Home Page for Physics 218 (Advanced Quantum Field Theory) for the 2016 Winter Quarter

This page contains copies of the class handouts, and other items of interest to the Physics 218 class. This course is being offered during the 2016 winter quarter at the University of California, Santa Cruz.


new!!! All student presentations are now available and have been posted to Section V of this website.

Solutions to Problem Set 4 have been posted to Section IV of this website.

In perturbation theory, only the Lie algebra of the gauge group is relevant. However, in some applications, the global aspects of the gauge group are important. Remarkably, the gauge group of the Standard Model is not quite SU(3)xSU(2)xU(1). For a clarification, see the new links added at the end of Section VIII of this website.

Table of Contents

[ I. General Information and Syllabus | II. Links to the Web Site for the Textbook | III. Problem Sets and Exams | IV. Solutions to Problem Sets and Exams | V. Final presentations of the student projects | VI. Other Class Handouts | VII. Free Textbooks and Lecture Notes on Quantum Field Theory | VIII. Useful articles and reviews]

I. General Information and Syllabus

The General Information and Syllabus handout is available in either PDF or Postscript format     [PDF | Postscript]
Some of the information in this handout is reproduced here.

General Information

Instructor Howard Haber
Office ISB 326
Phone 459-4228
Office Hours Mondays and Tuesdays, 2--3 pm
e-mail haber@scipp.ucsc.edu
webpage scipp.ucsc.edu/~haber/

Class Hours

Lectures: Tuesdays and Thursdays, 12--1:45 pm, ISB 231.
No lectures on Thurs Jan 7, Thurs Jan 21, Thurs Feb 4, Tues Feb 16, Thurs Feb 18 and Thurs March 10.
Make-up lecutres in ISB 231 from 5:15--7:00 pm on Mondays Jan 11, 25, Feb 1, 8, 29, and March 7.

Required Textbook

Quantum Field Theory and the Standard Model , by Matthew D. Schwartz (Cambridge University Press, 2014).

Course Requirements

The requirements of this course consist of problem sets and a final project. Problem sets will be handed out on a regular basis. The homework problem sets are not optional. There will be no midterm or final exam. A list of suggested topics for the final project is provided in the next two pages. Some of the topics require only additional readings in Schwartz. Others will require some consultation with outside sources.

The project may be presented orally or in written form at the end of the academic quarter. Oral presentations are encouraged since they will benefit all members of the class. In choosing your project, you should plan on meeting the following deadlines:

The oral presentations will take place on Wednesday March 16, 2016 from 6--8 pm and on Thursday March 17, 2016 from 12--3 pm in ISB 231.

All projects should include a one page bibliography (containing references pertinent to the project). For those projects presented orally, a digital copy of the powerpoint slides (or equivalent) and a brief set of notes will be acceptable in lieu of a full written version. If an oral presentation is not possible (not the preferred option), a full written version of the project is an acceptable substitute.

Course Syllabus

  1. Path Integral Formulation of Quantum Field Theory
  2. The anomalous magnetic moment of the muon
  3. Implications of Unitarity
  4. Non-Abelian Gauge Theory
  5. Spontaneous Symmetry Breaking and Goldstone's Theorem
  6. Spontaneously Broken Gauge Theories and the Higgs Mechanism
  7. Two-component fermion notation; Dirac and Majorana fermions
  8. The Standard Model of particle physics
  9. Gluon scattering and the spinor-helicity formalism

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II. Links to the Website of the Textbook

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III. Problem Sets

Problem sets and exams are available in either PDF or Postscript formats

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IV. Solutions to Problem Sets

The problem set solutions are available in either PDF or Postscript formats.

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V. Final presentations of the student projects

Students are required to give half hour presentations on a project involving a topic in advanced quantum field theory not treated in the class syllabus. These presentations are collected here.

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VI. Other Class Handouts

1. While our textbook discusses generating functionals for the Green functions of quantum field theory, it does not introduce generating functionals for connected Green functions and one particle irreducible (1PI) Green functions. A purely diagrammatic approach is provided by Predrag Cvitanović in Chapter 2 of his Field Theory lecture notes entitled Generating Functionals.     [PDF]

2. An elementary but detailed introduction to path integral methods in quantum mechanics and quantum field theory is provided in Walter Greiner and Joachim Reinhardt, Field Quantization (Springer-Verlag, Berlin, Germany, 1996).

3. A gaussian functional integral over real Grassmann variables yields a pfaffian. In this handout, I discuss some important properties of an antisymmetric matrix, and then define and explore the properties of the pfaffian. A number of different proofs of the famous result, [pf M]2=det M, are provided including one that makes direct use of the gaussian functional integral over real Grassmann variables.     [PDF | Postscript]

4. In computing one-loop amplitudes in quantum field theory, certain integrals appear. In a handout entitled Useful formulae for computing one-loop integrals, I have provided a collection of some of the most useful formulae used in one-loop computations. I also provide some useful Dirac gamma matrix relations, and some Feynman parameter formulae for combining denominators (an important step needed to obtain the loop integrals in the form given in this handout). The formulae in this handout are actually given in a form where the number of spacetime dimensions is a free parameter (which can be taken to be a continuous variable). This is especially useful in cases where the integral in question is divergent in 4 spacetime dimensions, in which case one can use the procedure of dimensional regularization to deal with the infinities that arise (a procedure that is especially useful in renormalization theory). Of course, if the integral in question is convergent, one can simply use the formulae provided in this handout by taking n=4.     [PDF | Postscript]

5. Appendices D and E of H.K. Dreiner, H.E. Haber and S.P. Martin, Two-component spinor techniques and Feynman rules for quantum field theory and supersymmetry, Physics Reports 494, 1 (2010), provide reviews of matrix decompositions for mass diagonalization in quantum field theory and Lie group theoretical techniques for gauge theories. These two appendices comprise this handout.     [PDF]

6. In a handout entitled Polarization sum for massless spin-one particles, I exhibit explicit expressions for the polarization vectors of a massless spin-1 particle with either positive or negative helicity. I then provide a derivation of the expression for the product of two massless spin-1 polarization vectors summed over the two physical helicity states. These results are contrasted with the corresponding results for a massive spin-1 particle which possesses three possible helicity values.     [PDF | Postscript]

7. In a handout entitled Useful relations involving the generators of SU(N), I provide numerous identities involving the generators in the defining and the adjoint representations of SU(N). These identities are sufficient to work out the color factors for any scattering process involving quarks and gluons. Although the color factors should be computed for the case of N = 3, it is useful to first evaluate the color factors for an SU(N) gauge theory, since the results allow one to identify sets of independent color factors that arise for a given process.     [PDF | Postscript]

8. I am providing a preliminary version of a chapter entitled Gauge Theories and the Standard Model, which will eventually appear in a textbook on supersymmetry by Herbi Dreiner, Howard Haber and Stephen Martin. Your comments and suggestions for improvement are most welcome.     [PDF]

9. The diagonalization of a charged Dirac fermion mass matrix employs the singular value decomposition of a complex matrix, whereas the diagonalization of a neutral Majorana fermion mass matrix employs the Takagi-diagonalization of a complex symmetric matrix. The mathematics of fermion mass diagonalization is treated in a short review article by S.Y. Choi and H.E. Haber.     [PDF | Postscript]

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VII. Free Textbooks and Lecture Notes on Quantum Field Theory

1. Introduction to Quantum Field Theory I, lectures by Horatiu Nastase (June 2012)     [PDF]

2. Predrag Cvitanović, Field Theory (Nordita Classics Illustrated, Copenhagen, Denmark, 1983).     [PDF | webpage]

3. Advanced Quantum Field Theory: Renormalization, Non-Abelian Gauge Theories and Anomalies, lecture notes by Adel Bilal (October, 2014)     [PDF]

4. Diagrammar, by G. `t Hooft and M. Veltman [CERN Yellow Book, CERN-73-09] is an idiosyncratic treatment of quantum field theory. Despite its age, you can still find many useful things in this review.     [PDF]

5. Markus Luty produced some lecture notes for an advanced quantum field theory course that he taught at the University of Maryland in 2007. The following pdf files complement nicely some of the subjects covered in Physics 218.

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VIII. Useful articles and reviews

1. My generation of particle physicists learned about the path integral formulation of quantum field theory and the theory of non-abelian gauge fields from a classic review article by Ernst S. Abers and Benjamin W. Lee, entitled Gauge Theories.     [PDF]

2. The other key set of lecture notes produced in the 1970s that educated a generation of particle physicists were the Erice lectures given by Sidney Coleman. These have been collected in a book entitled Aspects of Symmetry. The lecture of greatest impact, called Secret Symmetry: An Introduction to Spontaneous Symmetry Breakdown and Gauge Fields, still is one of the best introductions to the subject. The section entitled functional integration (vulgarized) provides a very nice elementary treatment. You can also find Coleman's lemma in section 4.3 which provides an alternative starting point for deriving Feynman rules.

3. One of the most insightful treatment of ghosts in quantum field theory appears in lecture notes for the Basko Polje Summer School (1976) by Benny Lautrup entitled Of Ghoulies and Ghosties. These lecture notes are written with much wit and served as an inspiration to the Field Theory book by Predrag Cvitanović. Benny Lautrup has provided a link to the original preprint of his lecture notes here:   [PDF]

4. A comprehensive review of the properties of two-component spinors and their applications in quantum field theory can be found in H.K. Dreiner, H.E. Haber and S.P. Martin, Two-component spinor techniques and Feynman rules for quantum field theory and supersymmetry, Physics Reports 494, 1 (2010). The most recent version of this review (which incorporates the latest known errata) can be found here:   [PDF | Postscript]

5. The helicity formalism for scattering and decay is extremely useful in treating relativistic particles. The formalism was first proposed by M. Jacob and G.C. Wick in a classic paper published in the Annals of Physics 1959. This paper was reprinted by the Annals of Physics in 2000 in order to have a readable copy online. It can be found here:   [PDF]

In 1984, Jeffrey D. Richman published a set of notes entitled An Experimenter’s Guide to the Helicity Formalism, which has served the particle physics community well. These notes can be found here:   [PDF]

In 1993, I a gave a set of lectures at the 21st SLAC Summer Institute on Particle Physics entitled Spin Formalism and Applications to New Physics Searches. These lectures provide a brief introduction to the helicity formalism for relativistic scattering and decay. The written version of these lectures can be found here:   [PDF | Postscript]

There are a number of textbook treatments of the helicity formalism. One such treatment appears in Chapter 13 of V. Devanathan, Angular Momentum Techniques in Quantum Mechanics (Kluwer Academic Publishers, New York, 2002). A copy of this chapter is provided here:   [PDF]

6. The Lie algebra of the Standard Model is the Lie algebra of SU(3)xSU(2)xU(1). But, the Lie group corresponding to this Lie algebra is not unique. The true Lie group of the Standard Model can be determined by requiring that the corresponding group representations are faithful. A mathematical discussion of this point can be found in John C. Baez, The True Internal Symmetry Group of the Standard Model     [PDF | Postscript | HTML]

Given the particle content of the the Standard Model, the Lie group of the Standard Model is in fact SU(3)xSU(2)xU(1)/Z6. Details can be found in:

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Last Updated: June 6, 2016