This course is devoted to a discussion of nonpertubative methods in quantum field theory, usually not covered in regular classes on the subject. We will discuss solitons, domain walls, vortices, monopoles, and instantons as well as quantum fluctuations and fermions in their background. Further, we will discuss the relation of instantons to quantum anomalies, their role in the breaking of global symmetries, and the generation of mass gap and confinement. Physical examples will include QCD, QCD-like theories as well as examples relevant to cosmology and condensed matter systems.
Quantum Field Theory I.
Quantum Field Theory II OR familiarity with basics of path integral, renormalization and one-loop calculations.
Grades will be based on 6-8 homework problem sets. Please, submit the homework on the due date. Late homeworks will be penalized 4% per day. No homework will be accepted if is more than a week late.
No textbook is required. I will regularily post lecture notes on this web page. I will heavily depend on the following references:
Advanced Topics in Quantum Field Theory: A Lecture Course ( Cambridge University Press), M. Shifman, 2012.
Classical Solutions in Quantum Field Theory: Solitons and Instantons in High Energy Physics (Cambridge Monographs on Mathematical Physics), E. Weinberg, 2012.
Solitons and Instantons: An Introduction to Solitons and Instantons in Quantum Field Theory (North-Holland Personal Library), R. Rajaraman, 1987.
Aspects of Symmetry: Selected Erice Lectures (Cambridge University Press), S. Coleman, 1988.
Classical Theory of Gauge Fields (Princeton University Press), V. Rubakov, 2002.
Magnetic Monopoles (Springer), Y. Shnir, 2010.
Magnetic Monopoles, Duality, and Supersymmetry arxiv.org/abs/hep-th/9603086 , J. Harvey.
Lectures on Instantons arxiv.org/abs/0802.1862 , S. Vandoren and Peter V. Nieuwenhuizen.
Lecture 1 (Jan 15): Classical solitons in 1+1 d, the kinks
Lecture 2 (Jan 16): Quantization about solitons (1)
Lecture 3 (Jan 22): Quantization about solitons (2) (renormalization)
Lecture 3 (Jan 22): Dirac fermions in the kink background (1)
Lecture 4 (Jan 23): Dirac fermions in the kink background (2)
Lecture 4 (Jan 23): Sine-Gordon model (1)
Lecture 5 (Jan 29): Sine-Gordon model (2) (quantization and duality)
Lecture 6 (Jan 30): Domain walls
Lecture 7 (Feb 5) : Vortices (1) (Global vortices and Derrick's theorem)
Lecture 8 (Feb 6) : Vortices (2) (ANO vortices)
Lecture 9 (Feb 12) : Vortices (3) (ANO vortices)
Lecture 9 (Feb 12) : Magnnetic monopoles (1) (Dirac monopoles)
Lecture 10 (Feb 13) : Magnnetic monopoles (2) (Dirac monopoles)
Lecture 10 (Feb 13) : Review: Non-Abelian gauge theory and Lie-algebra
Lecture 11 (Feb 19) : Phases of gauge theory
Lecture 13 (Feb 26) : 't Hooft-Polyakov monopoles (3) (Moduli space and dyons)
Lecture 14 (Feb 27) : 't Hooft-Polyakov monopoles (4) (non-critical monopoles and singular gauge)
Lecture 14 (Feb 27) : 't Hooft-Polyakov monopoles (5) (SU(N) monopoles)
Lecture 15 (March 5) : Instantons (1) (0+1 D)
Lecture 16 (March 6) : Instantons (2) (0+1 D)
Lecture 18 (March 19) : BPST instantons
Lecture 19 (March 20) : Index theorem (1)
Lecture 20 (March 26) : Index theorem (2)
Lecture 21 (March 27) : One-loop corrections and instanton measure
Lecture 21 (March 27) : Anomalies
Lecture 22 (April 2) : Fermions in the instanton background, and the QCD U(1) problem
Lecture 23 (April 3) : Instantons in the electroweak theory, and baryon and lepton number violations
Lecture 23 (April 3) : Confinement, the Polyakov model of confinement
Homework_1 , Solutions of Homework_1
Homework_2 , Solutions of Homework_2
Homework_3 , Solutions of Homework_3