Shifting Operators in Geometric Quantization

Cushman, Richard; Śniatycki, Jędrzej

doi:10.3390/axioms9040125

Cited by 1 publication

(1 citation statement)

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“…Thus, Dirac's quantization conditions allow us to interpret the lowering operators as quantizations of e −iϕ 1 and e −iϕ 2 only in the complement of L −1 (0) in E M −1 (R), where R is the set of regular values of the energy-momentum map. In [14] the Bohr-Sommerfeld quantization of a completely integrable Hamiltonian system is extended to a full quantum theory by constructing shifting operators, which give transitions between the quantum states.…”

Section: Introductionmentioning

confidence: 99%

Classical and Quantum Spherical Pendulum

Cushman

Śniatycki

2022

Symmetry

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The seminal paper by Niels Bohr followed by a paper by Arnold Sommerfeld led to a revolutionary Bohr–Sommerfeld theory of atomic spectra. We are interested in the information about the structure of quantum mechanics encoded in this theory. In particular, we want to extend Bohr–Sommerfeld theory to a full quantum theory of completely integrable Hamiltonian systems, which is compatible with geometric quantization. In the general case, we use geometric quantization to prove analogues of the Bohr–Sommerfeld quantization conditions for the prequantum operators Pf. If a prequantum operator Pf satisfies the Bohr–Sommerfeld conditions and if it restricts to a directly quantized operator Qf in the representation corresponding to the polarization F, then Qf also satisfies the Bohr–Sommerfeld conditions. The proof that the quantum spherical pendulum is a quantum system of the type we are looking for requires a new treatment of the classical action functions and their properties, which is given in Section 2.4 and 2.5. For the sake of completeness we have provided an extensive presentation of the classical spherical pendulum. In our approach to Bohr–Sommerfeld theory, which we call Bohr–Sommerfeld–Heisenberg quantization, we define shifting operators that provide transitions between different quantum states. Moreover, we relate these shifting operators to quantization of functions on the phase space of the theory. We use Bohr–Sommerfeld–Heisenberg theory to study the properties of the quantum spherical pendulum, in particular, the boundary conditions for the shifting operators and quantum monodromy.

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Section: Introductionmentioning

confidence: 99%