Introduction; Part I. Basics: 1. Preliminaries; 2. Basics of non-relativistic quantum mechanics; 3. Non-relativistic quantum fields; 4. The Lorentz group and the Poincaré group; 5. The massive scalar free field; 6. Quantization; 7. The Casimir effect; Part II. Spin: 8. Representations of the orthogonal and the Lorentz group; 9. Representations of the Poincaré group; 10. Basic free fields; Part III. Interactions: 11. Perturbation theory; 12. Scattering, the scattering matrix and cross sections; 13. The scattering matrix in perturbation theory; 14. Interacting quantum fields; Part IV. Renormalization: 15. Prologue – power counting; 16. The Bogoliubov-Parasiuk-Hepp-Zimmermann scheme; 17. Counter-terms; 18. Controlling singularities; 19. Proof of convergence of the BPHZ scheme.
A lively and erudite introduction for readers with a background in undergraduate mathematics but no previous knowledge of physics.
Michel Talagrand is the recipient of the Loève Prize (1995), the Fermat Prize (1997), and the Shaw Prize (2019). He was a plenary speaker at the International Congress of Mathematicians and is currently a member of the Académie des Sciences (Paris). He has written several books in probability theory and well over 200 research papers.
'This book accomplishes the following impossible task. It explains
to a mathematician, in a language that a mathematician can
understand, what is meant by a quantum field theory from a
physicist's point of view. The author is completely and brutally
honest in his goal to truly explain the physics rather than
filtering out only the mathematics, but is at the same time as
mathematically lucid as one can be with this topic. It is a great
book by a great mathematician.' Sourav Chatterjee, Stanford
University
'Talagrand has done an admirable job of making the difficult
subject of quantum field theory as concrete and understandable as
possible. The book progresses slowly and carefully but still covers
an enormous amount of material, culminating in a detailed treatment
of renormalization. Although no one can make the subject truly
easy, Talagrand has made every effort to assist the reader on a
rewarding journey though the world of quantum fields.' Brian Hall,
University of Notre Dame
'A presentation of the fundamental ideas of QFT in a manner that is
both accessible and mathematically accurate seems like an
impossible dream. Well, not anymore! This book goes from basic
notions to advanced topics with patience and care. It is an
absolute delight to anyone looking for a friendly introduction to
the beauty of QFT and its mysteries.' Shahar Mendelson, Australian
National University
'I have been motivated to try and learn about quantum field
theories for some time, but struggled to find a presentation in a
language that I as a mathematician could understand. This book was
perfect for me: I was able to make progress without any
initial preparation, and felt very comfortable and reassured
by the style of exposition.' Ellen Powell, Durham University
'In addition to its success as a physical theory, Quantum Field
Theory (QFT) has been a continuous source of inspiration for
mathematics. However, mathematicians trying to understand QFT must
contend with the fact that some of the most important computations
in the theory have no rigorous justification. This has been a
considerable obstacle to communication between mathematicians and
physicists. It is why despite many fruitful interactions, only very
few people would claim to be well versed in both disciplines at the
highest level. There have been many attempts to bridge this gap,
each emphasizing different aspects of QFT. Treatments aimed at a
mathematical audience often deploy sophisticated mathematics.
Michel Talagrand takes a decidedly elementary approach to answering
the question in the title of his monograph, assuming little more
than basic analysis. In addition to learning what QFT is, the
reader will encounter in this book beautiful mathematics that is
hard to find anywhere else in such clear pedagogical form, notably
the discussion of representations of the Poincaré group and the
BPHZ Theorem. The book is especially timely given the recent
resurgence of ideas from QFT in probability and partial
differential equations. It is sure to remain a reference for many
decades.' Philippe Sosoe, Cornell University
'a wonderful resource for anyone who wants to understand exactly
what a quantum field theory is.' Peter Woit, Not Even Wrong
blog
'The text has many exercises and sixteen (!) appendices from which
one can learn quite a bit. This shows the dedication of the author
to the subject and his wish to share his knowledge with others. The
book hits the point between mathematics and physics where the first
is not too abstract and the second not too phenomenological … In
short, the book is exceptional and might set standards.' Marek
Nowakowski, MathSciNet
'The text covers the fundamentals: quantum mechanics, spin, second
quantisation, interactions and renormalisation. The book focuses
not only on mathematical proof, but justifications of how and why
we do things in quantum field theory. The text is well written
enough that even a graduate physics student would benefit greatly
from reading it.' Kymani Tieral Keden Armstrong-Williams,
https://physicsbookreviewer.blogspot.com/
'Assuming an undergraduate background in mathematics and just the
fundamentals of physics, the reader will profit from a very broad
and deep exploration of QFT. … Recommended.' E. Kincanon, Choice
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