Report on the independent study paper

“The Spectral Decomposition and its Use in Finite Dimensions”

By Jaclyn R. Sanders at Budapest Semesters of Mathematics

In the fall semester of 2007 Jaclyn R. Sanders took my course in Mathematical Physics. The course was about the non-classical probability theory behind Quantum Physics. In the first part some necessary mathematical tools (operators on Hilbert spaces and non-distributive probability theory) were introduced, while in the second part of the course the introduced tools were applied to discuss some Quantum Physical “paradox” (such as the EPR paradox) from the mathematical point of view.

In previous semesters Mathematical Physics was about Special Relativity. The new topic, being kept for the first time, lacked printed lecture notes. When Jaclyn asked me to give her something to work on (that could become her independent study), I suggested her the somewhat unusual project of producing a part of the lecture notes. Among the proposed subjects she finally chose the one regarding spectral decompositions of normal operators.

The spectral theorem is essential for understanding (at the conceptual level) why physical quantities in Quantum Physics are described by self-adjoint operators. However, during the course we went through the spectral decomposition in a rather abstract way: due to limited time, we did not consider how to find such decompositions in actual examples, nor we discussed further mathematical applications.

Though the first sections of the paper of Jaclyn are simply summations of what was explained during the course about the subject, she then goes over the presented material in two ways. First, she works out this decomposition in several examples, second, she discusses some nice applications (for example to multi-variable differential equations).

The paper is well written and has very few mistakes. However, this is also due to the numerous rounds of corrections she made with my supervision, which we repeated until the level of a lecture notes was achieved. In the beginning, Jaclyn lacked the necessary mathematical bases, and she needed to work hard to catch up and fill her gaps. While appreciating this, in the final grade I also had to take account of the fact that to work out the examples and applications, Jaclyn needed much more help than what is desirable for an independent study. Thus I award the grade B.

Budapest, 25/02/2008,

Mihály Weiner

Advisor

Report on the independent study paper

Guest

2007-06-15T02:46:00