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12 of 12 people found the following review helpful:
5.0 out of 5 stars
I practically owe my today's academical self to this work..., October 11, 2004
As someone working in the field created by Prof. Haag - Local Quantum Physics, aka Algebraic Quantum Field Theory - I feel somewhat oblidged to write a review on this book. This is all the more true when a large amount of misunderstandings about this subject among, so to speak, "outsiders", pervade the community of theoretical physics. As for me, I had the good luck of having Local Quantum Physics as my entering door to Quantum Field Theory, after my undergraduate involvement with Nuclear Physics. Learning this through (in a major part) Prof. Haag's book, alongside with a conventional course in QFT, has helped me clear several conceptual issues underlying QFT tools and calculations - specially renormalization - which alone seemed to me more witchcraft than physics.
The aims of Local Quantum Physics, even when linked to computational issues, are eminently structural and conceptual, going beyond particular models. These concerns are transparent in this book, where only the essentials of the Lagrangian approach are mentioned, and even these with a conceptually clean and deep purpose (just to cite an example, quantization of free fields are treated in a covariant way by using the Peierls' bracket, instead of canonical quantization), and with no predilection whatsoever by any particular quantization technique (for instance, path integrals are only mentioned "en passant", with no formulas, in Section VIII.1, in the discussion on the Euclidean/Lagrangian approach to QFT). This last proviso, which is a common source of complaint, actually (at least, it looks so to me) bears the following message under the aegis of the aims above: the physical concepts of QFT have nothing to do with the quantization method chosen. Although the justification for this is somewhat subtle, it ends up being a natural consequence of the line of thinking along which this book proceeds.
Most of the things about which Prof. Haag writes in this book seem to have been thought about for a pretty long time. It's thanks to this that the formalism of Local Quantum Physics acquired a remarkably flexible and synthetic language. The underlying idea, present in almost every topic treated in the book, is the principle of locality ("Nahwirkungsprinzip" = "Principle of local action", i.e., no action at a distance). Namely, that physical procedures are all localized in finitely extended regions of spacetime, as it "usually" happens in experimental situations, and that the matter of choosing a Hilbert space on which these procedures act (often based on global criteria such as the concept of a vacuum state) is mainly a matter of convenience. The abstract framework of C*- and von Neumann algebras is what allows one to work independently of a particular representation. This is strengthened by Einstein causality - physical procedures localized at causally disjoint regions commute with each other (This is quite distinct from locality in the sense of the EPR phenomenon, which is intrinsically linked to the notion - here generalized - of states, this one still highly nonlocal, as restrictions of a state to two causally disjoint local algebras of procedures can, and do, present quantum entanglement if this state is suitably prepared), and Poincaré covariance.
The principle of locality, when applied to the myriad of inequivalent representations of the local algebras which is characteristic of QFT, lead to enormous achievements (most of them described in the book), such as: the meaning of internal global symmetries and fermion degrees of freedom, and how these emerge from the observables alone, independently of the assumption of an underlying field theory (superselection sectors); the meaning of infinities and renormalization in perturbation theory (disjointness and quasi-equivalence of representations); a natural setting for QFT at finite temperature and its thermodynamics (KMS condition, modular techniques, phase space conditions); when moving to curved spacetime, the clarification of the (still open) issue of the choice of physical states from nonessentials and how this forces us to "unlearn" several concepts of Minkowski QFT (Unruh effect, etc.). Recent developments by the schools of Wald and Fredenhagen show the growing importance of the latter and related problems.
Finally, other two admirable aspects of Haag's book are the honest treatment of latest developments regarding conceptual open issues such as the meaning of local gauge invariance in quantum theory, the infrared problem, and questions regarding the interpretation of quantum mechanics and the meaning o spacetime itself. Haag's closing personal views on the latter, in the light of the mathematical formalism of Local Quantum Physics, bear an intriguing resemblance with modern ideas by Rovelli, Ashtekar, etc. on loop quantum gravity.
The book as a whole takes quite some time to digest, due to the mathematical machinery involved (functional analysis and an acquaintance in C*-algebra theory are a rather strongly recommended background) and the subtlety of the physical ideas. But, to sum up, for me it was, in due time, the ultimate temptress.
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11 of 12 people found the following review helpful:
5.0 out of 5 stars
Deserves 10 stars, April 16, 2002
Quantum field theory is a subject that has occupied the time of an enormous number of researchers, both in physics and in mathematics. Those who have studied perturbation methods in quantum field theory have no doubt run acroos "Haag's theorem" that is usually loosely stated as saying that "the interactive representation does not exist". The statement of this theorem, and many other results in quantum field theory, particularly the procedure of renormalization, have been viewed by many as unsound from a mathematical standpoint, and so efforts were begun to put quantum field theory on a rigorous mathematical foundation. Going by the names of axiomatic or constructive quantum field theory, these approaches are interesting, but also a little troubling from a scientific perspective. Axiomitization is usually appropriate in mathematics when a subject has matured to the point where it can be "closed off", and this usually happens when the theory is very well understood and so its essence can be codified in a few well-forumlated axioms. But quantum field theory is no where near that stage; indeed one can say that it continues to be a theory that, oddly, has immense predictive power but whose rigorous mathematical formulation remains elusive. Not only that, quantum field theory is still in a course of evolution, and any attempt at axiomitization might become obsolete as soon as it is put down on paper. In addition, physical insight, as much as mathematical understanding, must not be sacrificed in any resulting axiomatization of quantum field theory. Frequently, the result of axiomatization is to divorce a physical theory from its physical roots, and beginning students of the theory then have difficulty in acquiring intuition of the essential physics of the theory. One of the best attributes of this book is that the author realizes this, and early on he refers to "general", rather than "axiomatic" QFT as being more appropriate since it allows flexibility in relation to future discoveries. Not only that, the author endeavors to explain the formalism that he is expousing in the book, and he succeeds brilliantly. Anyone interested in the mathematical physics behind quantum field theory, and not just doing bread-and-butter perturbation calculations, will gain a lot from the reading of this book. It is packed full of insight, a rare occurence in books that employ the heavy mathematical formalism that this one does. One will need a strong background in operator theory, abstract theory, and several complex variables to read the book, but a lot of this is developed impromptu as the text unfolds. When it is not, the author gives references for those readers who need more in-depth discussion. There are so many ineresting discussions in this book that space does not permit an evaluation of all of them, but the following is a short list of points in the book that I found particularly well-written: 1. The Wigner analysis of irreducible unitary representations of the Poincare group. This is not a mathematically rigorous discussion, but the author points out the physical relevance of the fact that the spectrum of the 4-momentum operator must be concentrated on a single orbit. This fact ensures the stability of matter. And, as frequently happens in physics, several mathematical consequences of a particular physical theory are discarded as not being relevant; in this case the other three classes of the irreducible representations. That being said, the author does include as of possible physical relevance the idea of parastatistics. He points out his reasons for this, namely that a strict adherence to the Bose-Fermi alternative is not operationally justified. 2. The role of fields in implementing the principle of locality and not as observable particles. This fact is usually not emphasized in books on quantum field theory. 3. The author clarifies the distinction between the notion of locality that deals with the commutation of two observables that are space-like separated, and the one dealing with the Einstein-Podolsky-Rosen paradox and Bell's inequality. 4. The discussion on the Bose-Einstein alternative, in particular the suggestion that parastatistics can be replaced by Bose or Fermi statistics in the presence of a non-Abelian unbroken global gauge group. 5. The discussion on topological charges and their prohibition by the Doplicher-Haag-Roberts selection criterion. The Doplicher-Haag-Roberts criterion was used in scattering theory and thought to be reasonable, but the author shows that its use is problematic in this case also, as well as in prohibiting topological charge. Purely massive fields can, it turns out, have measurable correlations at large distances, and Borcher's selection criterion, also discussed along these lines, gives topological charges. 6. The treatment of the Tomita-Takesaki theorem, modular automorphisms, and their connection to the KMS-condition. 7. The discussion on the need for type III-1 von Neumann algebras in relativistic quantum field theory versus type I in ordinary quantum mechanics. Such a von Neumann algebra is hyperfinite and is unique. 8. The discussion on the impossibility of coherent wave packets of one-electron states in quantum field theory, as contrasted with the usual practice in quantum mechanics. This is dues to superselection rules and the "infraparticle" nature of electrically charged particles, which are not associated with discrete eigenvalues of the mass operator. The author asks the reader to justify electron interference experiments in quantum field theory.
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6 of 7 people found the following review helpful:
5.0 out of 5 stars
The most important book about algebraic qft by its founder, May 1, 1999
In spite of the succes of quantum field theory it became very early clear that this theory needed a new mathematical formulation. Haag was one of the founders of this new theory which was later called algebraic quantum field theory but Haag himself preferred "local quantum physics". The algebra of observables is designed as the C*-inductive limit of a net of von Neumann-algebras the index set of which is formed of open subsets of space-time. The book discusses the DHR-selection criterion as well as the BF-criterion of Buchholz and Fredenhagen that is more adequate to massive fields. Furthermore Haag gives a short introduction to statistical qft in the algebraic framework. Especially the KMS-condition which was formulated in the sixties by Haag, Hugenholtz and Winnink is discussed. A highly recommended book!
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