PHY5667: Quantum Field Theory A (Fall 2013)
- My temporary office during the Keen bldg construction: IMB 401 (across the bridge from Keen). Also new office hours. (Posted on Oct 17, 2013)
- Office hours changed. (Posted on Sep 23, 2013)
- Lecture notes now available (to be irregularly updated). (Posted on Aug 30, 2013)
- Welcome! (Posted on Aug 26, 2013)
- Instructor: Takemichi Okui (email)
- Time: Tuesdays & Thursdays, 11:00–12:15
- Location: UPL 107
- Teaching Assistant: Sam Bein (email)
- Office Hours: Mondays and Fridays, 12:30–13:30
Pre-requisites, Co-requisites, and Recommendations
- Pre-requisites: Theoretical Dynamics, Quantum Mechanics A&B, Electrodynamics A&B
- Strongly Recommended: Special/General Relativity, Quantum Manybody Physics
- De facto standard:
- An Introduction to Quantum Field Theory, by M. E. Peskin & D. V. Schroeder
- To learn more:
- The Quantum Theory of Fields, Vols. I & II, by S. Weinberg
- Quantum Field Theory, by M. Srednicki
- Modern Quantum Field Theory, by T. Banks
- For bedtime reading:
- Quantum Field Theory in a Nutshell, by A. Zee
- Theory of Fundamental Processes, by R. P. Feynman
- Quantum Electrodynamics, by R. P. Feynman
- Grade Breakdown
- 90% Homework
- 10% Class Participation
- A tentitive guideline for the grades (subject to adjustment based on final grade distribution (only to benefit you)):
- A: ≥ 95%
- A−: ≥ 90%
- B+: ≥ 80%
- B: ≥ 75%
- B−: ≥ 70%
- C+: ≥ 60%
- C: ≥ 55%
- C−: ≥ 50%
- D/F: < 50%
- It will be assigned every Thursday, due on the following Tuesday, 11:00, collected at the beginning of the lecture.
(Exception to this pattern will apply to the very first and last HW sets.)
- (Important!) You are permitted and even encouraged to discuss homework problems with your classmates, friends, and colleagues,
but your solutions must be entirely written by yourself in your own words, based on your own understanding.
If solutions by two or more people are found suspiciously too similar, they will get zero points, and more serious consequences may follow.
To avoid unintended troubles, never show any part of your homework draft to others.
Instead, use blackboard or separate sheet of paper for discussions. You should then be fine—two people never write similarly.
You also don't want to copy solutions from books, the internet, or past solution sets, etc., either.
- HW will be graded on a 5-point scale:
- 5: Essentially perfect, i.e., no conceptual mistakes at all, and no or only a very few trivial math errors.
- 4: Certainly good, but contains a minor conceptual mistake, and/or a worse than trivial math error, and/or more than a few trivial math errors.
- 3: Sort of okay, but contains a major conceptual mistake, and/or more than a few minor conceptual errors, and/or math is rather sloppy all over the place. Work harder!
- 2: Not good. Many serious mistakes. You may be in a big trouble. We'll see how you do in the next few assignments.
- 1: Seriously? You may want to drop the course.
- 0: Looks like you looked at someone else's work (or let someone look at yours).
- Each day the HW is late, one point will be deducted.
- Your final grade will be based on your top-12 HW scores out of the total of 14 assignments (including HW0).
This is to account for your absence due to illness, research, conference, etc.
- Problem sets:
Just sitting in the classroom is not sufficient. You have to actively participate in the class by asking questions, making comments, pointing out my errors, etc.
Lecture notes: (to be irregularly updated)
- How to do QFT—Feynman rules and diagrams
- What does QFT describe? —Particles.
- What does QFT calculate? —Amplitudes.
- How? —Draw diagrams.
- Particles = Lines
- Charge arrow — Particle vs antiparticle
- Momentum arrow
- Interactions = Dots (aka "vertices")
- Spin indices
- 4-momentum conservation
- Initial states and final states = Lines with a dot only on one end (aka "wavefunctions")
- Initial vs final —Positive energy going in or coming out?
- Particle vs antiparticle — Is direction of positive energy the same as or opposite to charge arrow?
- Describing spins
- Intermediate states = Lines with dots on both ends (aka "propagators")
- On-shell vs off-shell momenta
- Drawing diagrams and computing amplitudes = Connecting all lines
- Respect charge arrows
- Fully connect all lines and vertices
- Amputate external legs
- Only connections matter
- All spin indices must be contracted. Contractions are always "local"
- 4-momentum conservation at every vertex. Watch for momentum arrows
- Integrate over each undetermined 4-momentum
- Symmetry factors and fermionic minus signs
- Origin of Feynman rules and diagrams
- Poincaré invariance + QM → Particles
= states labelled by 4-momentum and by angular momentum in rest frame (aka "spin", aka "polarization")
- Describing spins of initial and final particles = wavefunctions
- εμ(p,s) as a Lorentz-covariant description of "spin-1 in rest frame"
- Lorentz-covariant constraints on εμ(p,s)
- Final-state polarization vector from analytic continuation of initial-state εμ(p,s)
- Representations of Lorentz algebra
- Left-handed and right-handed spinors
- uL(p,s) and uR(p,s) as a Lorentz-covariant description of "spin-1/2 in rest frame"
- Dirac equation as a Lorentz-covariant constriant on uL(p,s) and uR(p,s)
- Dirac spinors and Gamma matrices
- v(p,s) from analytic continuation of u(p,s)
- Charge conjugation matrix as "metric" of spinor space
- u(p,s) and v(p,s) as uT(-p,-s) and vT(-p,-s) times "metric"
- Additional quantum numbers — Charge arrows and existence of antiparticles
- In-states, out-states, and S-matrix
- Unitarity of S-matrix: Completeness of out-states (or in-states) → S†S=1 (or SS†=1)
- Normalization conventions for one-particle states and multiparticle states
- Symmetries and conservation laws
- Spacetime translation symmetry and 4-momentum conservation
- Decomposition of S-matrix into "connected parts", definition of amplitudes iMfi
- Internal symmetries and charges
- Inputs to S-matrix perturbation theory: Zeroth order and first order
- Inputs to theory (Zeroth order): Symmetries and list of particles
- Inputs to theory (first order): 1st-order connected S-matrix elements (aka "vertices")
- 1st-order unitarity → vertices are "real" ("hermiticity" of vertices)
- Using Lorentz invariance and analyticity to constrain vertices
- Relevant, marginal, and irrelevant couplings
- Dimensional analysis for amplitudes and couplings
- Irrelevant couplings are irrelevant at low energy
- Outputs of S-matrix perturbation theory: Propagators
- Propagator for spin-0
- Sum over intermediate states → Propagator + c.c.
- Positive energy principle picks "+iε"
- Terms undetermined by unitarity → Can be absorbed into vertices
- Propagators for particles with spin
- Spin-Statistics Relation
- Bosons and fermions — Can two particles be in exactly the same state?
- Interacting spin-0 particles must be bosons
- Interacting spin-1/2 particles must be fermions
- Interacting spin-1 particles must be bosons
- CPT Theorem
- "PT=C" for initial/final-state wavefunctions
- Complexified Lorentz transformations
- Analytic continuation & complexified Lorentz invariance → "PT=C" for whole amplitudes
- Loop diagrams
- Origin of loops
- Amputation of externa legs
- Symmetry factors
- (-1) for a spin-1/2 loop
Gauge theories — Profound implications of Ward identity
- Massless spin-0 — No subtleties, no reduction of degrees of freedom
- Massless spin-1/2 — Can have only half of degrees of freedom
- Wavefunctions split in two
- Propagator also splits in two
- Massless spin-1 — Must have reduction of degrees of freedom
- Must satisfy Ward inidentities
- Must have only two degrees of freedom
- Propagator of massless spin-1
The standard model of particle physics — Gauge theory of Nature
- A 3-point vertex of massless spin-1 bosons — Defines "fabc"
- WI dictates this fabc must be the "fabc" of a Lie algebra
- WI determines 4-point vertex of these massless spin-1 bosons
- A 3-point vertex between a massless spin-1 and two spin-0 particles — Defines "Ta"
- WI dictates this Ta must satisfy the same Lie algebra
- WI dictates these spin-0 particles must be degenerate in mass
- WI determines a 4-point coupling between two of these spin-1 and two of these spin-0 particles
- A 3-point vertex between a massless spin-1 and two spin-1/2 particles — Defines "Ta"
- WI dictates this Ta must satisfy the same Lie algebra
- WI dictates these spin-1/2 particles must be degenerate in mass
- WI allows a γμγ5 coupling only if these spin-1/2 particles are massless
- Lie algebra
- U(1), SU(N), SO(N), Sp(2N) algebras
- Fundamental representaions
- Adjoint representations
- The standard model without masses
- Particle content
- Three Lie algebras
- Where is electric charge?
- Where is photon?
- Introcucing masses and the Higgs boson
- Data 1: Fermions' interactions — Exactly as expected from WI for W±/Z, in particular γμγ5 couplings.
- Data 2: Fermions have masses → γμγ5 couplings actually not allowed by WI!
- Data 3: W±/Z have masses → Actually no WI in the first place, so Data 2 is not a problem.
- Could still explain Data 1 by applying WI at high energy where masses can be ignored, if we are allowed to go high energy.
- Are we? — No, W±/Z couplings are secretly irrelevant!
- The Higgs boson — the simplest way to render W±/Z couplings marginal.