## Departmental PhD Thesis Exam – Jennifer Vaughan

Everyone is welcome to attend.  Refreshments will be served in the Math Lounge before the exam.

Monday, May 9, 2016
11:10 a.m.
BA6183

PhD Candidate:  Jennifer Vaughan
Co-Supervisors:  Yael Karshon
Thesis title:  Quantomorphisms and Quantized Energy Levels for Metaplectic-c Quantization

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Abstract:

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 $(M,\omega)$ and a function $H\in C^\infty(M)$, we propose a condition under which $E$, a regular value of $H$, is a quantized energy level for the system $(M,\omega,H)$.  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.

The draft to the thesis can be found here: Vaughan-Draft