UC Santa CruzUC Santa Cruz
Department of Physics
View of the Bay from Campus

Home Page for Physics 222 (Quantum Field Theory III) for the 2020 Spring Quarter

This page contains copies of the class handouts, and other items of interest to the Physics 222 class. This course is being offered remotely during the 2020 spring quarter at the University of California, Santa Cruz.


new!!! Final grades have now been posted. You may now enjoy your well-deserved summer break!

As promised, the final handout that provides the explicit calculation of the O(αs) correction to the γqq vertex that was used in class to derive the one-loop QCD correction to σ(e+e-  → hadrons) has been posted to Section V of this website.

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

A revised version of the class handout, which gives details on the functional integral derivation of the one-loop effective potential, and provides a number of illustrative examples, has been re-posted to Section V of this website.

Slides from the final project presentations have been posted to Section IV of this website.

The corrected slides of Michael Peskin's June 9 lecture on electroweak radiative corrections have been re-posted to Section V of this website.

Table of Contents

[ I. General Information and Syllabus | II. Problem Sets | III. Solutions to Problem Sets | IV. Final presentations of the student projects | V. Other Class Handouts | VI. Free Textbooks and Lecture Notes on Quantum Field Theory | VII. 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 Hours by appointment
e-mail haber@scipp.ucsc.edu
webpage scipp.ucsc.edu/~haber/
zoom by invitation

Class Hours

Lectures: Tuesdays and Thursdays, 4:00--5:35 pm, via zoom

Recommended Textbooks

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 (CRC Press, 2019)

Quantum Field Theory (Second Edition), by Lewis H. Ryder (Cambridge University Press, Cambridge, UK, 1996)

Course Requirements

The requirements of this course consist of four problem sets and a final project. The homework problem sets are not optional. There will be no midterm or final exam. A list of suggested topics for the final project can be found here. Some of the topics require only additional readings in the recommended textbooks. 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 strongly encouraged since they will benefit all members of the class. In choosing your project, you should plan on meeting the following deadlines:

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. Brief review of generating functionals in quantum field theory
  2. Introduction to Regularization and Renormalization
  3. One-Loop Renormalization of Scalar Field Theory
  4. Renormalization of Quantum Electrodynamics
  5. Charge Conservation in QED (and the Ward Identities)
  6. The Renormalization Group
  7. Renormalization of Yang-Mills theory
  8. Quantum Chromodynamics and Asymptotic Freedom
  9. The One-loop Effective Action and Effective Potential
  10. Anomalies in Perturbation Theory

Back to the Top

II. Problem Sets

Problem sets are available in either PDF or Postscript formats

Back to the Top

III. Solutions to Problem Sets

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

Back to the Top

IV. 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.

Back to the Top

V. Other Class Handouts

1. Generating functionals for connected Green functions and one particle irreducible (1PI) Green functions are treated using a purely diagrammatic approach by Predrag Cvitanović in Chapter 2 of his Field Theory lecture notes entitled Generating Functionals.     [PDF]

2. In this handout, the Wick Expansion in the functional integration formalism is given. This involves a formula for the perturbative expansion of the generating functional for the Green functions of a quantum field theory. Two different (equivalent) formulae for the Wick Expansion are given. The second formula, based on Coleman's lemma, allows one to write an elegant formula for the perturbative expansion of the n-point Green function.     [PDF | Postscript]

3. Two of the famous Erice summer school lectures of Sidney Coleman are especially pertinent to this class:

Coleman's lemma and its application to the perturbative expansion of the generating functional for Green functions are discussed in Section 4.3 of the Secret Symmetry lecture.

4. This handout, entitled Generalized Functions for Physics, provides a practical introduction to generalized functions that is particularly useful in mathematical physics applications, focusing on identifying the most common generalized functions employed in physics and the methods for manipulating them.     [PDF | Postscript]
These notes are based on four handouts taken from a previous class:

5. This handout, entitled Analytic formulae for the Feynman propagator in coordinate space, consists of a careful derivation of the expression for the Feynman propagator of scalar field theory, ΔF(x) in coordinate space (in four spacetime dimensions). Two different treatments are given and various subtleties of the calculations are emphasized.     [PDF | Postscript]
These notes are based on the solution to problem 4 of Problem Set 1.

6. 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). In these formulae, the number of spacetime dimensions is a free parameter. 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. 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]

7. In class, two derivations were provided for the Ward identity of QED that relates the vertex function to the inverse fermion propagators. In this handout, I provide a third proof that demonstrates the relationship between the Ward identity of QED and current conservation.     [PDF | Postscript]

8. In this handout, I compute the renormalized 1PI two-point Green function for electrons in QED at one loop order. Renormalization is carried out in the modified minimal subtraction scheme (MS) and the on-shell (OS) schemes, in a general covariant gauge using dimensional regularization. The case of a massless electron is also treated.     [PDF | Postscript]

9. In problem 4 of problem set 2, you were asked to compute the rate for a scalar particle to decay into two photons, in the one-loop approximation. The matrix element for this process can be expressed in terms of an integral, whose explicit form was provided in the problem set without derivation. This integral also arises in the one-loop decay of the Higgs boson to two photon. An explicit derivation of the explicit form for this integral is provided in this class handout, entitled Evaluating the one-loop function arising in H→γγ     [PDF | Postscript]

10. In gauge theories, the gauge fixing term spoils the gauge symmetry. Nevertheless, if the Faddeev-Popov term is included in the Lagrangian density, then the theory exhibits a generalized gauge symmetry called BRST symmetry. This symmetry can be exploited to derive Ward identities and is critical for the renormalizability of the theory. An introduction to the BRST transformation laws is presented in this class handout.     [PDF | Postscript]

11. This handout entitled The Running Mass in QCD derives an explicit expression for the QCD running quark mass in the one-loop approximation.     [PDF | Postscript]

12. In order to regulate potential singularities in one-loop QCD computations due to infrared divergences and mass singularities (corresponding to collinear emission of gluons by a massless quark), one can use dimensional regularization. This requires one to compute phase space integrals in n spacetime dimensions. In this handout, I provide the computation of two-body and three-body phase space integrals in n spacetime dimensions for the case of massless final state particles.     [PDF | Postscript]

13. In the class lectures, I used an expression for the O(αs) correction to the γqq vertex in the derivation of the one-loop QCD correction to σ(e+e-   →   hadrons). This handout provides an explicit calculation of the expression given in class. The computation makes use of the Passarino-Veltman one-loop function that was introduced in Problem Set 3 along with some of its friends.     [PDF | Postscript]

14. In class, two different methods were presented for deriving the one-loop effective potential. There is a third method which is based on the functional integral derivation of the effective action, which is most commonly used in the physics literature today. In this handout, I present the functional integral derivation of the one-loop effective potenital and illustrate its use with a number of examples.     [PDF | Postscript]

15. Michael Peskin has provided a lecture entitled Radiative Corrections to Electroweak Observables. The slides from this lecture are provided here:     [PDF]

Back to the Top

VI. Free Textbooks and Lecture Notes on Quantum Field Theory

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

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

3. Quantum Fields From the Hubble to the Planck Scale, by Michael Kacherlriess (a draft version from 2016), which has since been published by Oxford University Press. For more details, click here.

4. Quantum Field Theory, Lecture Notes by Jan Ambjørn and Jens Lyng Petersen is a draft of a book that does not seem to have been published. Nevertheless, you may find it quite useful.     [PDF]

5. A Stroll Through Quantum Fields, by François Gelis. This appears to be a slightly different version of a book by the same author reccently published by Cambridge University Press     [PDF]

6. 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]

7. 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 222.

8. James Cline recently uploaded to the arXiv his lecture notes on an Advanced Quantum Field Theory course that he teaches. There is a significant overlap between his course and Physics 222. Check out his notes here:     [HTML | PDF]

Back to the Top

VII. Useful articles and reviews

1. In 1990, Fred Jegerlehner gave a series of lectures at the TASI-1990 summer school entitled Renormalizing the Standard Model. His notes (slight updated, with corrections) can be found here:     [PDF]

2. Wolfgang Hollik has also made significant contributions to the renormalization of the electroweak Standard Model.

3. My two lecturse on anomalies are based on Section 6.26 of a review article entitled Two-component spinor techniques and Feynman rules for quantum field theory and supersymmetry by Herbi K. Dreiner, Howard E. Haber and Stephen P. Martin. Although the material given in Section 6.26 employs two-component fermion notation, my lectures in class will employ the more familiar four-component fermion notation. Translating between the two formalisms is given in Appendix G of the review article cited above.     [PDF]

4. Two lectures cannot do justice to the topic of anomalies in quantum field theory. If you are looking for a pedagogical treatment of this topic, have a look at Adel Bilal, Lectures on Anomalies     [HTML | PDF]

5. Adel Bilal also gave a course entitled Introduction to Anomalies in Quantum Field Theory in 2019. Bilal states that "the aim of the course is to give a self-contained, pedagogical introduction to anomalies." Links to videos of five of the lectures can be found on YouTube.

Back to the Top

Last Updated: June 24, 2020