An Introduction to Non-Perturbative Foundations of Quantum Field Theory (International Series of Monographs on Physics)

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Quantum Field Theory (QFT) has proved to be the most useful strategy for the description of elementary particle interactions and as such is regarded as a fundamental part of modern theoretical physics. In most presentations, the emphasis is on the effectiveness of the theory in producing experimentally testable predictions, which at present essentially means Perturbative QFT. However, after more than fifty years of QFT, we still are in the embarrassing situation of not knowing a
single non-trivial (even non-realistic) model of QFT in 3+1 dimensions, allowing a non-perturbative control. As a reaction to these consistency problems one may take the position that they are related to our ignorance of the physics of small distances and that QFT is only an effective theory, so that
radically new ideas are needed for a consistent quantum theory of relativistic interactions (in 3+1 dimensions).

The book starts by discussing the conflict between locality or hyperbolicity and positivity of the energy for relativistic wave equations, which marks the origin of quantum field theory, and the mathematical problems of the perturbative expansion (canonical quantization, interaction picture, non-Fock representation, asymptotic convergence of the series etc.). The general physical principles of positivity of the energy, Poincare' covariance and locality provide a substitute for canonical
quantization, qualify the non-perturbative foundation and lead to very relevant results, like the Spin-statistics theorem, TCP symmetry, a substitute for canonical quantization, non-canonical behaviour, the euclidean formulation at the basis of the functional integral approach, the non-perturbative
definition of the S-matrix (LSZ, Haag-Ruelle-Buchholz theory).

A characteristic feature of gauge field theories is Gauss' law constraint. It is responsible for the conflict between locality of the charged fields and positivity, it yields the superselection of the (unbroken) gauge charges, provides a non-perturbative explanation of the Higgs mechanism in the local gauges, implies the infraparticle structure of the charged particles in QED and the breaking of the Lorentz group in the charged sectors.

A non-perturbative proof of the Higgs mechanism is discussed in the Coulomb gauge: the vector bosons corresponding to the broken generators are massive and their two point function dominates the Goldstone spectrum, thus excluding the occurrence of massless Goldstone bosons.

The solution of the U(1) problem in QCD, the theta vacuum structure and the inevitable breaking of the chiral symmetry in each theta sector are derived solely from the topology of the gauge group, without relying on the semiclassical instanton approximation.

An Introduction to Non-Perturbative Foundations of Quantum Field Theory (International Series of Monographs on Physics) Review

Quantum field theory (QFT) is a theory of elementary particles combining quantum mechanics and special relativity. The synthesis of these two is far from straightforward. In fact, the synthesis of quantum mechanics and *general* relativity, known as 'quantum gravity,' is still an open problem. Excluding string theory, on which the jury is still out, QFT is the most fundamental and comprehensive theory of physics today, encompassing all known physics except gravity. (Questions related to the recent discovery of the Higgs Boson had to be analyzed using QFT, as string theory proved to be of little help with specifics.) It is therefore not surprising that there is a huge number of books on the subject. However, due to the great difficulty of making reliable calculations, most such books contain many 'formal' equations which are widely understood to lack mathematical sense. Frequently such difficulties are due to the use of perturbation theory with questionable convergence properties. In the 1950s, work began by A. S. Wightman, L. Garding, R. Jost, R. Haag and others to base the theory on a few fundamental principles (axioms) and derive as much of the physics as possible from these principles by rigorous mathematical arguments without using perturbation methods. This approach, known as `axiomatic QFT,' has turned out to be very fruitful. In the 1970s it resulted in a proof of the complete equivalence between QFT and its imaginary-time counterpart, known as `Euclidean QFT' because the imaginary time is indistinguishable from the three (real) space dimensions, so that relativistic spacetime becomes simplified to four-dimensional Euclidean space. The `weirdness' of special relativity (where time behaves like an imaginary space dimension) and the weirdness of quantum mechanics (where observables do not commute, hence their measurements interfere) seem to "cancel" and become subsumed in a theory of random (probabilistic) classical (commuting) fields in four-dimensional Euclidean space. In this realm, rigorous non-perturbative computations can be performed and then mapped to the corresponding results in relativistic spacetime. This approach, known as `constructive QFT,' is a deep validation of the axiomatic method.

Due partly to the great popularity of string theory, only a handful of books on rigorous (axiomatic and constructive) QFT have appeared since the 1980s. Franco Strocchi's book is the latest, and one of the best. It is written with great care and clarity. Furthermore, it brings the rigorous approach to topics of current interest such as the infrared problem, the Higgs mechanism, and the theta vacua in Quantum Chromodynamics, which often suffer from mathematical inconsistencies in the standard textbooks. It is also an excellent text for self-study. I highly recommend it.

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