Bohr-Sommerfeld-Heisenberg quantization of the mathematical pendulum

Abstract

In this paper we give the Bohr-Sommerfeld-Heisenberg quantization of the mathematical pendulum.arXiv:1507.05674v1 [math.SG]

Lectures on Geometric Quantization

“…Since the map ξ → J ξ is a homomorphism of g to the Poisson algebra C ∞ (P ), the map ξ → (i/ )P J ξ is a linear representation of the Lie algebra g on the space S ∞ (L), which we call the prequantization representation of g. Since Hamiltonian vector fields X J ξ are complete, each operator (i/ )P J ξ is skew-adjoint on the Hilbert space H obtained by the completion of S ∞ 0 (L) with respect to the norm given by (17). Recall that the action of g on L is given by vector fields X J ξ on L, see equation (14). We assume that this action integrates to an action of G on L that covers the action of G on P .…”

“…The level set H −1 (0) is a stable equilibrium at (0, 0) and the level set H −1 (2) is the union of an unstable equilibrium at (0, π) and two homoclinic orbits. This section is based on [14]. The homoclinic orbits are the only non-compact orbits of X H .…”

Shifting Operators in Geometric Quantization

“…In our earlier papers [9][10][11][12], we followed an algebraic analysis, similar to that used by Dirac [8], supplemented by heuristic guesses about the behaviour of the shifting operators at the points of singularity of the polarization. In particular, we assumed that a X ϑ vanishes on the states concentrated on a set of limit points of e t X ϑ (p) as t → ∞.…”

“…This singularity is so well known that we do not have to use the language of differential spaces to get results. It should be noted that the results in [9,11] rely on the theory of differential spaces.…”

Shifting operators in geometric quantization

“…In a series of papers on Bohr-Sommerfeld-Heisenberg quantization of completely integrable systems [4], [5], [6], [11], we interpreted shifting operators as quantization of functions e ±iϑ j , where (I j , ϑ j ) are action angle coordinates. The aim of this paper is to show how these operators occur in prequantization, which is the first step of geometric quantization.…”