb{R})$ discrete series representations.

Holomorphic Quantization in Constant Curvature Backgrounds

Abstract

We present a holomorphic quantization scheme for free point particles on two-dimensional constant curvature Riemannian backgrounds. The procedure is based on a Lagrangian embedding of the particle configuration space into a product of coadjoint orbits of the background isometry group. Examples are provided by particles on the plane, torus, sphere, and hyperbolic plane, with or without a monopole field. We elaborate the method by recovering the Hamiltonian spectrum and the wave functions on such spaces. As a by-product, we obtain a geometric and physical interpretation of Repka's result on the decomposition of tensor products of discrete series representations.
Paper Structure (37 sections, 186 equations, 3 figures, 1 table)

This paper contains 37 sections, 186 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. The plane $\mathbb{C}$ and $\widehat{\mathbf{ISO}}(2)$ orbits
  3. The Landau problem.
  4. A 'first order' oscillator formulation.
  5. $\widehat{\mathbf{ISO}}(2)$ coadjoint orbits.
  6. A quick aside: Fock space from orbit quantization.
  7. Holomorphic quantization of the particle.
  8. Dual $\widehat{\mathfrak{iso}}_2$ algebra.
  9. Case of vanishing magnetic field.
  10. Magnetic wavefunctions.
  11. The torus $\mathbb{T}^2$ and quotient of $\widehat{\mathbf{ISO}}(2)$ orbits
  12. Zero magnetic field.
  13. Non-zero magnetic field.
  14. Lowest Landau level.
  15. The sphere $\mathbb{S}^2$ and $\mathbf{SU}(2)$ orbits
  16. ...and 22 more sections

Figures (3)

  • Figure 1: The product of two copies of the Lobachevsky plane $\mathbb{H}_{p}^+ \times \mathbb{H}_{p+\mathfrak{q}}^-$, viewed as hemispheres of $\mathbb{CP}^1$ determined by $\lVert z\rVert^2>0$, $\lVert w\rVert^2<0$.
  • Figure 2: The product of two copies of the Lobachevsky plane $\mathbb{H}_{p}^+\times\mathbb{H}_{p+\mathfrak{q}}^+$, viewed as hemispheres of $\mathbb{CP}^1$ determined by $\lVert z\rVert^2>0$, $\lVert w\rVert^2>0$.
  • Figure 3: The deformed integration contour $\mathcal{C}$ is shown in green. The radius of the green circle is eventually sent to infinity. The cut $[1, \infty)$ is shown in red.