## DEPARTMENTAL PHD THESIS EXAM – Lindsey Shorser

Wednesday, February 24, 2010,
3:10 p.m. - 4:00 p.m.,
in BA 6183, 40 St. George Street

Ph.D. Candidate:  Lindsey Shorser

Ph.D. Supervisor:  Joe Repka

Thesis Title: Scalar and Vector Coherent State Representations
of Compact and Non-Compact Symplectic Groups in
a Unitary Basis

Thesis Abstract:

When solving problems involving quantum mechanical systems, it is
frequently desirable to find the matrix elements of a unitary
representation T of a real algebraic Lie group G. This amounts to defining
an inner product on the Hilbert space H that carries the representation T.
In the case where the representation is determined by a representation of
a subgroup containing the lowest weight vector of T, this can be achieved
through the coherent state construction. The scalar and vector coherent
state methods simplify the process of finding these overlaps by
introducing the coherent state triplet (F_H, H, F^H) of Bargmann spaces.
Coherent state wave functions - the elements of F_H and of F^H - are used
to define the inner product on H in a way that simplifies the calculation
of the overlaps. This inner product and a particular group action Gamma of
G on F^H are used to formulate expressions for the matrix elements of T
with coefficients from the given subrepresentation.

The process of defining Gamma relies on matrix factorizations in the
complexification of G even though the representation T does not extend to
the complexification. It will be shown that these factorizations are, in
fact, justified, that the overlaps and Gamma action can be expressed in
terms of the given subrepresentation, and that it is possible to find a
numerical values for the inner product in H. The scalar and vector
coherent state methods will both be applied to Sp(n) and Sp(n,R) to
formulate the coherent state representations in U(n) bases for both. The
issue of expressing the result in terms of a U(n)-basis is discussed.

A link to the thesis is here:www.math.toronto.edu/lshorser/LSThesis.pdf