Abstract
The original Bohr-Sommerfeld theory of quantization did not give operators of transitions between quantum quantum states. This paper derives these operators, using the first principles of geometric quantization.
Highlights
Even though the Bohr–Sommerfeld theory was very successful in predicting some physical results, it was never accepted by physicists as a valid quantum theory in the same class as the Schrödinger theory or the Bargmann–Fock theory
The reason for this was that the original Bohr–Sommerfeld theory did not provide operators of transition between quantum states
The first step of geometric quantization of a symplectic manifold ( P, ω ) is called prequantization. It consists of the construction of a complex line bundle π : L → P with connection whose curvature form satisfies a prequantization condition relating it to the symplectic form ω
Summary
Even though the Bohr–Sommerfeld theory was very successful in predicting some physical results, it was never accepted by physicists as a valid quantum theory in the same class as the Schrödinger theory or the Bargmann–Fock theory. The aim of this paper is to derive operators of transition between quantum states in the Bohr–Sommerfeld theory, which we call shifting operators, from the first principles of geometric quantization. We derive shifting operators in the framework of geometric quantization, and extend our result to cases with a variable rank polarization. The main singularity encountered here corresponds to the fact that the polarization F spanned by the Hamiltonian vector fields of a completely integrable system does not have constant rank. This singularity is so well known that we do not have to use the language of differential spaces to get results. Experts may omit this section and proceed directly to the section on Bohr–Sommerfeld theory
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