Quantum Field Theory for Mathematicians (Encyclopedia of Mathematics and its Applications Book 72) 1st Edition by Robin Ticciati (PDF)

Description

The approach to quantum field theory in this book is part way between building a mathematical model of the subject and presenting the mathematics that physicists actually use. It starts with the need to combine special relativity and quantum mechanics and culminates in a basic understanding of the standard model of electroweak and strong interactions. The book is divided into five parts: 1. Canonical quantization of scalar fields; 2. Weyl, Dirac and vector fields; 3. Functional integral quantization; 4. The standard model of the electroweak and strong interactions; 5. Renormalization. This should be a useful reference for anybody with interests in quantum theory and related areas of function theory, functional analysis, differential geometry or topological invariant theory.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐I preface my comments by stating that this book is not intended as an introduction to QFT.The student should have a solid understanding of SR, QM, tensor analysis, group theory including Lie Groups, and Hilbert spaces.I will not regurgitate what the book covers, one need only use the “search inside” tab to look at the contents.Having said this, this book is an excellent and indispensible to tool to BROADEN and DEEPEN your understanding of QFT. If all you want to do is calculate scattering amplitudes and decay rates I would not recommend this book, there are plenty of better applied QFT books available for this.This books fills in the gaps other books fail to close. There is no “hand waving” of results which was refreshing. As a consequence you begin to understanding the subtle points of QFT and why the theory is the way it is.As mentioned in the title of the review there are plenty of “pearls”. For example, there is an entire chapter on internal and external symmetries and their representations by groups of matrices ( lie groups ). There is a complete description of the importance of Lie alegbras and how the generators of the Lie Algebra create conserved currents and quantities ( operators ) which help one study the evolution of states since these quantities are conserved. By studying the structure of the lie algebra one gains importance insights into the commutative properties of the corresponding conserved current and quantity operators. There is a great section on the derivation of the S matrix and the relations between the “Schrodinger ” ” Heisenberg ” and “Interaction” pictures of QM. We see that the evolution of the interacting state can be entirely derived from the free field hamiltonians with certain restrictions. One thing I really liked about this section is that it explains the limitations of the S matrix approach ( has to do with the assumptions of turning “on and off” interactions )which I have not come across in other standard QFT texts. This motivates the need for functional integral quantization.Another point of contention I have had with standard presentations of QFT is that they just assume that Noether’s theorem from classical field theory can be applied after the quantization process. This book explains mathematically why it can be.Succinctly, the defects in QFT presentation in other texts is explained, which makes understanding the material more difficult. However, the payoff is that one understands the motivation behind the IDEAS of QFT.The book is also filled with little “homework” assignments to solidfy knowledge.The logical and organized presentation of the material made it very difficult for me to put this book down for any length of time until it was finished.

⭐This book is far from perfect, but I think it begins to fill an important niche in the world of QFT books: it presents most aspects of the theory, from basic principles to Feynman rules, gauge fields and renormalization, in a form that is unusually accessible to mathematicians. I’m coming at this from the perspective of a mathematician who has tried and failed to learn QFT from a variety of other books, and I wish I had discovered this one before even opening Weinberg or Peskin & Schroeder. Ticciati doesn’t completely avoid the kind logical sleight of hand that is commonplace among physicists, but when doing manipulations whose mathematical basis is questionable, he’s usually at least honest enough to point this out to the reader. I especially enjoyed the chapter on Lie algebra representation theory, which is closer to a mathematician’s presentation of this subject than a physicist’s, yet not without plenty of physical motivation. I’d criticize this book only for two things: (1) it’s riddled with misprints (some obvious, some not) and (2) some topics are explained rather more concisely than they deserve, and not always in the most logical order; Ticciati has a tendency to use certain subtle concepts implicitly a few sections before he defines them precisely. One may hope that such errors will be corrected in a future edition.

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