J. Rahman: Thinking Aloud: Literature Recommendation: Quantum Field Theory

Quantum Field Theory (QFT) is one of the marvellous accomplishments of the last century, but even today it is shrouded in mystery and hysterical infamy to newbies. For a beginner, it can be absolutely daunting, given the sheer abundance of books, formalisms and conventions to choose from. The paradox of choice, I believe!

Let me guide you through my idea of how one should approach QFT. To begin with, you will need absolute fluency in Special Relativity and Quantum Mechanics. After all, we intend to conjoin these theories in a holy union. Here are some topics you should be familiar with:

Special Relativity:

Tensor Calculus, Transforming contra- / covariant indices with the metric.
Lorentz Group, Poincare Group.
Energy-momentum dispersion relation: $ E^2 = (mc)^2 + (\vec{p}c)^2$.
What Lorentz transformations do to points in the above hyperboloid.
Maxwell's theory in covariant notation: $ \partial_\alpha F^{\alpha \beta} = \mu_0 J^\beta; \quad \partial_\alpha \epsilon^{\alpha\beta\gamma\delta} F_{\gamma\delta}= 0$.
(Optional) Classical Relativistic Field Theories – the language of Lagrangians.
(Optional) Lorentz Algebra, Poincare Algebra.

Quantum Mechanics:

Natural Units: $\hbar = c = k_B = 1; [Energy] \equiv 1; [AB] \equiv [A] + [B] \Rightarrow [Length] = [Time] = -1$ etc.
Quantum Harmonic Oscillator, Second Quantisation, Fock Space, Coherent States.
Heisenberg Picture, Feynman Path Integrals, Propagators.
Relativistic Quantum Mechanics: Dirac equation, Gamma matrix jugglery.
Angular Momentum and Spin: Representations of $SO(3)$ and $SU(2)$.
Clebsch–Gordan Coefficients, Tensor products of $SO(3)$ or $SU(2)$ representations.
Scattering Theory, S-Matrix, decay rates.
(Optional) Clebch–Gordan Coefficients for $SU(3)$.

[Attention: If you want to quickly jump to the summary of all the recommendations posted below, please click here.]

If you are unfamiliar with any of the topics listed above, please go study them elsewhere. For example, the MIT OCW courses 8.04, 8.05 and 8.06 provide the necessary Quantum Mechanics background. In addition, the lectures notes on Relativity, Particles and Fields by Prof. Dr. Nora Brambilla provide an advanced resource for the above topics, in addition to a glimpse of actual QFT (chapters 1-5 plus the appendices cover most of what you need, rest is QFT).

As part of your mathematical background, you also need an understanding of Green's functions, a bit of complex analysis and knowledge about distributions and special functions (such as gamma functions etc.). The resources at Perimeter Institute should help you with that.

The above is the absolute minimum you need to start with QFT. As an a-la-carte, you could do a course on Lie Algebras and its Representation Theory (in order to enrich your understanding of the projective unitary representations of $so(3,1)$, the Lorentz algebra). In fact, that is your ultimate goal: to find such representations, because the particles you see in the world are nothing but representations of some symmetry group. In order to have a relativistic quantum theory, one needs to find all the unitary representations of the Lorentz symmetry $SO(3,1)$ (we already have the translation part of Poincare as propagators). The prerequisite for undertaking Lie Groups and Algebras is a solid foundation in Linear Algebra. The books I recommend for this study are Symmetries, Lie Algebras and Representations by J. Fuchs and C. Schweigert, Notes on Lie Algebras by Hans Samelson and Lie Algebras in Particle Physics by Georgi. Another useful book, particularly for physicists, would be Quantum Theory, Groups and Representations by Peter Woit. An absolute classic reference, although not recommended for beginners, is Geometry, Topology and Physics by M. Nakahara. Please note that the mathematical study of Lie Groups & Algebras and their representation theory is not necessary for an introduction to QFT. It, however, enhances your understanding of the theory manyfold.

After acquiring sufficient knowledge (actually bare minimum) of Lie Algebras, you should read The Algebras of Grand Unified Theories by John Baez and John Huerta, an absolutely delightful article rich with physical insights that are hard to find elsewhere. It was originally intended for an audience of mathematicians, but I think it presents the physics beautifully. Even if you have not undertaken a course in Lie Algebras, you should read this article at some point in your life–probably, after studying the representations of the Lorentz algebra.

Now, there are two pedagogical approaches to QFT (not talking about mathematical formalisms): one is to prepare yourself for a career in experimental particle physics (for whom knowing how to compute phenomenological results is important), and another is to make yourself a theoretical physicist (striving for a better understanding of the theoretical/mathematical construct). As you might have guessed, I will take the latter approach.

Therefore, I will shun classic textbooks such as Peskin and Schroeder at the very outset. I personally do not like the book because one often tends to get lost in calculations of scattering amplitudes etc. and forgets the broad picture of what one is actually trying to do. However, I recommend that you keep it as a reference which you can go to whenever needed. On the formalism side, there are two: (1) field quantization and (2) the path (functional) integral approach. I prefer the latter to be introduced first because it is much faster to get kickstarted into studying Feynman diagrams for QFT's. One can always learn field quantization at a later stage. This is also why I avoid Peskin and Schroeder or other books that begin with field quantization.

Anyway, as an introductory remark, you should know that Feynman diagrams are not inherently quantum. They come whenever you try to perturbatively solve a partial differential equation. They are basically a bookkeeping device to keep track of all the terms in your integrals. Solving Classical Field Equations by Robert C. Helling is a good text explaining in detail what I just said. This should be your first dose of Feynman diagrams.

Once you are equipped with that knowledge, you should begin with Quantum Field Theory in a Nutshell by A. Zee and Quantum Field Theory by M. Srednicki. Zee provides an excellent treatise on the conceptual foundations of QFT in a very relaxed way that reminds me of Feynman's lectures. To be pedantic, I concede that Zee lies about some things once in a while: but that is okay for an introductory read. A good teacher knows to lie ("dumb down") when it is needed. You have been doing this all your life: learning half-truths, then unlearning whatever you learnt to this stage, only to replace it with something factually more accurate and conceptually more advanced.

A. Zee:
"Then there is the person who denounces the book for its lack of rigor. Well, I happen to know, or at least used to know, a thing or two about mathematical rigor, since I wrote my senior thesis with Wightman on what I would call “fairly rigorous” quantum field theory. As we like to say in the theoretical physics community, too much rigor soon leads to rigor mortis. Be warned. Indeed, as Feynman would tell students, if this ain’t rigorous enough for you the math department is just one building over. So read a more rigorous book. It is a free country."

Mark Srednicki, on the other hand, introduces the subject more formally in bursts of small chapters. He states the prerequisites at the beginning of every chapter and ends with further references to expand your knowledge of the topic at hand. I like Srednicki because I like anything concise (brevity is the soul of wit, remember?). Follow this book religiously. Do all the exercises. There are online solution keys if you need to verify your results.

You should always do this with all the subjects you study. Take one book and follow it religiously. Do all the exercises. I cannot emphasize how much important it is to do the exercises. Make your own notes. One cannot learn just by reading these books casually. You have to get involved. Use other books for reference.

Srednicki is not comprehensive. You will definitely need to supplement it with other books. For more worked out examples at this stage, you can refer to Field Quantization by Greiner. For the advanced reader, I will recommend Weinberg I for more physical insights.

When you start studying fermions, for example, you should learn about the representations of the Lorentz algebra, if you haven't already. I found the first three chapters of Maggiore enlightening in this regard. Furthermore, this article adds much more physical insight. Here is a summary of whatever you have learnt so far about representations of $so(3,1)$. This is probably a good time to study the article by John Baez I mentioned before.

To supplement your study of renormalization, read Renormalization of $\phi^4$–theory by Prahar Mitra which additionally discusses the renormalizability of QED, and for a detailed calculation of renormalization group flows and beta functions, read Running of the Coupling in the $\phi^4$–theory and the Standard Model by Olli Koskivaara. For detailed notes on perturbative calculations in QED and QCD, study Andrey Grozin's Lectures on QED and QCD.

NOTE: There is no conceptual framework (none yet) in which a divergent series can be meaningfully "summed" to a finite result. It is NOT true that the sum of all natural numbers is minus one-twelfth. It is, however, true that whenever a divergent sum is rendered convergent by means of a process known as regularization (such as multiplying each term of the series by a decaying function), the series converges to a sum of two parts: a part that blows up when the regularization is removed, and a finite part that is always the same regardless of the scheme of regularization. All physics is somehow encoded in this finite part (minus one-twelfth, in case of the sum of all natural numbers).

To see a visualization of the Riemann-Zeta function (how the sum of all natural numbers "metaphorically equals" minus one-twelfth), please watch this wonderful demonstration.