Phys 232A: Quantum Field Theory I

Fall 2020

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Basic Info

Discussion sessions:
Time: Thu 1:00-2:00pm, Fri 4:00-5:00pm.
Place: Discussions live via Zoom (note that the discussions have their own Zoom link, separate from the lectures).

Lecturer: Petr Hořava (email: horava@berkeley.edu)
Office: Pre-covid in 401 Le Conte Hall, now at home.
Office hours: Tuesdays 2:00-3:00pm (the permanent Zoom link for the office hours was sent to the registered students by email.)

GSI: Kevin Langhoff (email: klanghoff@berkeley.edu)
Office hours: Mondays 4:00-5:00pm, Wednesdays 11:00am-12noon.

Quantum field theory (QFT) and, more generally, many-body theory, represents the leading paradigm in modern theoretical physics, and an absolutely essential ingredient in our current understanding of the Universe on an astonishingly diverse range of scales. The basic ideas and techniques of QFT are at the core of our understanding of high-energy particle physics and cosmology, as well as phenomena in condensed matter, statistical mechanics, and even finance. QFT also naturally leads to its logical extension -- string theory -- which in turn provides a unified framework for reconciling the quantum paradigm with the other leading paradigm of the 20th century physics: that of general relativity, in wich gravity is understood as the geometry of spacetime.

At the core of the modern understanding of QFT is the so-called Wilsonian framework: A way of understanding how interacting systems with many degrees of freedom reorganize themselves as we change the scale at which we observe the system. This makes concepts and techniques of QFT remarkably universal, and applicable to just about every area of physics. As a result, a solid understanding of the basic structure, ideas and techniques of QFT is indispensable not only to high-energy particle theorists and experimentalists, or condensed matter theorists, but also to string theorists, astrophysicists and cosmologists, as well as an increasing number of mathematicians.

This course will provide the introduction to the principles of QFT, mostly in -- but not limited to -- the special case of the relativistic regime. The focus will be two-fold: First, on developing a "big-picture" understanding of the basic ideas and concepts of QFT, and equally on developing the techniques of QFT, including renormalization and the renormalization group.

The two main textbooks are going to be:

M.E. Peskin and D.V. Schroeder, An Introduction to Quantum Field Theory (Perseus, 1995);

and

A. Zee, Quantum Field Theory in a Nutshell. 2nd edition (Princeton U.P., 2010).

I also strongly recommend

M.D. Schwartz, Quantum Field Theory and the Standard Model (Cambridge, 2014).

There are now many many more texts on QFT, some excellent, some not so much. We will try to focus on the first two listed above, while adding some additional material of interest at least occasionally, from Zee and other sources. (In the case of Tony Zee's book, it is definitely worth buying the 2nd edition. It is substantially expanded compared to the 1st; and, notably, many many typos of the 1st edition have also been corrected in the 2nd.)

The rough plan for what will be covered in this semester: [Peskin & Schroeder] (or [PS] for short) consists of Parts I, II and III. We will NOT cover anything from Part III (which deals mostly with Yang-Mills and the Standard Model), but will cover Part I, and a big portion of Part II. Luckily, there is also 232B, 234B and 233A in Spring 2021, and those courses will cover miscellaneous advanced material, including -- but not limited to -- Part III of [PS].

I am personally particularly excited about the fact that this year, I get to teach both QFT I in Fall 20, and 232B QFT II in Spring 21! That will give us a great chance to build on the material from the first semester, and to cover some really exciting advanced and modern QFT topics (with focus on interdisciplinary connections) in the second semester. Should be a lot of fun!

The final grade will be based on three things: 1. homeworks, 2. participation in discussions (during discussion sessions as well as in lectures), and 3. the final exam. As I have done with other classes in previous years, I will again apply the "two-out-of-three" rule: Briefly, it means that for a good grade (say an A) it is sufficient to do really well on two out of 1., 2. and 3. listed above; for example, if you do great on homeworks and you interact well in discussion sessions and lectures, you will be exempt from the final exam. (Other permutations work as well.) I am really hoping that all students will do so great on the weekly homeworks that most will be exempt from the final exam. (Again, further details about the grading process were in Kevin's email from Sept 8.)

On a weekly basis, I will also post here some additional references or comments related to the material covered in the lectures.

Week 1: Two references that I recommend for extra reading for those who are interested.
First, we talked about Klein's paradox. A beautiful book on paradoxes in quantum mechanics is:
Y. Aharonov and D. Rohrlich, Quantum Paradoxes. Quantum Theory for the Perplexed (Wiley-VCH, 2005).
Truly a great read, and it even briefly mentions Klein's paradox itself!
For those students who are very mathematically inclined and will wish to see more rigorous details behind some of our physics constructions (when available), there is an interesting resource which I can recommend, and which contains lots of good mathematical details and insights:
R. Ticciati, Quantum Field Theory for Mathematicians (Cambridge UP, 1999).

HW1 (posted on Sept 10, due in one week): Two problems from [Zee]: Problem I.2.1 and Problem I.2.2 (they appear on p. 16 of the second edition of the book).

HW2 (due on Thu, Sept 24): Problems 2.1(a,b) and 2.2(a,d) from [Peskin-Schroeder] (on pages 33-4).

HW3 (due on Thu, Oct 1): Problems I.3.1, I.3.2, and I.3.3 from [Zee] (on pp. 24-5).

HW4 (due on Thu, Oct 8): Problems I.4.1 (on p. 31) and I.5.1 (on p. 39) from [Zee], and Problems 2.2(b), 2.2(c) from [PS] (on p. 34).

HW5 (due on Thu, Oct 15): Problems I.8.1, I.8.2 and I.8.3 from [Zee] (on p. 69). Then consider a real scalar field coupled to an external source, as discussed on p. 32 of [PS]; derive Eqns. (2.65) and (2.66) on p. 33 of [PS], and solve Problem 4.1(a,b) from [PS] (on p. 126).

HW6 (due on Thu, Oct 22): Problems I.7.1, I.7.2 and I.7.3 from [Zee] (p. 60). Problem 4.1(c,d,e) from [PS] (p. 126-7), and finally Problem 4.3(a) (on p. 127-8 of [PS]) except: please do not (yet) evaluate any cross-section as you are asked to at the end of this problem, focus just on deriving the propagator and the vertex for now.

HW7 (due on Thu, Oct 29): First, finish Problem 4.3(a) from [PS] by calculating the differential cross section. Then, solve Problems 4.3(b,c), and Problem 4.2 from [PS] (they are on pages 127-9), and also one problem from [Zee]: Problem II.6.7 (on p. 143). You will see that this problem from Zee is very closely related to Problem 4.2 from [PS].

This week, we discussed cross-sections (and, to some small extent, scattering kinematics), especially for the 2-on-2 scattering. If you wish to read more on more complicated kinematics and phase-space factors for the scattering to more than 2 particles, there is an excellent review, which can be found here:
T. Han, Collider Phenomenology: Basic Knowledge and Techniques," 2004 TASI Lectures, arXiv/hep-ph/0508097.
See, in particular, Appendix A of this review.

HW8 (due on Thu, Nov 5): First, use the defining property of the Clifford algebra to prove that the matrices listed in Eqn. (3.23) of [PS] (on p. 40) indeed satisfy the commutation relations of the Lorentz algebra. Then, solve Problems 3.1(a), 3.1(b) from [PS] (on pp. 71-2), and Problems II.1.11, II.1.12 from [Zee] (p. 106).

Here are some useful resources on the structure of Clifford algebras and spinors in various dimensions:
A simple introduction aimed at physicists can be found in Appendix B of Volume II of Polchinski's book of strings:
J. Polchinski, String Theory. Volume 2 (Cambridge U.P., 2005).
A beautiful mathematical presentation of this subject is in Chapter 1 of:
H.B. Lawson and M.-L. Michelsohn, Spin Geometry (Princeton U. Press, 1990).
(This was also the resource I was mostly relying on in the lectures.)
Finally, more details about Clifford algebras and spinors can be found also in:
F.R. Harvey, Spinors and Calibrations (Academic Press, 1990).

HW9 (due on Thu, Nov 12): Problems II.1.2 (on p. 105) and II.2.1 (on p. 113) from [Zee], and Problems 3.4(a), 3.4(b), 3.4(c) and 3.4(e) from pages 73-4 of [PS].

HW10 (due on Thu, Nov 19): Problem 4.4(a,b,c) from [PS] (on pp. 129-130); Problems II.5.1 and II.5.2 from [Zee] (on page 131); Problem II.6.3 from [Zee] (on page 143).

HW11 (due IN TWO WEEKS, on Thu, Dec 3): First, a few less technical problems from [Zee]: III.1.1, III.1.3 (on p. 168), III.2.1 (p. 172), III.3.1, III.3.2 (p. 181). Then read Chapter VI.6 of [Zee], which contains an example of a nonrelativistic field theory; with this system in mind, solve problems VI.6.2 (p. 349 of [Zee]) and VI.8.5 (p. 368 of [Zee]). Finally, try Problem 5.1 from [PS] (on p. 169 -- this problem requires some calculational tricks that we will only discuss in class on Dec 1).

HW12 (due on Thu, Dec 10): First, if you haven't completed your work from HW11 on Problem 5.1 from [PS] yet, you can complete it now. Then solve Problem 5.2 from [PS] (p. 170). Finally, Problem VI.8.4 from [Zee] (p. 368). This is the final homework set of the semester.

This week we talked extensively about the basics of the field theory of Fermi liquids. Here is my favorite reference to read about the renormalization-group treatment of Fermi liquids:
R. Shankar, Renormalization Group Approach to Interacting Fermions, Rev. Mod. Phys. 66 (1994) 129-192.
It is truly a lot of fun to read, perhaps this would be a suitable reading material for the upcoming holidays!

horava@berkeley.edu