Quantomorphisms and Quantized Energy Levels for Metaplectic-c Quantization

Metaplectic-c quantization was developed by Robinson and Rawnsley as an alternative to the classical Kostant-Souriau quantization procedure with half-form correction. This thesis extends certain properties of Kostant-Souriau quantization to the metaplectic-c context. We show that the Kostant-Souriau results are replicated or improved upon with metaplectic-c quantization. We consider two topics: quantomorphisms and quantized energy levels. If a symplectic manifold admits a Kostant-Souriau prequantization circle bundle, then its Poisson algebra is realized as the space of infinitesimal quantomorphisms of that circle bundle. We present a definition for a metaplectic-c quantomorphism, and prove that the space of infinitesimal metaplectic-c quantomorphisms exhibits all of the same properties that are seen in the Kostant-Souriau case. Next, given a metaplectic-c prequantized symplectic manifold ( M, ω) and a function H ∈ C ∞(M), we propose a condition under which E, a regular value of H, is a quantized energy level for the system (M, ω, H). We prove that our definition is dynamically invariant: if two functions on M share a regular level set, then the quantization condition over that level set is identical for both functions. We calculate the quantized energy levels for the n-dimensional harmonic oscillator and the hydrogen atom, and obtain the quantum mechanical predictions in both cases. Lastly, we generalize the quantization condition to a level set of a family of Poisson-commuting functions, and show that in the special case of a completely integrable system, it reduces to a Bohr-Sommerfeld condition.

Department dc:contributor.department

Mathematics

Year dc:date.issued

2016

Author dc:creator

• Vaughan, Jennifer Jaye

Advisor dc:contributor.advisor

• Karshon, Yael

Handle dc:identifier.uri

http://hdl.handle.net/1807/77389

OAI identifier oai:identifier

oai:utoronto.scholaris.ca:1807/77389