Details

Autor(en) / Beteiligte

• Bradshaw, Zachary P

Titel

Explorations in Mathematical Physics: Special Functions in Quantum Theory and Feynman Integrals by the Method of Brackets

Ort / Verlag

ProQuest Dissertations Publishing

Erscheinungsjahr

2023

Link zum Volltext

• ProQuest

Quelle

ProQuest Dissertations & Theses A&I

Beschreibungen/Notizen

• In the first part of this thesis, Ramanujan’s master theorem and the method of brackets are dealt with. The method of brackets has not been given a rigorous treatment, and this problem is the first to be examined. A natural class of functions is given for which the method holds by providing a proof of the first two formerly heuristic rules. The remaining rule deals with the integration of series with fewer indices than the dimension of the integral being considered. In this situation, the method commonly produces series which diverge, and it asserts that these series should be discarded, while series which converge in a common region should be summed to produce a representation of the relevant integral. An operational extension of Ramanujan’s master theorem which transforms the evaluation of an integral to the computation of a Laplace transform is produced (joint work with C Vignat). Moreover, Ramanujan-like master theorems are obtained by adjusting Hardy’s proof. The method employed to accomplish this can be used to formally obtain a similar result for any meromorphic function with poles at the non positive integers. The method of brackets portion of the thesis comes to a close with an application of the method to the evaluation of Feynman diagrams. In the second part of the thesis, a one-to-one correspondence between finite groups and some quantum separability algorithms is given. In doing so, one shows that the acceptance probabilities of the constructed algorithms are given by the cycle index polynomial of the corresponding group. This is joint work with M. LaBorde and M. Wilde. These concepts are then linked with the zeta function of a density matrix (joint with M. LaBorde). A generalization of Dirac’s factorization procedure is discussed in the final chapter. The resulting coefficients are shown to exhibit the structure of a generalized Clifford algebra, motivating the move to a more general setting. The archetypal example in this setting is a partial differential equation with non-constant coefficients. This is ongoing joint work with students mentored by the author in 2022, including E. Albertin, K. Kirt, K. Long, and A. Nguyen.