Phys 232B: Quantum Field Theory II

Spring 2018

shortcut to Miscellaneous References
shortcut to the List of Reading Assignments

Basic Info

Time: Tue and Thu, 9:40-11:00am.
Place: 402 Le Conte Hall.

Discussion sessions:
The primary discussion time and place is:
Wed 10:10-11:30am (402 Le Conte Hall).
The Wednesday discussions will happen every week (except during Spring Break). Later on, we will also have two discussion sessions on Thursday, 3:40-4:30pm (one in March and one in April).
The Wednesday discussions in February will present additional material complementary to the lectures. Then, starting from the first week of March, all discussion sessions will be devoted to the Student Presentations of their Reading Assignments. The precise schedule of those discussions/presentations is here.

Lecturer: Petr Hořava (email:
Office: 401 Le Conte Hall.
Office hours: Thursdays 2-3pm.

GSI: Chris Mogni.


Physics 232A: QFT I, for example at the level of the course I taught in Fall 2017. Also, basics of general relativity will be very useful in the second half (or final third) or the course.

Goal & Scope

This semester will continue in the style of my Fall 2017 Physics 232A course, focusing again on the two-fold goals: Expanding our understanding of the technical structure of QFT, but also keeping the "big picture" in sight, with applications across various areas of physics ranging from particle physics to phenomenology, cosmology, string theory, quantum gravity and even pure mathematics.

More specifically, on the frontier of "theories", we will focus on two major areas: 1. Gauge theories, and 2. Effective field theories. On the frontier of "techniques", we will also focus on two main themes: A. Calculus of renormalization, universality and Renormalization Group (RG); and B. Nonperturbative techniques, especially those involving geometry and topology.

Grading & Homeworks

Since this is the more advanced, second semester of QFT, there will be no Homework Assignments: We need to move from the ability to solve problems already solved, to the next level, of being able to read, understand and interpret original research papers. Therefore, Homeworks will be replaced by Reading Assignments, which will be posted on this page later. Once they are posted, each student will choose one (and sign up for it with me by email, on a first-come, first-served basis). Then the student will read the chosen paper, prepare a typed summary of at least two pages, and later in the semester present the essential results of the paper at the blackboard in one of our Discussion Sessions. My grading policy is again "two-out-of-three": In order to earn an A, you need to do well on two out of the following three -- (1) class attendance and activity, (2) reading assignment, (3) final take-home exam. Since the final take-home exam will be intentionally very difficult, I hope that everyone will do well on (1) and (2), and therefore will be exempt from the final exam :)

Reading Assignments

The official list of reading assignments will appear on this website sometime in the second half of February. Update: The List was posted here on Thursday, February 15.

Textbooks and Reference Books

In the list of Miscellaneous References below, I will often refer to particular Chapters of several useful reference textbooks, which I list here, indicating also the canonical way in which I will refer to them below:

[PS]: M.E. Peskin and D.V. Schroeder, An Introduction to Quantum Field Theory (Perseus, 1995);
[Zee]: A. Zee, Quantum Field Theory in a Nutshell. 2nd edition (Princeton U.P., 2010);
[Schw]: M.D. Schwartz, Quantum Field Theory and the Standard Model (Cambridge, 2014);
[W:I], [W:II], [W:III]: S. Weinberg, The Quantum Theory of Fields, Vols. I, II, III (Cambrdige U.P., 1996).

Miscellaneous References

Here I will post, on a weekly basis, the links to references which are either directly relevant to the material presented in lectures, or represent particularly nice opportunities for further reading on related topics. (Just to be clear: These additional "Miscellaneous References" represent extra material, and will NOT overlap with the list of Reading Assignments.)

Week 1. We introduced bosonization in 1+1 dimensions. My favorite discussion of the subtleties in the QFT of free massless scalars in 1+1 dimensions is in
M. Stone, The Physics of Quantum Fields (Springer, 2000) (see Ch. 17).
A very nice and careful treatment (with applications) can also be found in
J. Zinn-Justin, Quantum Field Theory and Critical Phenomena (4th edition, Oxford, 2002) (especially in Ch. 32).
An overview of bosonization aimed mostly at condensed matter is in
D. Senechal, An Introduction to Bosonization, arXiv:cond-mat/9908262.
Another great overview, followed by condensed matter applications, is in the recent book
R. Shankar, Quantum Field Theory and Condensed Matter (Cambridge, 2017), especially in Ch. 17 and 18.
Additional condensed-matter applications are in Ch. 20 of
S. Sachdev, Quantum Phase Transitions (2nd edition, Cambridge, 2011).

Week 2.The equivalence between the sine-Gordon theory and the Thirring model was discovered on the particle-theory side in
S. Coleman, Quantum sine-Gordon Equation as the Massive Thirring Model, Phys. Rev. D11 (1975) 2088,
and in
S. Mandelstam, Soliton Operators for the Quantized Sine-Gordon Equation, Phys. Rev. D11 (1975) 3026.
The Schwinger model was solved in
J. Schwinger, Gauge Invariance and Mass. II, Phys. Rev. 128 (1962) 2425.
This result was some 12 or more years ahead of its time, as can be seen from the discussions in
A. Casher, J. Kogut and L. Susskind, Vacuum Polarization and the Absence of Free Quarks, Phys. Rev.D10 (1974) 732,
and in
S. Coleman, R. Jackiw and L. Susskind, Charge Shielding and Quark Confinement in the Massive Schwinger Model, Ann. Phys. 93 (1975) 267.
A review of the partial success in attempts to generalize bosonization to dimensions greater than 1+1 can be found here:
A. Houghton, H.-J. Kwon and J.B. Marston, Multidimensional Bosonization, arXiv:cond-mat/9810388..

Week 3. For an introduction to classical Yang-Mills gauge theories, the following chapters in the above-mentioned textbooks are good: Ch. IV.5 of [Zee] (read also Ch. IV.4 for a refresher on differential forms); Ch. 25 of [Schw]; Ch. 15.1-15.3 of [W:II] (read also Appendix A and B of Ch. 15); Ch. 15 of [PS].
A good comprehensive resource on the basics of group theory aimed at high-energy physicists is the (relatively) recent book:
A. Zee, Group Theory in a Nutshell for Physicists (Princeton U.P., 2016).
The mathematically correct language for non-Abelian Yang-Mills gauge fiels is that of connections on principal bundles. My favorite mathematically rigorous introduction to this subject can be found in the old but elegant book
S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vols. 1 and 2 (Wiley Classics).
In our Hamiltonian quantization of Yang-Mills, we will follow the elegant (and correct!) original treatment in the spirit of Faddeev and Popov, which can be found in the sleek book
L.D. Faddeev and A.A. Slavnov, Gauge Fields: Introduction to Quantum Theory (Perseus, 1991).

Week 4. For the Killing-Cartan classification of simple compact Lie groups/algebras in terms of Dynkin diagrams, see for example the very readable and concrete discussion of all the necessary ingredients in Part VI of Zee's "Group Theory in a Nutshell" book cited above. (This Part VI involves a detailed discussion of roots and weights, and the specifics of where Dynkin diagrams come from; it is definitely worth going over those ~60 pages!)
The specific subclass of so-called "simply laced" groups in the Killing-Cartan classification scheme is an example of an "ADE classification" -- a celebrated and somewhat mysterious classification that appears over and over in so many different (and apparently disconnected) parts of mathematics (and lately physics as well); a good source of basic info on ADE classification, with many intriguing links to further reading, can be found here.
A good but somewhat old review of the universality of ADE classification in Math is
M. Hazewinkel et al, The Ubiquity of Coxeter Dynkin Diagrams (An Introduction to the ADE Problem), Nieuw A. Wiskunde 25 (1977) 257.
A more modern review is
K. Siegel, The Ubiquity of ADE Classifications in Nature.
In the previous week, we saw another example of an ADE classification: The classification of all relativistic CFTs in 1+1 spacetime dimensions whose "number of degrees of freedom" (as measured by the cetral charge of the Virasoro symmetry algebra) is the same as that of a single real scalar; that classification is based on
P. Ginsparg, Curiosities at c=1, Nucl. Phys. B295 (1988) 153; see in particular the Figure on Page 158 in that article.

Week 5. The discussion sessions are starting this week! On Wednesdays, February 14, 21 and 28, from 10:10am to 11:30am (in 402 Le Conte), Chris will present additional supplementary material related to what is being discussed in lectures, plus will answer your possible questions about the material.
Starting from Wednesday March 7 (and also on some Thursdays 3:40-4:30pm), the discussion sessions will consist solely of the Student Presentations of their Reading Assignments.
The best reading material for this week is still [Schw], Chapters 25 and 26.
A great comprehensive resource on quantization of systems with all kinds of constraints is
M. Henneaux and C. Teitelboim, Quantization of Gauge Systems (Princeton U.P. 1994).
An excellent consise description of quantization of constrained systems (such as Yang-Mills) is in Chapters 9.3 and 12 of
C. Itzykson and J.-B. Zuber, Quantum Field Theory (originally from 1980; reprinted by Dove r).
The List of Reading Assignments has been posted! If you are registered for the course, you can now sign up for your paper of choice with me, by email.

Week 6. I have been asked to post some of the original references to Dirac's work on quantization of constrained systems; here they are:
First of all, the story is explained first-hand in his brilliant lectures,
P.A.M. Dirac, Lectures on Quantum Mechanics (available as reprinted by Dover).
The original work is:
P.A.M. Dirac, Generalized Hamiltonian Dynamics, Proc. Roy. Soc. A246 (1958) 326;
and its applications to quantum gravity are in the follow-up:
P.A.M. Dirac, The Theory of Gravitation in Hamiltonian Form, Proc. Roy. Soc. A246 (1958) 333.
This week's Wednesday discussion will be devoted to the introduction to instantons, and their basic properties. An excellent introduction to the role of instantons in quantum mechanics and QFT is offered by S. Coleman's Erice lectures on The Uses of Instantons, reprinted in a nice form as Chapter 7 of
S. Coleman, Aspects of Symmetry (Cambridge, 1985).
My favorite presentation of the essential aspects of BRST symmetry is in Chapter 4.2 of Volume 1 of
J. Polchinski, String Theory (Cambridge, 1998).
In lectures, we discussed BRST quantization in the Lagrangian formalism. If you are interested in the Hamiltonian version, you can consult
D. Nemeschansky, C. Preitschopf and M. Weinstein, A BRST Primer, Ann. Phys. 183 (1988) 226.
Topological Yang-Mills gauge theories in 4 dimensions were introducted in
E. Witten, Topological Quantum Field Theory, Commun. Math. Phys. 117 (1988) 353.

Week 7. The important class of topological QFTs known as "cohomological field theories" (which includes topological Yang-Mills in four dimensions) is reviewed beautifully in
E. Witten, Introduction to Cohomological Field Theories, Int. J. Mod. Phys. A6 (1991) 2775.
Our upcoming discussion of BRST-BV formalism will follow the great lectures by Marc Henneaux,
M. Henneaux, The Antifield BRST Formalism for Gauge Theories, in: Quantum Mechanics of Fundamental Systems 3, ed: C. Teitelboim and J. Zanelli (1992).
(If more details are needed or desired, they can be found in the book by Henneaux and Teitelboim cited above.)
Another useful and concise intro to the BRST-BV formalism is in Chapter 15.9 of [W:II].

Week 8. Guest lectures by Stephen Randall on Supersymmetric Yang-Mills gauge theories in 3+1 dimensions.

Week 9. Guest lectures by Ori Ganor on Chern-Simons gauge theories in 2+1 dimensions.

Week 10. We continue the discussion of the BRST-BV "antifield" formalism, and its applications to quantization of gauge theories. Given the material covered in lectures, the students now have enough background to understand the applications of the antibracket-antifield formalism to quantum Yang-Mills, as presented for example in Chapters 17.1-17.3 of [W:II], which is the recommended reading for this week.
Besides the Henneaux and Teitelboim references mentioned above, the geometric nature of the antibracket and the antifields was further clarified in a very short and elegant paper by Witten,
E. Witten, A Note on the Antibracket Formalism, Mod. Phys. Lett. A5 (1990) 487.

Week 11. Spring break: No lectures, discussions or office hours. Enjoy the break!

Week 12. Guest lectures by Grant Remmen, on the cosmological constant problem and on his recent work on the infrared implications of structures in effective field theory.

Week 13. The idea of a large N expansion for QCD (and similar theories) was presented in
G. 't Hooft, A Planar Diagram Theory for Strong Interactions, Nucl. Phys. B72 (1974) 461.
Over the next 1/4 century, this idea culminated in the celebrated AdS/CFT correspondence, to which a great introduction is:
J. Maldacena, TASI 2003 Lectures on AdS/CFT, arXiv:hep-th/0309246.
In stat-mech, the large N idea was applied very successfully earlier,
H.E. Stanley, Dependence of Critical Properties on the Dimensionality of Spins, Phys. Rev. Lett. 20 (1968) 589.
A good pedagogical review of the large N approach to QCD is:
A. Manohar, Large N QCD, arXiv:hep-ph/9802419.
An inspiring review of large N as a semiclassical limit is in:
L.G. Yaffe, Large N Limits as Classical Mechanics, Rev. Mod. Phys. 54 (1982) 407.
Some other introductions to large N are:
Chapter VII.4 of [Zee], and
Chapter 8 of Coleman's collected Erice lectures: S. Coleman, Aspects of Symmetry (Cambridge, 1985).

Week 14. Holographic renormalization in the context of AdS/CFT correspondence is reviewed in
K. Skenderis, Lecture Notes on Holographic Renormalization, arXiv:hep-th/0209067,
and in
J. de Boer, The Holographic Renormalization Group, arXiv:hep-th/0101026.
Another interesting perspective on holographic renormalization was presented in
I. Heemskerk and J. Polchinski, Holographic and Wilsonian Renormalization Groups, arXiv:1010.1264.
The discussion of anomalies, with a particular emphasis on the Wess-Zumino consistency conditions and the usefulness of the BRST as well as BRST-BV formalism in the understanding of anomalies, is in Chapter 22 of [W:II] (especially, see Ch. 22.6).
Applications of the antibracket formalism to the proof of renormalizability of Yang-Mills are discussed in detail in Chapter 21 of J. Zinn-Justin's book cited above (in Week 1).

Week 15. Two good papers on the c-theorem in the holographic context are:
R.C. Myers and A. Sinha, Seeing a c-Theorem with Holography, arXiv:1006.1263,,
R.C. Myers and A. Sinha, Holographic c-Theorems in Arbitrary Dimensions, arXiv:1011.5819.
The influential 2011 result on the a-theorem in 3+1 dimensions is in
Z. Komargodski and A. Schwimmer, On Renormalization Group Flows in Four Dimensions, arXiv:1107.3987.
A nice review of the effective field theory approach to quantum gravity is:
J.F. Donoghue, The Effective Field Theory Treatment of Quantum Gravity, arXiv:1209.3511;
see also the more recent lectures
J.F. Donoghue, M.M. Ivanov and A. Shkerin, EPFL Lectures on General Relativity as a Quantum Field Theory, arXiv:1702.00319.
A great introduction to inflation and its treatment in the framework of effective field theory can be found in Baumann's TASI lectures,
D. Baumann, TASI Lectures on Inflation, arXiv:0907.5424,
and in his Cambridge lectures:
D. Baumann, The Physics of Inflation, Cambridge graduate course.

Week 16. Another good review of gravity as an effective field theory is
C.P. Burgess, Quantum Gravity in Everyday Life: General Relativity as an Effective Field Theory, arXiv:gr-qc/0311082.
Interesting comments on the EFT approach to modifications of general relativity are in
C.P. Burgess, Effective Theories and Modifications of Gravity, arXiv:0912.4295.
Many aspects of the effective field theory of inflation, and its possible connections to string theory, are reviewed in:
D. Baumann and L. McAllister, Inflation and String Theory, Cambridge University Press, 2015;
(see also arXiv:1404.2601 for the arXiv version of that same book, sans Appendices).
A very nice, recent and fresh set of lectures on the effective field theory of inflation is:
C.P. Burgess, Intro to Effective Field Theories and Inflation, arXiv:1711.10592;
besides providing a very up-to-date review of the story, these lecture notes point in the direction of many interesting original references, such as the papers by Cheung et al and by Weinberg from 2007-8 which triggered much of the interest in the subject over the past decade.
For an alternative point of view on whether or not inflation really solves the horizon and flatness problems, see (for example):
A. Albrecht, Cosmic Inflation, in: Post-Planck Cosmology: Les Houches Lecture Notes, Volume 100 (2013).

Path-integral and QFT methods in application to financial markets are nicely reviewed in Chapter 20 of the 5th edition of Hagen Kleinert's path integral book:
H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 5th Edition (World Scientific, 2009)
(see here for the pdf file of the book).
The use of the Boltzmann distribution and the existence of a temperature of financial markets was proposed in
H. Kleinert and X.-J. Chen, Boltzmann Distribution and Temperature of Stock Markets, arXiv:physics/0609209.
Finally, I thank Zoltan Ligeti for bringing to my attention during this week that Mark Wise co-authored a book on finance:
M.B. Wise and V. Bhansali, Fixed Income Finance: A Quantitative Approach (McGraw Hill, 2010).
I haven't seen the book yet, but I am sure it is brilliant, if it approaches the brilliance of some of Mark's other writings, such as for example the great thin book with Manohar on effective field theory:
A.V. Manohar and M.B. Wise, Heavy Quark Physics (Cambridge U.P., 2007).