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arXiv:1808.04002 · Axioms

Shifting Operators in Geometric Quantization

Richard H. Cushman, Jędrzej Śniatycki

Published Oct 28, 2020• 0 citations
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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.
DOI
10.3390/axioms9040125

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In the framework of geometric quantization we extend the Bohr-Sommerfeld rules to a full quantization theory which resembles Heisenberg's matrix theory. This extension is possible because Bohr-Sommerfeld rules not only provide an orthogonal basis in the space of quantum states, but also give a lattice structure to this basis. This permits the definition of appropriate shifting operators. As examples, we discuss the 1-dimensional harmonic oscillator and the coadjoint orbits of the rotation group.