Physics 230A - Quantum Field Theory I
shortcut to Homework Assignments
and a shortcut to Miscellaneous References
Tue and Thu, 11:10 - 12:30, 430 Birge Hall
Discussions: Fri, 2:10-3:00, 75 Evans, for the time being. At some point
later in the semester, we may move the discussions to the Oppenheimer room.
Instructor: Petr Hořava
Office: 441 Birge (usually on Tue, Wed afternoon, and Thu);
50A-5107 LBNL (usually on Mon, Wed morning, and Fri)
The course is designed as a logical continuation of the material covered in
229A, with three major themes: the path integral method, renormalization in
quantum field theory, and quantization of Yang-Mills theories. My intention
is to keep some balance between the techniques of and conceptual insight into
(both perturbative and non-perturbative) quantum field theory in general.
In this course, we will primarily use the excellent textbook by Tony Zee:
Quantum Field Theory in a Nutshell (Princeton U. Press, 2003)
with occasional supplements from texts such as
M.E. Peskin & D.V. Schroeder, An Introduction to Quantum Field
S. Weinberg, The Quantum Theory of Fields I - III,
J. Zinn-Justin, Quantum Field Theory and Critical Phenomena,
will be posted weekly on this website, every Thursday before (or around :-)
2 pm. The homeworks will be due in seven days, on Thursday in class.
Once due, they will be graded that same day on a crude scale (1 for reasonable
effort, 0 for not turning the homework in or for not exhibiting sufficient
effort). Then they will be returned to the class and discussed in the
discussion session on Friday, eight days after the original assignment.
(Uncollected homeworks can be picked up later from the envelope next to my
Unless stated otherwise, the problems are from Tony Zee's book.
Week 1: Problems I.2.1, I.2.2, I.3.1 and I.3.2. In addition, one
problem which is not from Zee's book is
here (as a pdf
file). And, in addition to that, an Opening Quiz has been handed out on
Tuesday 1/24 in class. This Quiz is also due in class on Thu, Jan 26.
Week 2: Three problems for this week can be found
here (as a pdf
file) (due Thu, Feb 2).
Week 3: Problems I.4.1 and I.5.1 (due Thu, Feb 9). In addition, read
Chapter 1 of S. Weinberg's book The Quantum Theory of Fields (Vol. 1),
which gives a fascinating historical account of the early days of quantum
Week 4: Problems I.7.1, I.7.2, I.7.3, I.7.4, I.8.1, I.8.3 and I.8.4.
In addition, try to identify all the Feynman diagrams that Zee missed in
Figures I.7.2 and I.7.3, and check your result against the
errata. This homework is due on Thu, Feb 16 as usual; however, the
discussion session on Fri, Feb 17 is cancelled, and the homework will instead
be discussed in a make-up discussion on Tue, Feb 21 at 5pm in the Oppenheimer
room (4th floor Birge).
Week 5: Problems I.9.2, I.9.3, I.9.4, IV.1.1, IV.1.2, II.1.1, II.1.2,
II.1.9 (due Thu, Feb 23). Also, read Problem I.9.1. (The three problems from
Chapter II are a brief reminder of things covered in 229A.)
Week 6: Problems II.3.4, II.4.1, III.1.1 and III.1.3. In addition,
make sure that you understand how to calculate the constant C that
appears in Eqn. (III.1.2); for this calculation, you may find Appendix 1
to Chapter III.1 particularly useful.
Week 7: Problems III.3.3, III.3.4, IV.3.1, IV.3.3, IV.3.4 and
Week 8: Problem III.5.2 (courtesy of Max :-), and Problems
V.1.1 and V.6.1.
In addition, work through the calculations in the Appendix of Chapter V.1.
Week 9: Problems V.6.2, V.7.4 and V.7.12. In addition
(courtesy of I-Sheng :-), read Chapter V.5 and, using the results of
Problem IV.3.3 from Week 7, rederive the consequences discussed below
Eqn.(V.5.4) on page 275.
Week 10: There is no homework assigned this week, so that people can
enjoy their Spring break :-) Wherever you are going during that time,
however, I would still recommend that you take Zee's book with you, and
re-read the parts of the book that have been covered in our lectures so
far. In your spare time, you can also read
Harvey's lectures on
monopoles and duality as recommended below.
(Various types of dualities in quantum field theories will be the main focus
of Physics 230B,
which I am scheduled to teach this Fall as a continuation of this semester
Week 11: Spring break - no homework.
Week 12: Problems IV.5.1, IV.5.2, IV.5.3 and IV.5.4. (As a bonus,
you can also try IV.5.5.) This homework is due Thu, April 13 in class.
There is no discussion on Fri, April 7.
Week 13: This week, no problems from Zee will be assigned.
Instead, there are three problems specified below. The material covered
in class this week corresponds roughly to Chapter III.4 and (the first four
pages of) Chapter VII.1. Unfortunately, Zee does not discuss BRST symmetry;
a good discussion of BRST can be found in Chapter 4.2 of the first volume
of Polchinski's String Theory book, or in the beginning chapters of Weinberg's
second volume. (For more references on BRST, see also the list in
Miscellaneous References below.)
Here are this week's three problems:
Problem 1: One possible gauge fixing condition in non-Abelian
Yang-Mills theory simply sets the third spatial component of the gauge field
to zero. Show that the Faddeev-Popov determinant for this gauge choice is
independent of the gauge field, and explain why.
Problem 2: Show that the BRST charge Q (as defined in class)
squares to zero.
Problem 3: Consider free electromagnetism (i.e., pure U(1)
gauge theory with no matter fields) in four spacetime
dimensions. Introduce the Faddeev-Popov ghost and antighost associated with
(say) the Lorentz gauge fixing condition. The BRST cohomology on the
full (Fock) Hilbert space of the gauge fields and ghosts is defined as the
set of states annihilated by Q, modulo adding states that are
themselves Q of some other states. Show that the BRST cohomology
conditions on this Hilbert space reproduce the usual two physical
polarizations of the electromagnetic field. [Hint: Work with the creation
and annihilation operators of the gauge field and the ghost fields, in
Week 14: No homework will be assigned this week. Consequently,
there will be no official discussion on Friday, April 28 in Evans Hall.
Instead of that, I will be available on April 28 from 1:30 to 3:30 in my
office in Birge Hall to answer individual questions about the final grading
for this course.
In the remaining weeks of this semester, two more homeworks will be assigned.
Based on the homework performance so far, most students will qualify for a
grade ranging from A+ to A-, and will have their final written take-home exam
waived. If you have missed (or anticipate missing) more than one homework,
or are concerned about your grade for whatever reason, you should come to my
office on April 28 in the time window specified above, and we can discuss the
possibility of improving your grade by taking the final take-home exam.
Week 15: Problems VI.1.1, VI.1.2, VI.1.4, and VI.1.7 (due Thu,
May 4). On Fri April 28, I will be available in my office from 1:30 to 3:30
to discuss any aspects of the class. Students who wish to sign up for a
formal appointment in that time window (instead of just showing up) can do
so on a sign-up sheet posted on the board next to my office 441 Birge.
Week 16: Problems VI.6.2, VI.8.3, VII.4.1, VII.4.4, and VII.4.6
(due Thu, May 11, by 2pm in my office 441 Birge). (For a solution of
VII.4.6, see the reference quoted at the bottom of the "Miscellaneous
As the semester progresses, I will post various useful references here,
in response to questions from the class, or as extra material outside of the
main focus of the lectures.
Week 1: This is in response to questions in the discussion on Fri.
A good review of the (non)summability of the weak-coupling expansion
(and the accuracy of the asymptotic series) in QM and QFT is
J. Zinn-Justin, Phys. Rep. 70 (1981) 109.
The nature of nonperturbative effects and their relation to the asymptotic
growth of the terms in the weak-coupling expansion are discussed (and
contrasted) for string and field theory in
S. Shenker, The Strength of Nonperturbative Effects in String Theory.
(This is a paper written at Rutgers in 1990, but not published in a journal
- the original can be found for example
Week 2: It is worth pointing out that Zee's book has an
extensive list of errata
Week 3: This is in response to a question raised in class. Some recent
references which discuss the current experimental bounds on (large) extra
I. Antoniadis, The
Physics of Extra Dimensions (hep-ph/0512182),
G. Landsberg, Collider
Searches for Extra Dimensions (hep-ex/0412028).
The most up-to-date, "cutting-edge" collider data can then be found at the
(Thanks to Michele Papucci for recommending these references and
Week 5: A nice readable summary of the structure of Clifford algebras
in general dimensions of signature (p,q), and the implications
for the existence of various types of spinors and their properties, can be
P.G.O. Freund, Introduction to Supersymmetry (Cambridge U. Press,
For the mathematically sophisticated readers, there is also
H.B. Lawson and M.L. Michelsohn, Spin Geometry (Princeton U. Press,
(You could also check out Appendix B of the second volume of Joe Polchinski's
String Theory book.)
Week 6: Some students have asked for sources on representation theory
of Lie algebras/groups. Here are some, ordered from the most accessible by
physicists to more abstract or comprehensive:
H. Georgi, Lie Algebras in Particle Physics (Addison-Wesley, 1982)
W. Fulton and J. Harris, Representation Theory. A First Course
A.O. Barut and R. Raczka, Theory of Group Representations and
Applications (World Scientific, 1987).
There are many more; if you have a favorite book on representations of Lie
groups that you wish to recommend to the rest of the class, please let me know
and I will post the reference here.
Week 8: A brilliantly written in-depth review of the concepts of RG in
nonrelativistic Fermi liquids, and their comparison to RG in relativistic
field theory, can be found here:
Renormalization Group Approach to Interacting Fermions,
Mod. Phys. 66 (1994) 129.
About the homework problem (IV.3.4) from last week, asking to interpret the
one-loop effective potential in phi^4 theory as an infinite sum of one-loop
diagrams: This issue is very nicely discussed for example in Chapter 6.4 of
T.-P. Cheng and L.-F. Li, Gauge Theory of Elementary Particle Physics
(Oxford U., 1984).
Week 9: Here are some references related to the material discussed
this week in class:
One of the most readable and physically insightful sources on
the basics of topological defects (solitons, instantons, etc) are Sidney
Colman's lectures from Erice. Most of the lectures have been collected in
S. Coleman, Aspects of Symmetry (Cambridge U.P., 1985).
Another good source, devoted completely to solitons and instantons and
covering the early years of the subject, is
R. Rajaraman, Solitons and Instantons: An Introduction to Solitons and
Instantons in Quantum Field Theory (Elsevier, 1982).
The connection between D-branes and K-theory was found in
E. Witten, D-Branes and K-Theory, hep-th/9810188.
The connection was extended to Type IIA string theory in
P. Hořava, Type IIA D-Branes, K-Theory, and Matrix Theory,
The relation between stable Fermi surfaces in nonrelativistic Fermi liquids
and K-theory was found in
P. Hořava, Stability of Fermi Surfaces and K-Theory,
Week 10: A really nice set of lecture notes on
magnetic monopoles and duality is
J.A. Harvey, Magnetic Monopoles, Duality, and Supersymmetry,
Week 11: In response to our discussions in Week 10, here are some
references on topological methods, and on solitons.
A really nice, compact (but mathematically rather advanced) book on algebraic
R. Bott and L. Tu, Differential Forms in Algebraic Topology (Springer,
1982 - with many reprintings in later years).
A comprehensive source on differential geometry and algebraic topology in
physics can be found in:
T. Frankel, The Geometry of Physics. An Introduction (Cambridge U.P.,
1st ed: 1997; 2nd ed: 2004) - get the 2nd edition if you can, it contains a
lot more material than the 1st edition.
A book devoted to integrable field theories (and, in particular, their
L.A Takhtajan and L.D. Faddeev, Hamiltonian Methods in the Theory of
Solitons (Springer, 1987) - originally published in Russian a few years
Week 12: The emergence of a non-Abelian Yang-Mills connection in
rather unexpected places in theoretical physics (including the problem of
the falling cat) is one of the main themes of the following collection of
A. Shapere and F. Wilczek, Geometric Phases in Physics (World
Week 13: Here are some references on the BRST formalism.
The basic story can be found in a summarized form in J. Polchinski's
String Theory book, Vol. 1, Ch. 4.2. Also, the beginning chapters of the
2nd volume of Weinberg's book are useful. In addition, more details can be
found in the following sources, listed here (roughly) in the order of the
increasing number of relevant pages:
A. Fuster, M. Henneaux and A. Maas, BRST-Antifield Quantization:
A Short Review,
D. Nemeschansky, C.R. Preitschopf and M. Weinstein, A BRST Primer,
Ann. of Phys. 183 (1988) 226;
M. Henneaux, Lectures on the Antifield-BRST Formalism for Gauge
Theories, Nucl. Phys. Proc. Suppl. 18A (1990) 47;
L. Baulieu, Perturbative Gauge Theories, Phys. Rept. 129 (1985)
M. Henneaux and C. Teitelboim, Quantization of Gauge Systems
(Princeton UP, 1994).
Week 14: Topological quantum field theories (of the cohomological
type) were first introduced by Edward Witten in:
E. Witten, Topological Quantum Field Theory, Comm. Math. Phys.
117 (1988) 353 (which introduces topological Yang-Mills in four
spacetime dimensions), and
E. Witten, Topological Sigma Models, Comm. Math. Phys. 118
(1988) 411 (which presents a topological version of the
two-dimensional QFT of scalar fields, later used as a building block for
topological string theories).
A nice re-analysis of these theories from the precise point of view of BRST
gauge fixing of a trivial Lagrangian was developed shortly afterwards in
several papers, including
L. Baulieu and I.M. Singer, Topological Yang-Mills Symmetry, Nucl. Phys.
Proc. Suppl. 5B (1988) 12, and
L. Baulieu and I.M. Singer, The Topological Sigma Model, Comm. Math.
Phys. 125 (1989) 227.
Many reviews of and lectures on this topic (with varying levels of
difficulty, and beyond the scope of this course) can also be found in the
Week 15: A really nice review of the connection between
Chern-Simons gauge theory and the fractional quantum Hall effect can be found
for example in
Zee, From Semionics to Topological Fluids.
For another good review, see
A. Lopez and E. Fradkin,
Fermionic Chern-Simons Field Theory for the Fractional Hall Effect,
Week 16: A very elegant and readable review of various applications of
Parisi-Sourlas supersymmetry to stochastic equations, disorder, polymers,
localization, etc., can be found in
N. Sourlas, Introduction to Supersymmetry in Condensed Matter Physics,
Physica 15D (1985) 115.
The first original Parisi-Sourlas paper (of several they have written
together) is just two pages long, and
definitely worth the read:
G. Parisi and N. Sourlas, Random Magnetic Fields, Supersymmetry, and
Negative Dimensions, Phys. Rev. Lett. 43 (1979) 744.
Some very exciting recent progress in this field is reviewed in
Lectures on Branched Polymers and Dimensional Reduction,
Week 17: A derivation of the Vandermonde determinant using the
Faddeev-Popov trick can be found, for example, in Footnote 31, Chapter 7.1
(on Page 84 of the hep-th version) of
P. Ginsparg and G. Moore,
Lectures on 2D Gravity and 2D String Theory,