PCT, Spin and Statistics, and All That

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Summary: Wrong, but still needed
Review: Axiomatic Field Theory (AFT) is the worst of both worlds: it is not very productive, because it tries to be completely rigorous. But, it turns out, it is not rigorous either: the progress in QFT of the last thirty years has shown that assumptions of locality made by AFT don't survive renormalization.

On the other hand, many of the general results of AFT, like the PCT and spin-statistics theorems HAVE to be true and it should be possible to derive them from a rigorous point-of-view. There is a need for a replacement of this book, but none is in sight.

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Summary: Obsolete and full of incorrect statements
Review: Many new discoveries and advances in particle physics and quantum field theory have been made since 1964, the year when this book was originally published. In the 1960s, this book might have been a legitimate attempt to establish a new, albeit controversial paradigm in the research of Quantum Field Theory. However the authors were not too lucky because now, exactly 40 years later, we see that Axiomatic Field Theory has not led anywhere, unlike the specific, constructive, and old-fashioned quantum field theories. Axiomatic Field Theory has given no physical predictions and it has led to no conceptual developments. Today, Axiomatic Field Theory is not an active field of physics anymore. Moreover, most of its conclusions are believed to be incorrect.

Although some sections of the book may sound familiar and they are remotely related to some topics in contemporary physics, the book does not cover the most essential parts of current quantum field theory - such as gauge theories, path integrals, Higgs mechanism, confinement, asymptotic freedom, renormalization group, dualities, solitons, instantons, semiclassical treatment of quantum gravity, string theory, and many others - and even the topics that the book tries to cover are described in a confusing way. For example, the book claims that one can always define the parity operator, which everyone today knows to be wrong, and so on.

The discovery of the Renormalization Group (RG) showed that many exact - and seemingly rigorous - ideas about the operator algebras were too naive to be true. Today, a realistic quantum field theory must be given a distance scale, and all quantities are calculated with respect to this scale. There exists almost no useful quantum field theory that would satisfy the axioms of Axiomatic Field Theory, and therefore the "theorems" derived within the framework of Axiomatic Field Theory have almost no physical impact. Although there are many correct and useful statements in the book, the number of incorrect and misleading sections is too large and it makes the book useless. There are much better recent books written at a comparable level of difficulty, e.g. "Quantum Field Theory in a Nutshell" by Anthony Zee.

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Summary: What is a quantum field ?
Review: This advanced book exposes the mathematical framework of quantum field theory, as discovered by Wightman, who explained, in an article published in Physics Today, that he was led to such abstract language in order to understand the forces that bind the deuteron nucleus! This was only the beginning: present books and papers on the structure of quantum fields employ (and develop!) mathematics which make the Streater-Wightman book look like Harry Potter. Still, this book is assumed reading for all these advanced studies. It is extremely well written, for one thing. The initial chapter introduces the concept of free particle along ideas which I think belong to Wigner: particles are irreducible representations of the Poicare group. So, group theory is necessary for the grammar of quantum field theory. Fields require Schwartz distribution theory, as intuitively discovered by Bohr and Rosenfeld. Then follow general theorems for local field theory, like spin-statistics, CPT (and all that...). There is now an appendix where more modern topics are presented, but the core of the book is more or less what I described. This is a great book, written with great care and wit. It is a pity that most physicists, even quantum field theorists, did'nt read it.

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Summary: Related comments
Review: [...]
for a highly readable current overview on whether quantum field
theories 'exist,' and the unique importance of this book.
It describes a prize you can get if YOU figure it out.
Personally I cannot yet say whether a better definition of
'existence' is possible, but this one is important. Classic papers by Alfred Goldhaber and by 'tHooft etc (Phys Rev letters 76) and
Wilczek and Zee rely heavily on this book, and remain extremely important.