From classical to quantum fields (Q2827271)

The impossibility of describing relativistic particle interactions in terms of forces at a distance (by the so-called no-interaction theorems) strongly supports the need of rather considering local interactions mediated by fields. Thus, classical and quantum fields become the privileged, if not the unique, way for describing relativistic particle dynamics and qualify as a fundamental branch of theoretical physics.NEWLINENEWLINEThe book under review offers a very comprehensive treatment of quantum field theory, starting from the basic elements up to the recent achievements, including the successful Standard Model of fundamental particle interactions.NEWLINENEWLINEThe amount of material covered is impressive and the result goes beyond the declared aim of addressing to undergraduate students. The remarkable quality of the book is that every topic is carefully treated without missing the need of emphasizing the deep entanglement of physics and mathematics.NEWLINENEWLINEThe first four chapters present the main elements of classical field theory and of the underlying symmetry properties (electromagnetic field with Lorentz invariance and gravitational field with diffeomorphism invariance).NEWLINENEWLINERelativistic wave equations and their drawbacks for describing quantum relativistic particles are discussed in Chapters 5, 6, 7. The general problem of relativistic quantum mechanics is dealt with the functional integral approach, starting from Feynman path ``integral'' (Chapter 8) and then adopting the euclidean functional integral (Chapters 9, 10, 11) with much better mathematical properties.NEWLINENEWLINEThe treatment of these chapters is remarkably good, clarity, full control of the arguments and no disregard of the mathematical soundness being the distinguished features.NEWLINENEWLINEParticularly accurate and well founded is the scattering theory presented in Chapters 12, 13, starting from a brief account of general (so-called axiomatic) quantum field theory, and then rooting the (non-perturbative) S-matrix theory on the asymptotic limits of fields and on the reduction formulas. The perturbative expansion is then introduced in terms of Feynman diagrams and later fully discussed in Chapter 16.NEWLINENEWLINEGauge theories and their BRST quantization are outlined in Chapter 14; the main ideas of their (perturbative) renormalization by exploiting the BRST Ward identities are provided in Chapter 18.NEWLINENEWLINEThe non-perturbative construction of quantum field theory is an important foundational issue both for the question of mathematical existence and control of non-trivial quantum fields, as well as for the need of getting some light on non-perturbative effects. The praiseworthy Chapter 24 provides a readable and clear account of results whose treatment in the literature involves hard work and technical expertise; the wisdom of Roland Sénéor, a world expert on the subject, has likely been of help.NEWLINENEWLINEThe Standard Model and its extensions to grand unified theories are presented in Chapters 25, 26 and the (yet speculative) supersymmetry is discussed in Chapter 27. Useful technical tools and formulas are given in the appendices.NEWLINENEWLINEIn conclusion, the book covers an amazing amount of topics, a real \textit{tour de force}, which will be very useful to students, researches and even to scientists working in the field. It is hard to envisage a more comprehensive account in a single book at such a level of competence, depth and mathematical soundness. For these reasons it would be unfair, if not silly, to blame for the omissions.