Remarks on the geometric quantization of the Kepler problem
type: Item
instance of: scholarly article
title: Remarks on the geometric quantization of the Kepler problem
zbMATH Open document ID: 0711.58016
DOI: 10.1007/BF00398955
author: Giuseppe Gaeta
author: Mauro Spera
published in: Letters in Mathematical Physics
publication date: 1988-01-01T00:00:00Z
review text: In the geometric quantization of a system of \(n+1\) harmonic oscillators with equal frequencies the states of fixed energy \(E=p+(n+1)/2\) are identified with sections of the bundle pH over the associated classical orbit space \(P^ n\) \((H:=\) hyperplane bundle, \(pH:= H\otimes H\otimes...\otimes H\), p times). The authors show that this construction formally agrees with Bargmann's quantization. In the latter these states are represented by homogeneous polynomials of degree p of \(n+1\) complex variables, and creation and annihilation operators are identified with multiplication and partial differentiation with respect to these variables. Now, the regularized Kepler Hamiltonian can be identified with the Hamiltonian associated with 4 harmonic oscillators with identical frequencies, subject to a certain constraint [see, e.g., the reviewer, Commun. Math. Phys. 84, No.1, 133-152 (1982; Zbl 0496.70023)]. Combining this fact with geometric quantization of four \((n=3)\) oscillators the authors obtain a geometric quantization of the Kepler problem which agrees with \textit{D. J. Simms}'s original one [in Symp. Math. 14, 125-137 (1974; Zbl 0306.53038)]. Their essential tool is the Segre map S which imbeds the orbit space \(P^ 1\times P^ 1\) (or \(S^ 2\times S^ 2)\) of the Kepler problem with fixed negative energy into \(P^ 3\). It turns out that the space of holomorphic sections of \((N- 1)S^*H\) \((H=hyperplane\) bundle over \(P^ 3)\) can be identified with those polynomials of 4 complex variables \(z_ 0,z_ 1,w_ 0,w_ 1\) homogeneous of degree 2(N-1) which are spanned by the monomials \(z_ 0^{n_ 0}z_ 1^{n_ 1}w_ 0^{m_ 0}w_ 1^{m_ 1}\), where \(n_ 0+n_ 1=m_ 0+m_ 1.\) The authors also generalize the Segre map S from \(N=2\) to arbitrary N by showing that the association \[ (z_ 0,z_ 1,w_ 0,w_ 1) \to x_{ik}:=z_ 0^{i-1}z_ 1^{N-i}w_ 0^{k-1}\omega_ 1^{N-k}\quad (i,k = 1,2,...,N) \] realizes an imbedding of \(P^ 1\times P^ 1\) into \(P^{N^ 2-1}\) as an algebraic surface defined by…
Mathematics Subject Classification ID: 53D50
Mathematics Subject Classification ID: 70F05
zbMATH DE Number: 4169708
zbMATH Keywords: sections of hyperplane bundles
zbMATH Keywords: geometric quantization
zbMATH Keywords: harmonic oscillators
zbMATH Keywords: Kepler problem
zbMATH Keywords: Segre map
MaRDI profile type: Publication
cites work: On the regularization of the Kepler problem
cites work: COHERENT STATES AND KAHLER MANIFOLDS
cites work: Conformal regularization of the Kepler problem
cites work: Spinor regularization of the \(n\)-dimensional Kepler problem
cites work: Q5834477
description: scientific article; zbMATH DE number 4169708