Class Title: Quantum Field Theory II

The QFT II Group:

Session: Fall 2019 (9/3/2019-12/11/2019)
Days & Times: MoWe 2:30PM – 3:45PM
Location: Hasbrouck Laboratory room 136

Instructor: Tigran Sedrakyan
E-mail: tsedrakyan@physics.umass.edu
Office: HAS 403B
Phone:  413-545-2409
Office hours: Fr. 5-6. We can meet also after the lectures or at other times by appointment.

Course Description: This course is a continuation of Physics 811, covering topics in quantum field theory: renormalization of scalar field theory and scalar electrodynamics; the renormalization group; quantization of spin-1/2 fields and quantum electrodynamics; functional methods; and non-abelian gauge theory.

Prerequisite: PHYSICS 811

Reading: 

1. “Quantum Field Theory II” by Mikhail Shifman, World Scientific (2019). https://doi.org/10.1142/10825
2. “Quantum Field Theory and the Standard Model”, by Matthew D. Schwartz, Cambridge University Press (2014).
3. “An Introduction to Quantum Field Theory”, by Michael E. Peskin and Daniel V. Schroeder, Westview Press (1995)
4. “Field Theory: A Modern Primer”, by Pierre Ramond, Addison-Wesley Publishing (1990)
5. “Quantum Field Theory”, by Claude Itzykson and Jean-Bernard Zuber, McGraw-Hill (1980)
6. “Introduction to theory of Quantized Fields” by N. N. Bogoliubov and D. V. Shirkov, Izdatel’stva Nauka (1976)
7.  “Field theory of non-equilibrium systems” by Alex Kamenev, Cambridge University Press (2012)
8. “Field theories of condensed matter systems” by E. Fradkin, Cambridge University Press, Second Edition (2013)
9. “Condensed Matter Field Theory” by A. Altland and B. Simons, Cambridge University Press, Second Edition (2013)
10. “Advanced topics in quantum field theory” by M. Shifman, Cambridge University Press (2012)
11. “Aspects of Chern-Simons Theory” Les-Houches lectures by Gerald V. Dunne.

Grading: Homework solutions will determine the final grade.

For Disability Accommodation and Academic Honesty policy statements see:
Academic Honesty Policy Statement
Disability Statement

Reading — Homework assignments and solutions — Notes

Week 1
Reminder: Classical field theory, Hamiltonian and Lagrangian formulation, Euler-Lagrange equations, real scalar field theory, Klein-Gordon equation. Reading Ref. 2 Chapter 3.  Second quantization, field expansion, Fock space, time dependence, canonical commutators.  Reading Ref. 2 Chapter 2, paragraph 2.3.  Green’s function, time-ordered products, the Feynman propagator. Reading Ref. 2 Chapter 6, paragraph 6.2.  See also Week 1 Lecture notes.

Homework assignment 1 (Due Sept 11)

HW1 Solutions

Week 2
Reminder: Correlation functions in Phi^4 interacting theory. Perturbative expansion of correlation functions of interacting fields. Wick’s theorem. Reading: Ref 2, Chapter 7; Ref 3, Chapter 4.  See also Week 2 Lecture notes. Additional reading for this week: Feynman Diagrams; Cross sections and the S-matrix.

Homework assignment 2 (Due Sept 18)

HW2 Solutions

Week 3
Radiative corrections to the Green’s function: The Källén–Lehmann spectral representation for the time-ordered two-point function, field-strength renormalization.  Reading: Ref 3, Chapter 7.1. Lorentz and translational invariance, unitary representations of the Poincaré group, construction of Lagrangians for particles with single spins: s=0, massive s=1, massless s=1. Reading: Ref 2, Chapters 8.1 through 8.2.4.
 See also Week 3 Lecture notes. Additional reading

Homework assignment 3 (Due Sept 25)

HW3 Solutions

Week 4
Coupling the matter field to the gauge field: gauge invariance and covariant derivatives; scalar QED, gauge symmetries and conserved currents. Quantization and the Ward identities: massive quantum fields with s=1; massless quantum fields with s=1. The photon propagator in covariant gauges, ghosts and longitudinal fields, quantization of complex scalar fields. Reading: Ref 2, Chapters 8.3 through 8.7.1. See also Week 4 Lecture notes.

Homework assignment 4 (Due Oct 2)

HW4 Solutions

Week 5
Scalar Quantum Electrodynamics: Quantizing complex scalar field; Feynman rules for scalar QED; External states; Scattering in scalar QED; Ward identity and gauge invariance; Lorentz invariance and charge conservation. Reading: Ref 2, Chapter 9 Paragraphs 9.1 through 9.5.

Week 6
Spinors: Schroedinger-Pauli equation; Representations of Lorentz group; Group theory: generators of Lorentz group in a 4-vector basis; Lorentz algebra = so(1,3); General representations of the Lorentz group: (1/2,0); (0,1/2); and(1/2;1/2) irreducible representations of Lorentz algebra. Spinor representations: left-handed and right-handed Weyl spinors; unitary representations of the Lorentz group. Reading: Ref 2 Chapter 10, paragraphs 10.1 through 10.2.1.

Week 7
No class on Monday, Oct 14 ( Columbus Day). Instead, the Monday class schedule will be followed on Tuesday, Oct 15.

Lorentz-invariant Lagrangians; Dirac-mass Lagrangian term; Dirac spinor; Dirac Matrices (\gamma matrices); Dirac Lagrangian and Dirac equation of motion. Clifford algebra (of Dirac matrices); Weyl representation of Clifford algebra.  Reducible Dirac (1/2,0)+(0,1/2) representations of the Lorentz group. Majorana representation of Clifford algebra. Lorentz transformation properties, Coupling tothe photon. Reading: Ref 2 Chapter 10 paragraphs 10.2.2 through 10.5.

Week 8
Majorana and Weyl fermions: Majorana masses; Properties of Weyl fermions [Weyl spinor: (\psi_L,\psi_R)]. Reading: Ref 2 Chapter 10 Paragraph 10.6. Spinor solutions and CPT: Chirality, helicity, and spin; left-handed and right-handed [ (1/2,0) and (0,1/2)] representations of the Lorentz group; handedness of a spinor = chirality;  \gamma_5 matrix, projection operators. Reading:  Ref 2, Chapter 11 paragraph 11.1.

Homework assignment 5 (Due Oct 30)

HW5 Solutions

Week 9
Solutions of the Dirac equations; normalization. Majorana spinors, Majorana masses, and Majorana fermions. Charge conjugation, Parity, and Time reversal transformations. Applications to scalar fields, vector fields, and spinors. Symmetries of QED. Reading:  Ref 2, Chapter 11 paragraphs 11,2 through 11.6.1. Additional reading: paragraph 11.6.2.

Homework assignment 6 (Due Nov 6)

HW6 Solutions

Week 10
Spin and statistics: Identical particles, Spin-statistics from path dependence, anyons in 2+1 dimensions, particle exchange and the statistical angle. Quantization of spinors, Lorentz invariance of the S-matrix, spinors, free scalar and spinor fields, Stability and Causality, free fermions, spinors, and higher fields.  Reading:  Ref 2, Chapter 12 paragraphs 12.1 through 12.6.

Week 11
No class on Monday, Nov 11 (Veterans’ Day). Monday schedule will be followed on Wednesday, Nov 13.

Vacuum polarization in QED: scalar and spinor QED. Physics of vacuum polarization: effetive (renormalized) charge and running coupling. Reading: Ref 2, Chapter 16. Renormalization of QED: Mass renormalization, Vacuum expectation values, Electron self-energy, Pole mass, On-shell subtraction, Amputation, Minimal subtraction.  Required reading: Reading: Ref 2, Chapter 18. Renormalized Perturbation Theory: Counterterms, two-point functions, Photon self-energy, Three-point functions, Renormalization conditions in QED. Required reading: Ref 2, Chapter 19. Infrared divergences and Dimensional regularization: Ref 2, Chapter 20.

Week 12
Renormalizability of QED: Four-point functions, Five, six, and higher-point functions. Bogoliubov and Parasiuk, Hepp, and Zimmermann (BPHZ) theorem (AII divergences can be removed by counterterms corresponding to superficially divergent 1PI amplitudes). Non-renormalizable field theories. Divergences in non-renormalizable theories,  examples, and results for non-renormalizable theories. Ref 2, Chapter 21. Renormalization beyond one-loop. Reading: Ref. 3 Chapters 10,11.

Week 13

Thanksgiving recess.

Week 14
Classes canceled on Dec 2 (snowstorm).

Lie groups and representations (reminder). Yang-Mills Theory: Construction of Non-Abelian gauge theories; Fermion (quark) matter; Yukawa couplings. Reading: Ref 1, Chapter 1. Ref 3, Chapter 15.

Week 15
Quantization of Non-Abelian Gauge Theories, Feynman rules for Fermions and Gauge Bosons, Vertices in Yang-Mills with quarks. Quantum Chromodynamics (QCD). Higgs mechanism in Maxwell theory (reminder, see e.g. ref 11). Higgsing in Non-Abelian gauge theories. Weak interactions and the Standard Model.  Reading: Ref 1, Chapters 1 and 2; Ref. 3 Chapter 15, Chapter 15 paragraph 16.1.  Suggested reading: Ref 1, Chapter 16, Ref 2 Chapters 25 through 29, Ref 3. Chapter 16 paragraphs 16.2 through 16.7. Chapter 17.

Homework assignment 7