What Is a Quantum Field Theory?

I am a layman in both math and physics, self‑studying QFT with this book alongside Introduction to Quantum Field Theory by Anthony Williams. I’ve read carefully up to Chapter 6. This book serves as a reliable self‑learning guide—partly because it is the product of an esteemed mathematician’s own self‑study. As the author notes, it began as "an attempt to rewrite Folland's textbook" during his study of QFT. Talagrand is kind to the reader: he assumes little in the way of prerequisites and builds steadily. His candid, self‑deprecating humor is relatable too—for example, "To top it all, I was buried by the worst advice I ever received: to learn the topic from Dirac's book itself." I have enjoyed studying Chapter 3 (Non‑relativistic Quantum Field), which offers a clean progression that actually feels learnable: tensor products of Hilbert spaces → symmetrization → boson Fock space → universe in a box → quantization. Each step is clear with careful exposition, especially the part on the number basis. By putting the mathematical structure front and foremost, it reduces the mystery around QFT. I also appreciated Talagrand’s clarifications of the physicists’ point of view. A telling example appears on p. 99: when physicists say “attaching a harmonic oscillator to each point of space,” what’s really meant is one attached to each point in the set of allowable momenta. These translations clarify common slogans. The same can be said of his exposition of group theory (Appendix D). The balance between the physics and math perspectives makes the book approachable and provides a solid footing in a subject many find challenging. I believe it provides a foundation for tackling the standard textbooks (Peskin & Schroeder, Zee, Weinberg). It is accessible even to a layman and useful for physics students who want a healthy dose of the mathematical perspective.

I am a PhD student in mathematics. In order to prepare for my doctoral ABD defense in August, I began studying the connections between quantum field theory (QFT) and statistical mechanics last year. I looked for many books on QFT for self-study, but most of them were not very accessible — they either assume an extremely strong background in mathematics or an equally strong background in physics. Michel’s book is different. It doesn’t require much prior knowledge; it starts from a level that is basic enough for me and gradually introduces the main ideas of quantum fields. To be honest, I haven’t finished reading the entire book yet, but I have already found in it the answers I was looking for — what QFT really is, and more importantly, where it comes from. The book covers a wide range of topics in quantum field theory. My suggestion for first-time readers is simply to focus on understanding what QFT is; the details can be filled in later during subsequent readings.

1. Talagrand does not sweep anything under the rug (or when he does, he explicitly told you and explain why he did it). 2. There are a number of places in the book that cracked me up, it is full of subtle humors. 3. I know nothing about physics to speak of. I have to read Wheeler & Taylor's Spacetime Physics to learn SR while reading Part II of this book. But you can go through the book without knowing any serious physics. The book treat QM as a mathematical formalism which I find helpful. The author did not lie when he told you about the prerequisites for the book (which is so rare among math/physics book writers nowadays) 4. You need a solid background in Linear Algebra and Calculus (when in doubt, integrate by parts). But the rest of the math can be slowly (but surely) picked up from the appendix. 5. This is an extreme personal work for Talagrand and it shows, and it is such a joy to read, especially his comments between the maths. 6. If you email the author about your questions, he will reply promptly with helpful comments. There is an errata on his website which is also helpful. 7. I'm going to read it again soon. There are so many things I feel I can understand better on a second read.

It is difficult for me to give a sufficiently high praise to this book. I am a professional mathematician, working in partial differential equations and differential geometry, and my primary research focuses are different topics. Nonetheless I have encountered a large number of times colleagues in physics suggesting this and that about some things I am interested in, and inevitably one key bottleneck of all our conversations is that I could not make heads of what they were saying, computing, or suggesting when QFT became part of the conversation, which was the case in so many occasions. Like the author I was often intrigued, but I encountered enormous difficulties in the past to make sense of too many thing. This book has been a revelation. Finally so many past conversation have started to make (at least some :-)) sense!

This is an excellent book on quantum field theory. The physical quality is superb: solid binding, high-quality printing, and overall a pleasure to work with. You can immediately tell it’s built to last. The content is equally impressive. Talagrand writes with clarity, precision, and remarkable depth. The book is not only suitable for mathematicians but also for physicists who are looking for a rigorous yet well-structured introduction to quantum field theory. A book you enjoy using – and one you definitely won’t be throwing away after heavy use.

This book is undoubtedly the best resource I've found for truly understanding quantum fields and how they work. It is also a model of exemplary mathematical writing. As a mathematically inclined reader attempting to self-learn, I often found the treatment of many aspects of the theory in standard physics textbooks confusing. Michel Talagrand’s rigor and clarity have been the blessing I had given up hoping for. I highly recommend this book not only to mathematicians but also suspect that physicists would greatly benefit from it too. By rigor, I don't mean nitpicking over the inversion of integral orders, but rather clearly defining every mathematical object used, maintaining consistent notation throughout, not skipping any developmental steps, and being honest about where legitimate math ends and physics heuristics begin. The benefit of rigor in this context is not mere intellectual satisfaction but understanding. Another remarkable aspect of this book is that while Talagrand develops the theory as much as possible using only solid mathematical methods and definitions, he also presents how physicists typically approach the theory and explains how to reconcile the different approaches. To physicists, Talagrand does not propose divorce, but mediation. This is useful for readers intending to pursue further education in QTF theory. I found Chapter 10 on Basic Free Fields and Chapter 14 on Interacting Quantum Fields particularly enlightening. Chapter 10 develops a general framework for constructing quantum fields for different types of particles. Chapter 14 culminates with a justification of the LSZ reduction formula and introduces the challenging notion of renormalization. While this doesn’t make it easy, it certainly makes it understandable, which is a real feat. These are just primary examples, so many other parts of the book deserve praise: the chapters on representations, the various approaches to quantization, the intrinsic definition of a quantum field, to name a few. Throughout, Talagrand replaces the hand-waving and dubious symbol manipulation found in standard literature with lucid, precise development. It's a pleasure for the mind. In conclusion, this book is a gem, and I expect it to become a standard reference for many years to come.

Hat alles gut funktioniert.