Physics 523 & 524:  Quantum Field Theory
    M & W 5:30 to 6:45 in room 184,  Instructor:  Kevin Cahill

This web-page is arranged with the material of 523 at the top and that of 524 below it.

    On the first day of class, we will decide collectively what book to use as the principal text.  There is no ideal textbook. 
    Weinberg's book The Quantum Theory of Fields, Vol. I: Foundations (Cambridge University Press 1995, reprinted with corrections 1996, '98, '99, 2000, & '02, ISBN 0-521-55001-7) is excellent, but he derives everything in complete generality and from basic principles.  It is therefore a very hard book and narrow in scope: he deferred nonabelian gauge theory and supersymmetry to volumes II and III.
    Srednicki's book Quantum Field Theory (Cambridge University Press 2007) is broader and somewhat easier to read, but it skips over the elementary topics too quickly and tends to bore down too intensely on some subjects.
An early draft of his book is online.
    Incidentally, pdf's of scans of many books are available from Gigapedia.
    Pierre Ramond's book  Field Theory : A Modern Primer is clear and elegant.
    Mandl and Shaw's book Quantum Field Theory has many advantages but uses the old-fashioned Gupta-Bleuler formalism to quantize the electromagnetic field.
    Warren Siegel's book Fields ( arXiv:hep-th/9912205v3) is the broadest book of all, and it is free.  But it is the most advanced book I know of, and so probably is hard to read.
    The paper  "Introductory Lectures on Quantum Field Theory" by  Alvarez-Gaume and Vasquez-Mozo (arXiv:hep-th/0510040v3) also is free and is worth looking at.
   The book by Peskin and Schroeder An Introduction to Quantum Field Theory (Addison Wesley 1995) is a reasonable compromise.  Errata in their book.  One possibility is for me to follow Peskin and Schroeder, adding in some of the insights available in the other books, especially the ones by Weinberg and Srednicki.  Another possibility is for me to follow Srednicki's book, adding in material from Weinberg, etc.
    We decided to use the book by Peskin and Schroeder;  I will add extra material from Weinberg, Srednicki, Zee, and others, as well as from my book on mathematics for physics graduate students.

    Pages 1-27 of class notes.   Pages 28-34 of class notes.   Extract from my book on the rotation and Lorentz groups with the metric switched to that of P&S.   Notes on Wigner rotations and on how massive particles and states respond to Lorentz transformations as well as on how spinors must be defined if fields are to respond properly to Poincare' transformations with typos corrected.  A few pages from chapter 5 of Weinberg with two typos corrected.   Notes on Schur's lemma.   Notes on qed.  Brief review of the Dirac picture, also called the interaction picture.   Notes on the Feynman propagator.   Notes on Feynman diagrams.  The other eight yards on the quantization of massless vector fields.  Notes on massless particles and fields.  Notes on fermion-boson scattering and the Feynman propagator for spin-one-half particles.   Informal notes on neutrino oscillations.  Notes on the photon spin sum.  Notes on the photon propagator.  Notes on electron-positron scattering, a.k.a. Bhabha scattering, (signs are correct).  Notes on the Feynman rules for QED, on their application to e+e- ==> mu+mu-, on traces of gamma matrices, and on how to get the cross-section from the S-matrix amplitude.  The best treatment of Feynman diagrams is in sections 6.1, 6.3, and 8.6 of Weinberg's QFTI.  Notes on helicity.   Notes on e-e- ==> e-e- scattering.   Notes on Compton scattering.  Notes on functional differentiation and path-integration are in chapters 16 and 17 of my on-line book.   Notes on the path-integral derivation of the Feynman propagator from Coulomb-gauge QED.   Paper on coherent states for fermions.   Notes on Yang-Mills theory.   Notes on fermion-antifermion goes to two gauge bosons.  Weinberg on the Faddeev-Popov method.

Video of first lecture.   Video of second lecture.   Video of third lecture.   Video of fourth lecture.   Video of fifth lecture.   Video of sixth lecture.  Video of seventh lecture.   Video of eighth lecture.    Video of tenth lecture.   Video of eleventh lecture.    Video of twelfth lecture.  Video of thirteenth lecture.    Video of 14th lecture.   Video of 15th lecture.   Video of first 60 or so minutes of 16th lecture.   Video of 17th lecture.   Video of part one and part two of 18th lecture.   Video of 19th lecture.  Video of 20th lecture.  Video of 21st lecture (11/03/10).  Video of 22d lecture (11/08/10).  Video of 23d lecture (11/10/10).   Video of 24th lecture (11/15/10).   Video of 25th lecture (11/17/10).   Video of 26th lecture (11/22/10).  Video of 27th lecture (11/24/10).  Video of 28th lecture (11/29/10).   Video of 29th lecture.   Video of 30th lecture (12/06/10).  Parts 1, 2, and 3 of the 31st lecture (12/08/10).   Video of 32d lecture (12/13/10). 

    First homework assignment:  Do problems 1 and 2 of chapter 2 of P&S.  This assignment is due on Thursday, Sep. 9th.   Put it in Zhang's mailbox or send it to him at  Here is a pdf of chapter 2 of P&S for those who haven't yet gotten a copy.  Solutions to problems 1 and 2 of first homework.

    Second homework assignment due on Wednesday 29 September.  Solutions.

    Extra-credit homework assignment:  Find the other lowest-order amplitude S_2 for boson-antiboson scattering by aping what I did in class.  Due Monday 4 October at the start of class; I will do the problem in class.

    Third homework assignment due on Wednesday 13 October.   Solutions.

    Fourth homework assignment:  For the theory of a neutral boson interacting with a charged fermion, compute the lowest-order S-matrix amplitude for fermion-antifermion goes to two bosons (due Monday 25 October).   Solution.

     Fifth homework assignment:  Compute or write down the lowest-order S-matrix amplitude for the process in which two photons turn into an electron-positron pair (due Monday 1 November).   Solution.

     Sixth homework assignment:  Compute or write down the lowest-order S-matrix amplitude for the process in which two fermions scatter via an SU(2) interaction as described in this assignment.  Due Monday 15 November.  Solution.

    Seventh homework assignment:  Do problems 11-14 of the most recent version of chapter 17 of my book.  Due Wednesday 1 December.   Solutions.  In problem 14, students need not carry out the elaborate discussion of pages 5-8 of the solution; they can just repeat with suitable modifications the solution of problem 13.

    Eighth homework assignment:   For the action density on page 12 of my notes on Yang-Mills theory, show that the action is stationary if the gauge fields satisfy the equation of motion on page 13 of those notes.   Due Monday 6 December or Wednesday the 8th.   Solution.

    The last lecture will be on Monday at 5:30 in room 5 unless we somehow can use room 184.

First and second videos of the first lecture, 19 Jan. 2011, which was a review of gauge theory.
Video of second lecture, 24 Jan., on the Faddeev-Popov tricks and ghosts.
Video of third lecture, 24 Jan., on the Casimir effect.
Video of fourth lecture, 31 Jan., on dimensional regularization.
Video of fifth lecture, 7 Feb., on dimensional regularization.
Video of sixth lecture, 9 Feb., on the Ward-Takahashi identity.
Video of seventh lecture, 14 Feb., on poles in scattering amplitudes.
Video of eighth lecture, 16 Feb., on the Yukawa potential and propagators as a poles in scattering amplitudes.
First and second videos of ninth lecture, 21 Feb., on electromagnetic form factors.
Video of tenth lecture, 23 Feb., on the anomalous magnetic moment of the electron.
Video of eleventh lecture, 28 Feb., on Wilson's formulation of lattice gauge theory.
Video of twelfth lecture, 2 Mar., on Wilson's formulation of lattice gauge theory, the case of Z_2, left-invariant measures on groups, and the Monte Carlo method.
Video of 13th lecture, 21 March, on the renormalization group and the lattice spacing a in SU(N) lattice gauge theory.
Video of 14th lecture, 23 March, on the renormalization group in continuum quantum field theory.
Video of 15th lecture, 28 March, on the renormalization group in continuum quantum field theory and on Wilson's views.
Video of 16th lecture, 30 March, on the renormalization group in condensed-matter physics.
Video of 17th lecture, 4 April, on the renormalization group in condensed-matter physics, spontaneous symmetry breaking, Goldstone's theorem, and the Anderson-Higgs mechanism.
Video of 18th lecture, 6 April, on spontaneous symmetry breaking, Goldstone's theorem, the Anderson-Higgs mechanism, grand unification, and the non-relativistic limit of quantum field theory.
Video of 19th lecture, 11 April, on the renormalization group in condensed-matter physics, quark confinement, superfluidity, the Landau-Ginzburg approach to critical phenomena, and superconductivity.
Video of 20th lecture, 13 April, on the pion as a Nambu-Goldstone boson, and on solitons in two-dimensional space-time.
Video of 21st lecture, 18 April, on differential forms and magnetic monopoles.
Video of 22d lecture, 20 April, on solitons, vortices, and magnetic monopoles.
Video of 23d lecture, 25 April, on vortices, forms in Yang-Mills theory, and magnetic monopoles.   This video is incomplete because the camera ran out of memory.
Video of 24th lecture, 27 April, on instantons and anyons.
Video of 25th lecture, 2 May, on abelian and nonabelian Chern-Simons theory and on Hall fluids.
Video of 26th lecture, 4 May, on the quantum Hall effect.
Video of 27th lecture, 11 May, on electroweak unification and on grand unification.

First homework assignment:  (1)  Show that the area of a unit sphere in d dimensions is 2 pi^(d/2)/Gamma(d/2), which is Weinberg's (11.2.10).  Hint: Compute the integral of exp(-x^2) in d dimensions using both rectangular and spherical coordinates.  (2) Show that the contribution of the counterterm - (Z_3 - 1) F_ab F^ab is Weinberg's (11.2.15).  These equations may be found in the pdf, Weinberg on renormalization via dimensional regularization. 

Second homework assignment:  Do the three problems I discussed in the last class.  The first is to relate Wilson's formula for the action of a plaquette to the continuum action for a gauge theory.  The second is to do problems (9.23) and (9.24) of my online book.  The third (for extra credit) is to simulate the Z_2 gauge theory on a 5^4 (or bigger) lattice following
Creutz's paper (PRD 21 (1980) 1006.
Solutions to the Wilson-loop and invariant-measure problems.

Third homework assignment:  Run Creutz's C code (see below) for Z_2 lattice gauge theory and produce a rgaph showing strong hysteresis.   For extra credit, modify his code and produce a graph showing the coexistence of two phases at the critical coupling beta_t = 0.5* ln(1 + sqrt(2)).    Hint: do a cold start and then 100 updates at beta_t, then do a random start and do 100 updates at beta_t.  Plot the values of the action against the update number 1, 2, 3, ...100.

Fourth homework assignment:  For extra credit, modify Creutz's C code (see below) for Z_2 lattice gauge theory and vary the dimension.   Show that for d=2, there's no transition; for d=3, it's a second-order phase transition; and for d = 4, it's a first-order phase transition.   For extra, extra credit, find out and show what happens for d = 5?

Fifth homework assignment:  Let A = A_j dx^j be a 1-form.  Let F = dA + A^2 be the Maxwell-Yang-Mills 2-form.  Let Q be the trace Tr( A dA + 2A^3/3).  Show that dQ = Tr(F^2).  Hint:  look at page 4 of my notes on the applicattion of differential forms to Yang-Mills theory.

My notes on the Casimir effect.   My notes on the Faddeev-Popov tricks and ghosts.   Weinberg on renormalization via dimensional regularization.  My notes Weinberg's notes on dimensional regularization.  My notes on pi(q^2).  Weinberg on polology.  My notes on the renormalization of scalar and spinor fields.  My notes on symmetry.  My notes on the Ward-Takahashi identity.  My notes on the conservation of momentum and charge.  My notes on poles in scattering amplitudes.  My notes on Yukawa's example.   My notes on the renormalized propagator and self-energies.  My notes on form factors.   My notes on the anomalous magnetic moment of the electron.   My notes on the renormalization group on the lattice.  My notes on the renormalization group in continuum quantum field theory.   My succinct notes on the renormalization group in continuum quantum field theory.   Chapter 18 of my book describes the renormalization group.  My notes on Nambu-Goldstone bosons.   My notes on the Anderson-Higgs effect.   My notes on non-relativistic field theories, on superfluids, on the Landau-Ginzburg theory of critical phenomena, and on superconductivity.  My notes on the pion as a Nambu-Goldstone boson.   My notes on differential forms and magnetic monopoles.   My notes on solitons and on vortices and magnetic monopoles.  My notes on differential forms applied to Yang-Mills theory.  My notes on instantons.  My notes on the Kosterlitz-Thouless phase transition.   Two of my papers on gauge fields and geometry: I and II.   Two of Wilczek's papers on anyons:  I and II.   My notes on Wilczek's anyons.   My notes on Chern-Simons theory.  Zee on electroweak and grand unification.

Michael Creutz's talk on Z_2.   His C code for Z_2 gauge theory.   His makefile.  His C++ code for SU(3) gauge theory.   His papers PhysRevLett.42.1390 and PhysRevD.20.1915.

An article that may be of interest or may be just an effort to attract attention to the Perimeter Institute.
Possible discovery of a new particle with a mass of about 140 GeV at FNAL.