Quantum gauge theories. A true ghost story (Q2757766)

Comparison with other textbooks on quantum field theory shows that this one is an ``anti-Feynman book'' (so the author): no Feynman diagrams though it uses perturbation theory, no Feyruman path integrals though it quantizes concrete gauge theories. How does it work? What are the fundaments? The setting of the stage for what has come to be known as the causal method was begun, one may say, by \textit{H. Epstein} and \textit{V. Glaser} [Ann. Inst. Henri Poincaré, Nouv. Sér., Sect. A 19(1973), 211-295 (1974)]. The essence of this method is a careful handling of products of distributions, passing inductively from one order of the perturbation expansion to the next, so as to preserve the causal structure, a procedure meant to mimic ordinary renormalization. While in his previous book [Finite Quantum Electrodynamics: the Causal Approach, Springer (1995; Zbl 0844.53052)], Scharf restricted this method to QED, he now continues his work by applying similar methods to non-Abelian gauge theories and gravity. As the subtitle indicates, ghost fields form the new ingredient.NEWLINENEWLINENEWLINEThe books starts with discussing the basic properties of free fields in Chapter 1 and then turns to the construction of the \(S\)-matrix (i.e. time-ordered products of interacting fields) in Chapter 2. Gauge invariance of the perturbation series for massless gauge fields is adressed in Chapter 3 and defined through a gauge variation of the asymptotic (free) fields. This is where the ghost fields come in. Chapter 4 extends this discussion to deal with situations where the asymptotic gauge fields become massive like in the electroweak theory.NEWLINENEWLINENEWLINEChapter 5 constitutes the most ambitious part of the book where Scharf suggests another approach to general relativity, applying the principle of perturbative gauge invariance to spin-2 gauge fields. What he essentially finds is that the selfcoupling is unique and coincides with the one derived from the Einstein-Hilbert action. Since the coupling is nonrenormalizable, the problem remains how to fix an infinite number of free constants during the process of quantization. Here, Scharf offers no solution.NEWLINENEWLINENEWLINEFinally, the Bibliogaphical Notes at the end guide the reader to special references strongly related to the various topics of the book.NEWLINENEWLINENEWLINEThe book is a useful addition to the existing literature since it emphasizes an alternative and rarely adopted point of view.