Remarks on the geometric quantization of the Kepler problem
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 the intersection of \((1/2)N^ 4-((7/2)N^ 2+4N-1\) quadrics.
scientific article; zbMATH DE number 4169708