Department of Mathematics Syllabus

This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page.

MAT 266: Mathematical Statistical Mechanics and Quantum Field Theory
Approved: 2009-05-01, Albert Schwarz

Units/Lecture:

Winter, alternate years; 4 units; lecture/term paper or discussion

Suggested Textbook: (actual textbook varies by instructor; check your instructor)
Quantum Mathematical Physics, Thirring, ($120). Comment: Schwarz will give some handouts from the book, Quantum Field Theory and Topology.
Search by ISBN on Amazon: 3-540-43078-4

Prerequisites:

MAT 265 or consent of instructor. Comment: Formally the knowledge of quantum mechanics will not be necessary for understanding of the course, however, I expect that the students are familiar with the main principles of quantum mechanics.

Course Description:

Mathematical principles of statistical mechanics and quantum field theory. Topics include classical and quantum lattice systems, variational principles, spontaneous symmetry breaking and phase transitions, second quantization and Fock space, and fundamentals of quantum field theory.

Suggested Schedule:


Lectures Sections Topics/Comments
1
Main notions of classical and quantum statistical mechanics (Partition function, observables, correlation functions, entropy, etc.).
2
Harmonic oscillator. Multidimensional harmonic oscillator. Creation and annihilation operators. Representations of canonical commutation relations (of Weyl algebra).
3
Phonons and photons. Electromagnetic field.
4
Fock space. System of identical bosons.
5
Representations of canonical anticommutation relations (of Clifford algebra). Fermionic Fock space. System of identical fermions.
6
Symbols of operators. Functional integral. Integral over anticommuting variables. Perturbation theory.
7
Ideal Bose-gas. Bose-condensation.
8
Ideal Fermi-gas.
9
Phase transitions.
10
Lagrangians of relativistic quantum field theory. Gauge fields. Higgs effect. Standard model of electromagnetic, weak and strong interactions.