Course Information, Math 541, Fall 2023, TuTh 11-12:15

Instructor: Doug Pickrell

Office: Mathematics 703

Office Phone: 621-4767


Office Hours: TuTh 12:30-1:30

Text: Leon Takhtajan, Lectures on Quantum Mechanics for Mathematics Students. Other recommended books: Mathematical Methods of Classical Mechanics by V.I. Arnold, Quantum Mechanics for Mathematicians, Brian Hall, Lectures on Quantum Mechanics for Mathematics Students, L.D. Faddeev and O.A. Yakubovskii

About the course: This will be a basic course in mathematical physics. For math students my aim is to provide a sufficient foundation so that most of the talks in the (probability/)mathematical physics seminar make sense. For physics students, for whom the topics of the course will be familiar, my aim is to more fully explain the geometry that underlies classical mechanics, the rationale for Gibbs approach to statistical mechanics, and so on. By the end of the course, I hope the math students will appreciate how the physical perspective illuminates basic concepts from mathematics, and I hope the physics students will more deeply understand the mathematical underpinnings of their subject. As a rough outline, I will be following the book Quantum Mechanics for Mathematicians by Takhtajan, which is available online from the library. This book is quite sophisticated from a mathematical point of view, and I will fill in a significant amount of the background which the author assumes as we go along. There will be three or four parts to the course, which will be somewhat independent (general ideas, but not full details, from one part may be useful for other parts). The first part of the course (chapter 1 of the book) will be a geometric introduction to Lagrangian and Hamiltonian classical mechanics (Hermann Weyl referred to the underlying geometry as symplectic geometry, which is somewhat analogous to Riemannian geometry). This will involve some familiarity with tensors, especially differential forms, which we will review. I am hoping that physics students will have seen this before, in some form, so that the level of sophistication in the book is not too much of a shock. We may briefly mention some basic examples of field theories, which fit into this framework, but involve an infinite number of degrees of freedom. The second part of the course (not in the book) will be a brief introduction to statistical mechanics, i.e. classical systems with a large number of degrees of freedom. This will entail a review of some basic notions from probability theory, which may be unfamiliar to both the math and physics students. There are a number of important notions which we will discuss: the renormalization (semi-)group, universality, and so on. The third part will introduce quantum mechanics (chapters 2 and 3 in the book). From a mathematical point of view, this involves the spectral theorem (for unbounded operators), which we will do from scratch. If all goes well, students will present a number of standard examples: the harmonic oscillator, energy levels of the hydrogen atom, one dimensional scattering theory,... In the fourth part, assuming there is sufficient time, I will try to explain how (Euclidean) quantum field theory can be viewed as a continuum limit of statistical mechanics (at a critical parameter), or to put it another way, how quantum field theories can be viewed as `central limit theorems'. This fourth part will serve as an introduction to a topics course on statistical mechanics and quantum field theory which I will propose for Spring 2024.

Expectations: I will assign homework about every two weeks, and send out some solutions. I hope to recruit participants in the class to present some of the applications of the material in the context of the course. There will not be any exams.

Grades: To earn an A in the course, a graduate student will be expected to demonstrate understanding of the material at a graduate level. This can be done in multiple ways: by doing the homework, by doing a project, by giving a presentation (to the class, or to me),...

Attendance: Students are expected to attend every scheduled class. • The UA’s policy concerning Class Attendance, Participation, and Administrative Drops is available at: • The UA’s policy regarding absences for any sincerely held religious belief, observance or practice will be accommodated where reasonable. See: • Absences pre-approved by the UA Dean of Students (or Dean Designee) will be honored. See:

Classroom Behavior: To foster a positive learning environment, students and instructors have a shared responsibility. We want a safe, welcoming, and inclusive environment where all of us feel comfortable with each other and where we can challenge ourselves to succeed. To that end, our focus is on the tasks at hand and not on extraneous activities (texting, chatting, reading a newspaper, making phone calls, web surfing).

Communication: It is the student’s responsibility to keep informed of any announcements, syllabus adjustments or policy changes made during scheduled classes, by email.

Students with disabilities: Our goal in this classroom is that learning experiences be as accessible as possible. If you anticipate or experience physical or academic barriers based on disability, please let me know immediately so that we can discuss options. You are also welcome to contact the Disability Resource Center (520-621-3268) to establish reasonable accommodations. For additional information on the Disability Resource Center and reasonable accommodations, please visit If you have reasonable accommodations, please plan to meet with me by appointment or during office hours to discuss accommodations and how my course requirements and activities may impact your ability to fully participate. Please be aware that the accessible table and chairs in this room should remain available for students who find that standard classroom seating is not usable.

Students withdrawing from the course: Must be made in accordance with University policy

Incompletes: Must be made in accordance with University policies, which are available at

University Policies: • The UA Threatening Behavior by Students Policy prohibits threats of physical harm to any member of the University community, including to oneself. See • Students are encouraged to share intellectual views and discuss freely the principles and applications of course materials. However, graded work/exercises must be the product of independent effort unless otherwise instructed. Students are expected to adhere to the UA Code of Academic Integrity as described in the UA General Catalog. See: • The University is committed to creating and maintaining an environment free of discrimination; see

Note: Information contained in the course syllabus, other than the grade and absence policy, may be subject to change with advance notice, as deemed appropriate by the instructor.