The Quantum Theory of Fields: Volume 1, Foundations [ペーパーバック]

For those who are receptive to its charms, this book is simply indispensible to any high energy physicist. This book is not terribly "intuitive"(in the sense that things are derived heuristically just to the point that the result seeems plausible), nor does it take a purely mathematical standpoint, emphasizing the unbending rigour of all proofs. Instead, it offers something far, far more valuable to any physicist; namely it offers truly profound physical insight into the fundamental principles of nature. This book is so chock full of brilliant profound ideas that it seems as if Weinberg put into this book almost all of the insights he has had over the course of his long, productive, and Nobel Prize winning career. He offers a truly logical presentation of particle physics, starting from the fundamental principles of quantum mechanics (superposition principle especially) and the principle of invariance under the Poincare group modulo time and spatial inversion, as well as the principle that distant measurements do not affect each other, and derives, with a minimum of simplifying assumption, the whole, wonderful edifice of quantum field theory. This set of volumes contains almost all that we know about QFT, but somehow, magically, it is not encyclopedic; it is instead refreshingly original and, as I have said before, truly profound. Also, unlike many other QFT texts, it very clearly points out how the assumptions of the theory could be weakened, and also gives an indication of what sorts of theories come from these modified assumptions. The whole book is simply fascinating, but I found the chapter on general renormalization theory particularly enlightening, especially the section on "nonrenormlizable" theories. I learned, in a particularly clear, inspiring way, that these theories are not any more or less renormalizable than standard model theories, when all terms alowed by symmetry are included in the Lagrangian. Although these theories might seem as if they have little power of prediction (after all, there are an infinite number of parameters to the theory), but in fact Weinberg argues that the nonrenormalizalbe interactions are strongly suppressed at low momenta, so it is possible for low energies to create an effective perturbation theory, which yield in this regime astonishingly precise prediction. On the other hand, Weinberg is quick to point out that for large enough energies, this perturbation expansion simply does not make sense, and that THIS is the reason why quantum gravity based on the Einstein-Hilbert Lagrangian makes no sense for energies at the scale of unification.
Although this book is truly wonderful, I would not recommend reading it as an introduction to QFT. This book is simply too intense and profound for the uninitiated. Instead, I would recommend as a first introduction Ryder's fine text, which yields enough insights to give the reader a taste of the ideas behing QFT but not so many that the reader is overwhelmed at first, followed by Peskin-Schroeder, which gives the student all of the tools that he/she will need for almost any QFT calculation.

To put the review in perspective, My Background: I am a senior undergraduate engineering/physics student with an interest in mathematics and theoretical physics. This is my third QFT book.

Things I liked about the book:
- The book follows a very logical progression. I love how Weinberg presents a coherent argument based on simple physical principles (specifically Lorentz invariance and the cluster decomposition principle).
- Weinberg takes painstaking effort to avoid hand-waving, and is very careful to enumerate (and make plausible) his assumptions. In so doing, he avoids the sort of black-magic feeling I got when reading some less well written QFT books (see for example: Peskin and Schroeder, which makes a mockery of logical progression in an effort to teach you how to calculate as soon as possible).
- The book was very thorough, and often provided an original approach to the material. The coverage of renormalization seemed natural and coherent, and since the book is presented in a logical order (rather than a historical one) Weinberg avoids justifying renormalization as some mysterious subtraction of infinities, basing it instead on general non-perterbative methods (e.g. poles of the S-matrix, etc...)

What I didn't like about the book:
- As a result of his unwavering emphasis on logical progression, and his inclusion of a vast amount of material (almost all of which is necessary to understand in order to progress through the book), the book is somewhat painful to get through. Be prepared to re-read many of the sections a couple of times, and to make very slow progress.
- Weinberg chooses to present QFT in a very general form (i.e. abstracting it from a particular field such as particle physics or condensed matter physics). This is not necessarily a disadvantage, but I often found my interest waning after reading a few hundred pages without making any contact with phenomenology. Additionally, the excercises were similarly abstract, which makes it difficult (at least for me) to particularly care about their results. (More of a problem for self-study)
- The notation is very complete, which isn't normally a bad thing. However, the equations sometimes become very cumbersome when he includes every index, and every functional dependence regardless of how redundant they may be.
- In his coverage of path integrals, he derives things using functional determinants rather than through the more common generating functional methods. I think this hides a lot of the physical insight of the path integral approach, particularly, its equivalence to the 2nd-quantized approach, and its relation to Feynman diagrams.
- This book will drive the more mathematically inclined crazy, as the author admits, it makes very little attempt at rigour, and is very uncareful. He exchanges orders of limits willy-nilly, and often is not even clear about what sort of limiting process is taking place. There is not discussion of functional integration measures, or convergence, and there is very little justification provided for regularization methods (actually the coverage of dimensional regularization is extremely sparce, and would have been unfollowable, had I not already known it).

General Comments:
- I think that, contrary to some of the previous reviews, that the first few chapters of the book (through 6) would be a good first exposure to quantum field theory. I think the reader would have a much better understanding of the theory. However, the rest of the book is quite advanced, and would not be good for the uninitialized.
- I think that in an effort to make his coverage thorough and abstracting his discussion from phenomenology, the author sacrificed some of the readability of the book. That being said, if you're serious about learning the subject, this is a good resource.

I like this book because it explains everything from first principles to the most advanced results, and this is really the best way to master such a subject. Moreover, Weinberg managed to give full proofs of all intermediate steps using only maths available to the average physicist. I don't think it is a pedagogic text, like e.g. Peskin-Schroeder, it is too demanding at some places (however the most advanced sections that can be skipped at first reading are indicated by footnotes). Paradoxically, this can make the reader's task easier because he/she gets better equipped to tackle the difficult problems, and the systematic development does not tolerate any hiatus. The contrast is in the generality and completeness: Peskin-Schroeder discuss the representations of Lorentz group only in the context of spin 1/2. Weinberg discusses them in full generality. P-S do not explain canonical quantisation of the EM field; Weinberg explains it for any kind of field. On the other hand, the first QED process computation (Compton scattering) only appears at page 362 (page 131 in Peskin-Schroeder), and it appears as a unique example, while Peskin-Schroeder teaches you how to compute any known process. So this book is more focussed on the inner working and motivations or foundations of the theory than Peskin-Schroeder or other similar books, which have applications to particle physics in mind.