on October 30, 2012
This is a must-have book for all serious students of
relativistic quantum field theory. The author Anthony H.
Duncan is a professor in the Dept. of Physics and Astronomy
at the University of Pittsburgh known for his research on quantum chromodynamics. He has produced a magnificent volume on the conceptual ideas that underlie the most accurate description of microscopic reality we have. It will be of
immense value to graduate students, working physicists, and
even some philosophers of science.
The book is divided into four parts. The first is a short history of QFT starting with Planck's work on blackbody radiation through the contributions of Jordan, Dirac, Pauli,
Heisenberg, etc. up to the Shelter Island conference. The second part presents the foundations of the theory as derived
from quantum mechanics, special relativity, and the clustering
property. Here the influence of Steven Weinberg is clearly seen. Prof. Duncan was a student of Weinberg and his book can
be viewed as a useful companion to Weinberg's three volume
text. There are some minor differences in conventions : Duncan
uses the Minkowski metric with diag(1,-1,-1,-1), etc.
The climax of the second part is the insightful chapter 9
on the general aspects of interacting fields including Haag-Ruelle scattering theory and the LSZ formalism. The perturbative formulation is discussed in chapter 10 where path
-integral methods are introduced and an explanation is given
of why one may ignore the implications of Haag's Theorem.
Chapter 11 presents " perturbatively non-perturbative " processes - threshold bound states via the Bethe-Salpeter equation.
The third part of the book is devoted to symmetries in field theory. There is a chapter on discrete symmetries with
a discussion ( but no proof ) of the PCT and the spin-statistics theorems. Then chapters on continuous global and
local symmetries and the breaking of these symmetries. The
last part of the book is concerned with the implications of
varying distance-scales leading to a discussion of effective
field throries, the renormalization group, perturbative
renormalizability, the short and long distance structure of
the theory and concluding with an explanation of confinement
in gauge theory.
This book is written as a graduate level text book with
problems at the end of most chapters ( but no solutions are
provided). The price may be an unfortunate deterrence to
some potential readers. Even in a book of almost 800 pages, one cannot cover all the interesting aspects of QFT. No mention is made, for example, of the renormalization method
of Epstein and Glaser ( causal perturbation theory ) , of QFT
in curved space-time, nor of Algebraic Quantum Field Theory
( Local Quantum Physics ) which in any case would require
another book by itself.
Overall, I think that the author succeeds in his attempt "
... to arrive at a truly deep and satisfying comprehension of the most powerful, beautiful, and effective theoretical edifice ever constructed in the physical sciences ...".
When one engages in the learning of quantum field theory it is easy to be amazed by what typically can be a collection of mathematical tricks can result in such unprecedented agreement with experiments. At the same time, there are many aspects of the subject that can be troubling to both the physicist and the mathematician alike. Physicists can be troubled by the typically abstract nature of quantum field theory while mathematicians can be annoyed by the constructions that are not defined rigorously. These concerns have instigated an enormous amount of research and many questions still remain to be answered in quantum field theory. It is a very active field of research and no doubt this will be the case for years to come.
Some of the questions that are frequently asked by those first learning quantum field theory and even experienced researchers include:
1. What is really the meaning of Haag's theorem in terms of the interaction picture in quantum field theory?
2. Why is cluster decomposition important and should it be considered on the same level as for example the requirement of Lorentz invariance? Can cluster decomposition be established without using the creation-annihilation operator formalism (so-called "second quantization")? The author views cluster decomposition as being responsible for most of the interesting phenomena in quantum field theory, and Lorentz invariance as being a kind of incidental side constraint. This is an interesting comment and many who have learned quantum field theory (such as the reviewer) have not found this emphasis on cluster decomposition in other texts and monographs on quantum field theory. Along these same lines the author views cluster decomposition as a kind of search engine for finding sensible quantum field theories, in that those theories that incorporate it will have long-distance behavior that is more in line with physical reality.
3. Why is it so difficult to establish the existence of a bound state in quantum field theory? Should quantum field theory be viewed as a formalism to be used strictly for scattering phenomena with bound state calculations done using an extension of quantum field theory? Will insisting on the formation of bound states in quantum field theory entail a radical revision of the formalism? It would seem at first glance that the 'clustering principle' that the author views as being absolutely fundamental is really only of relevance for scattering phenomena. What is its applicability for lets say a proton and an electron who come close enough together but do not scatter, but instead form a bound state (a hydrogen atom)? The author gives a fine discussion of various Hamiltonians that lead to clustering theories. Can the same be done for Hamiltonians that lead to "bound-state theories" or is this not a meaningful question to ask within the scope of quantum field theory? Are quantum field theories that allow bound states 'ultra local' in the sense that the author uses this term in the book, or will they violate ultra locality? Do the 'particle-field duality axioms', which are deemed to be essential to the development of a "comprehensible" theory of particle scattering in quantum field theory have analogs in showing how particles (fields) come together to form bound states?
4. Do ideas from the renormalization group only make sense for quantum field theories that are renormalizable? Along these same lines, can a non-renormalizable quantum field theory have any kind of "predictive power"?
5. What is the nature of number-phase complimentarity for fields?
6. Did the founders of quantum field theory find it difficult intuitively to accept the concepts behind quantum field theory? One might ask this question in modern terms as wondering whether there was any kind of cognitive dissonance among the early practitioners of quantum field theory?
7. What is the micro causality principle in quantum field theory and why is it important? Can it be dispensed with without affecting the successful predictions of quantum field theory?
8. The "action-at-a-distance" effects that the author speaks of in the book are to be distinguished he says from the "entangled" states of EPR fame. But do these effects really predict phenomena that are of interest observationally or are they too small to be relevant from the standpoint of what is measurable from a technological perspective? Should this issue be viewed then in the context of "imperfect resolution" of measuring devices? If so, this might allow a resolution of the 'infrared problem' by approximating massless photons by massive photons and concentrating only on 'inclusive cross sections'.
9. What is really the mechanism behind the appearance of inequivalent representations of the canonical commutation relations in quantum field theory?
10. Can quantum fields "restore" broken symmetries at the classical level? For example, are there quantum field theories that when viewed from the standpoint of perturbation theory will recover momentum conservation when the classical counterpart manifestly breaks symmetry translation? This would imply that momentum convservation, although broken classically, could be recovered in quantum field theory (at least perturbatively).
11. What conditions guarantee that the vacuum state is cyclic for products of quantum field operators localized in a bounded Euclidean spacetime region at positive time?
12. Can inhomogeneous local gauge transformations be used to generate massive gauge bosons, and consequently avoid the need for the Higgs mechanism?
Not all of these questions are answered in this book, but it has enough discussion of elementary principles and concepts to allow the reader who is genuinely interested in understanding quantum field theory ample food for thought. In this regard, some of the highlights of the book include:
1. The historical introduction and the insight it offers on how quantum field theory arose as a methodology: (a) The role played by Pascual Jordan on showing that the view of Einstein who held that energy fluctuations of electromagnetic radiation has two "structurally independent" causes is not necessary if one uses an electromagnetic field description.The work of Jordan has recently been formalized in the framework of algebraic quantum field theory using a concept called 'modular localization' and is a highly interesting development in the foundations of quantum field theory. (b) The contrast between today's emphasis on the use of functional methods in quantum field theory versus what was the preferred method in the early years of quantum field theory, namely the interaction-picture methods of Julian Schwinger. It was the contributions of Freeman Dyson that apparently convinced many to switch over to the Feynman paradigm of path integrals, with a few physicists however hanging on to the Schwinger methods well into the 1970's.
2. The many discussions throughout the book on how to implement classical symmetries in quantum field theory and problems faced when attempting to do this (the famous anomalies). The author gives an interesting example in quantum mechanics involving (canonical) point transformations to motivate the occurrence of anomalies.
3. That the author believes that quantum field theory is manifested most powerfully as a scattering theory is readily apparent throughout the book. His discussion of the need for requiring more than just Lorentz invariance in building a scattering theory in relativistic quantum mechanics is very interesting and gives good motivation for the consequent discussion of cluster decomposition.
4. The treatment of the Majorana field, which lately has become very important from an experimental point of view in condensed matter physics.
5. The chapter on the classical limit of quantum fields. Those who are interested in whether or not superconductivity for example is a "macroscopic manifestation of quantum phenomena" as is claimed many times should find this chapter of great interest. In addition, the discussion of the anti cyclic permutation operator as being the quantum analog of the complex classical phase is very helpful from the standpoint of the number-phase uncertainty principle.
6. The author is not shy about discussing the bound state problem, and this is refreshing considering the extremely difficult nature of this problem. Even outside of the context of his discussion on bound states, the author gives information that indicates that many of the results in quantum field theory cannot be derived if bound states are present. One example of this is the discussion on the principle of asymptotic completeness. Another example is the discussion of the Heisenberg field, where certain 'auxiliary' fields are introduced to characterize the kinematical structure of asymptotic states. What is not so clear is whether or not these asymptotic states can indeed be bound states. i.e. can the "out-states" be bound states, and will the Heisenberg field be of assistance in constructing these states, conforming to the claim that "knowledge of the Heisenberg field" will always allow a construction of the out-states. The author is well-aware of the fact that the S-matrix philosophy concentrates on what happens in the distant past and future, and ignores what happens "in-between" to use his terminology.
The only major omission that the reviewer is aware of is that the author does not give an in-depth discussion of the Reeh-Schlieder theorem, but merely refers to this interesting result only in a footnote.