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Title: Anomalies in Quantum Field Theory (International Series of Monographs on Physics, No 91) by Reinhold A. Bertlmann ISBN: 0-19-850762-3 Publisher: Clarendon Pr Pub. Date: January, 2001 Format: Paperback Volumes: 1 List Price(USD): $75.00 |
Average Customer Rating: 4 (1 review)
Rating: 4
Summary: A good overview
Comment: An understanding of quantum field theory cannot be done without the consideration of anomalies. The occurrence of anomalies in quantum field theory is relatively new, if compared with the time period that the subject has been around. This book gives an excellent overview of anomalies and is suitable for those who have a background in quantum field theory. Mathematicians may also find it of great interest, even though their standards of rigour may not always be respected in the book. The study of anomalies will entail also a background in mathematics that may not be acquired by a reader who has taken a prior course in quantum field theory. Hence the author spends the first part of the book outlining the mathematics needed, such as differential geometry, homotopy theory, de Rham cohomology, Lie groups, and fiber bundles. With the presence of this level of mathematics, one can say indeed that a course in quantum field theory in the 21st century is very different from one just 20 years ago.
The author gives a motivation for the subject in the introduction to the book. He describes the origin of anomalies as the violation of a classically conserved current, which for gauge theories signals the breakdown of gauge symmetry, leading to an inconsistent theory. The elimination of anomalies then leads to a tightly constrained theory, and leads to physical predictions, such as the top quark. The author's concern in the book is the axial- or chiral anomaly that corresponds to an axial- or chiral fermion current.
Even though it would be rare these days to find a beginning student of quantum field theory who had learned it without the consideration of path integrals, the author gives an overview of them in chapter 3 of the book. This chapter also includes a thorough discussion of the Faddeev-Popov method in non-Abelian gauge theories. One can view this as a method of "Lagrange multipliers" in quantum field theory, with the famous Faddeev-Popov ghosts playing the role of these multipliers. The author does a fine job of discussing why they are so important for the physics of quantum field theory, such as preserving the unitarity of the S-matrix and the proof of Ward identities.
The study of anomalies begins in detail in chapter 4. The author discusses the origin of the Adler-Bell-Jackiw anomaly and why the various strategies of renormalization must lead to the same result. That radiative corrections or contributions from higher order perturbation theory will not change the nature of the anomaly is the statement of the Adler-Bardeen theorem, but the author does not prove this theorem. The author also relates the chiral anomaly to dispersion relations in the context of 2-dimensional quantum electrodynamics, and the connection of anomalies to the Dirac sea via the study of the Schwinger model of 2-dimensional quantum electrodynamics on a cylinder. The experimental consequences of anomalies are discussed by studying the decay of the neutral pion into two photons. A calculation of the transition amplitude for this decay using the LSZ reduction formula yields zero, and so anomalies are brought in force the decay rate to be in agreement with the experimental value. Lastly, the author discusses the non-Abelian anomaly in the last part of the chapter.
Chapter 5 studies anomalies in the context of path integrals, via the methods of Fujikawa, arising as they do in the consideration of the transformation properties of the path integral measure. What is most interesting about this discussion is the author's statment of an "uncertainty principle", namely that one cannot impose gauge and chiral symmetry simultaneously, due to the noncommutation of the Dirac operator and the gamma-five matrix. In addition, the author shows in detail the connection between the zeta function and the Fujikawa procedure.
After a brief review of differential forms in chapter 6, the author studies the relation of anomalies to the Chern-Simons form in chapter 7. The transgression formula, which shows that the difference of two invariant polynomials is exact, is proven because of its central role in the rest of the book. Using a notion of a "homotopy derivation", the first-order variaton of the Chern-Simons form is calculated and shown to give an anomaly formula.
The derivation of the Wess-Zumino consistency condition is the main subject of chapter 7. This condition determines the anomaly, in that any solution of it that is not a gauge variation of a local functional is an anomaly, and is derived via the BRS formalism. The Wess-Zumino consistency condition is then shown to to correspond to a cocycle condition in the space of gauge potentials, and more generally in the context of the cohomology of Lie algebras.
The author derives, from a stricly mathematical viewpoint, the Stora-Zumino descent equations in chapter 8, which show that a singlet anomaly in even dimensions determines the non-Abelian anomaly in two dimensons less.
The "covariant" anomaly is discussed in chapter 10, which as the name implies transforms covariantly, unlike the consistent ("Bardeen") anomaly considered so far. Noting that they arise from two different regularization schemes, the author then relates them via the Bardeen-Zumino polynomial.
The most interesting chapter is the next one, which discusses the index theory of anomalies. Having very important mathematical connections, the presentation is very understandable but not rigorous mathematically, due to the use of the path integral. The author shows that anomalies are intimately connected with the topology of gauge theories, in particular the non-Abelian anomaly characterizes the nontrivial topology of the determinant bundle over the 2-sphere X 2n-sphere.
Gravitational anomalies are the subject of the last chapter, wherein gauge transformations are the coordinate and Lorentz transformations. The work in preceeding chapters proceeds here without too much change, and the reader can see the fermionic contribution to the occurrence of anomalies, and the author again uses the BRS formalism to derive the descent equations.
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Title: Geometry, Topology and Physics by Mikio Nakahara ISBN: 0750306068 Publisher: Institute of Physics Publishing Pub. Date: October, 2003 List Price(USD): $55.00 |
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Title: Gauge Theories in Particle Physics: A Practical Introduction: From Relativistic Quantum Mechanics to Qed by Ian J. R. Aitchison, Anthony J. G. Hey ISBN: 0750308648 Publisher: Institute of Physics Publishing Pub. Date: 01 September, 2002 List Price(USD): $39.99 |
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Title: Mathematical Concepts of Quantum Mechanics (Universitext) by Stephen J. Gustafson, Israel Michael Sigal ISBN: 3540441603 Publisher: Springer Verlag Pub. Date: October, 2003 List Price(USD): $49.95 |
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Title: Geometry, Particles, and Fields (Graduate Texts in Contemporary Physics) by Bjrn Felsager, Bjorn Felsager ISBN: 0387982671 Publisher: Springer Verlag Pub. Date: January, 1998 List Price(USD): $84.95 |
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Title: Renormalization : An Introduction to Renormalization, the Renormalization Group and the Operator-Product Expansion by John C. Collins ISBN: 0521311772 Publisher: Cambridge University Press Pub. Date: 23 January, 1986 List Price(USD): $35.00 |
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