Class Title: Quantum Field Theory II
The QFTII group:
Session: Fall 2020 (Aug 24, 2020 – Nov 20, 2020)
Days & Times: MoWe 2:30PM – 3:45PM
Location: Fully Remote Class
Join Zoom Meeting 10 minutes in advance using the following link:
https://umass-amherst.zoom.us/j/94893359885
Meeting ID: 948 9335 9885
Instructor: Tigran Sedrakyan
E-mail: tsedrakyan@umass.edu
Office: HAS 403B
Phone: 413-545-2409, skype ID: sedrakyan
Office hours (tentative): Fr. 10:30-11:30. 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:
- “Quantum Field Theory II” by Mikhail Shifman, World Scientific (2019). https://doi.org/10.1142/10825
- “Quantum Field Theory and the Standard Model”, by Matthew D. Schwartz, Cambridge University Press (2014).
- “An Introduction to Quantum Field Theory”, by Michael E. Peskin and Daniel V. Schroeder, Westview Press (1995)
- “Field Theory: A Modern Primer”, by Pierre Ramond, Addison-Wesley Publishing (1990)
- “Quantum Field Theory”, by Claude Itzykson and Jean-Bernard Zuber, McGraw-Hill (1980)
- “Introduction to theory of Quantized Fields” by N. N. Bogoliubov and D. V. Shirkov, Izdatel’stva Nauka (1976)
- “Field theory of non-equilibrium systems” by Alex Kamenev, Cambridge University Press (2012)
- “Field theories of condensed matter systems” by E. Fradkin, Cambridge University Press, Second Edition (2013)
- “Condensed Matter Field Theory” by A. Altland and B. Simons, Cambridge University Press, Second Edition (2013)
- “Advanced topics in quantum field theory” by M. Shifman, Cambridge University Press (2012)
- “Aspects of Chern-Simons Theory” Les-Houches lectures by Gerald V. Dunne.
- Quantum Field Theory Lectures by Jan Ambjorn and Jens Lyng Petersen
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 part 1.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 1 Lecture notes part 2. Additional reading for this week: Feynman Diagrams; Cross sections and the S-matrix.
Homework assignment 1 (Due September 2 by email)
Week 2
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. Radiative corrections to the Green’s function: Additional reading: The Källén–Lehmann spectral representation for the time-ordered two-point function, field-strength renormalization (Ref 3, Chapter 7.1.). Required reading: Ref 2, Chapters 8.1 through 8.2.4. See also Week 3 Lecture notes. 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.Reading: Ref 2, Chapters 8.3, 8.4. See Ref. 12 for Feynman path integrals and functional methods, which may be helpful for homework problems.
Homework assignment 2 (Due September 9)
Week 3
Ward identities continued. 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. Scalar Quantum Electrodynamics: Quantizing complex scalar field; Feynman rules for scalar QED; External states; Scattering in scalar QED; Ward identity and gauge invariance from Feynman diagrams; Lorentz invariance and charge conservation. Reading: Ref 2, Chapter 9 Paragraphs 9.1 through 9.5. Spinors: Schroedinger-Pauli equation; Representations of Lorentz group; Basics of Group theory: generators of Lorentz group in a 4-vector basis. Ref 2 Chapter 10 Read through paragraph 10.1.1. Also, see this week’s lectures one and two in the box folder.
Homework assignment 3 (Due September 16)
Week 4
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. 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. Reading: Ref 2 Chapter 10 paragraphs 10.1.1 through 10.3.1. Also, see this week’s lectures one and two in the box folder.
Homework assignment 4 (Due September 23)*
*Note that problem 3 of HW 4 requires the knowledge of the matrix \Gamma_5, a definition of which you can find in any of the QFT textbooks (and we will discuss this matrix next week).
Week 5
Spinors, Lorentz transformation properties, and Dirac matrices; Coupling of the spinors to the photon. Majorana and Weyl fermions: Majorana masses; Properties of Weyl fermions [Weyl spinor: (\psi_L,\psi_R)]. Spinor solutions of Dirac equation 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. Solutions of the Dirac equations; normalization. Majorana spinors, Majorana masses, and Majorana fermions, Charge conjugation. Reading: Ref 2 Chapter 10, paragraphs 10.3.1, 10.4 10.6. Chapter 11. Paragraphs 11.1 through 11.5. Also, see this week’s lectures one and two in the box folder.
Homework assignment 5 (Due Oct 7)
Week 6
Charge conjugation continued, Parity transformations for scalars, vectors, and spinors. Time reversal transformations (including T-transformation and Wigner’s Time reversal). CPT theorem. Applications to scalar fields, vector fields, and spinors. Symmetries of QED. Reading: Ref 2, Chapter 11 paragraphs 11.5, 11.6. Also, see this week’s lectures one and two in the box folder.
Week 7
Spin and statistics; Identical particles; Spin statistics from path dependence: article exchange and statistical angle; Anyons. Quantizing spinors accounting for statistics;
Lorentz invariance of the S-matrix and particle statistics, Spin-0, Spinors. Stability of QFT and statistics: Free scalar fields, Free fermions, General spins. Causality: Spinor case and higher spins. Reading: Ref 2 Chapter 12, paragraphs 12.1 through 12.6.2. Also, see this week’s lectures in the box folder.
Week 8
Quantum Electrodynamics (QED): QED Feynman rules, taking care of signs in Feynman diagrams. Gamma-matrix identities, electron-positron annihilation with muon pair production (photon exchange diagram), unpolarized scattering, differential cross-section. Rutherford scattering, QED amplitude, corrections to Rutherford’s formula. Reading: Ref 2, Chapter 13, paragraphs 13.1 through 13.5.4. Also, see this week’s lectures in the box folder.
Week 9
Compton scattering, photon polarization sums, matrix element, Klein-Nishina formula, high-energy behavior. Reading: Ref 2, Chapter 13.5 (through 13.5.4), 13.6. Vacuum polarization. Scalar \phi^3 theory and its renormalization. Vacuum polarization in scalar QED. Reading: Ref 2, Chapter 16, Paragraphs 16.1 through 16.2.1. Also, see this week’s lectures in the box folder.
Homework assignment 6 (due Nov. 9). Compton effect: give a detailed derivation of Klein-Nishina formula for differential cross-section and analyze its high-energy behavior (derive Eqs 13.132 and 13.140 of Ref 2).
Week 10
Vacuum polarization in QED: Scalar QED continued. Dimensional regularization, the exact evaluation of integrals in d-dimensions, field dimensions, and \epsilon expansion. Vacuum polarization in spinor QED. Physics of vacuum polarization. Small momentum behavior of vacuum polarization, correction to the Coulomb potential, and Uehling potential. Lamb shift. Large momentum behavior: radiative correction to the potential, charge renormalization, and effective charge. Loop corrections and the Landau pole. Perturbation theory for the photon propagator. Running coupling, the renormalization group equation, and 1-loop \beta function in QED. Reading: Ref 2, Chapter 16, paragraphs 16.2.1 through 16.3.3 including Appendix. Suggested reading: Ref 2, Chapter 17. Also, see this week’s lectures in the box folder.
Week 11
Mass renormalization. Vacuum expectation values of fields in QED, tadpole diagram. Spontaneous symmetry breaking. 1-loop electron self-energy diagram and perturbation theory for electron Green’s function. Calculation of the 1-loop self-energy diagram using Pauli-Villars regularization and dimensional regularization. Renormalization: electron Green’s function. Mass renormalization and the Z-factor.
Diagram summation using 1-particle irreducible self-energy diagrams. Poll mass and on-shell subtraction scheme. Amputated diagrams. Minimal subtraction. Reading: Ref 2: Chapter 18. Renormalized perturbation theory. Couterterms. Two-point functions: Photon self-energy. Three-point functions. Renormalization conditions in QED. Reading: Ref 2: Chapter 19. Suggested reading: Infrared divergencies and 4-point functions in QED. Ref 2: Chapter 20. Also, see this week’s lectures in the box folder.
Week 12
Renormalizability. Renormalizability of QED. Four-point functions. Five, Six, and higher -point correlation functions. Renormalizability to all orders. Bogoliubov-Parsiuk-Hepp-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. Suggested reading: The renormalization group. Ref 2, Chapter 23. 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. Also, see this week’s lectures in the box folder.
Week 13
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. Also, see this week’s lectures in the box folder.
Homework assignment 7 (Due December 9)