Geometry, topology and quantum field theory.
Despite its very general title this book is a research monograph and not a textbook. In fact its main subject is a rather unorthodox attempt at formulating a theory of all the fundamental interactions of elementary particle physics. This approach has been developed by the author and a few collaborators over the past 15 years, starting with a stochastic quantization procedure of the Nelson type for relativistic fermions. A key ingredient is the introduction of an internal dynamical variable called ``direction vector'', which appears as the imaginary part of the coordinate in complexified space-time and is associated with an internal helicity identified as fermion number. With this interpretation the operator representation of the external and internal variables gives rise to the appearance of two \(\text{SL} (2,\mathbb{C})\) gauge fields (and moreover to an \(\text{SL} (2,\mathbb{C})\) gauge theory of gravitation). This construction is called the ``gauge theoretic extension of a fermion'' and entails via the chiral anomaly the topological generation of fermion mass, which is proposed here as an alternative to the Higgs mechanism of the standard model of elementary particles. As a pictorial explanation the author offers the fact that the motion of the particle in an anisotropic internal space gives rise to similar features as that of a charged particle in the field of a magnetic monopole. In particular a boson with vanishing free mass will acquire a nonvanishing mass and appear as a fermion when it carries an odd number of flux quanta. Another association pursued by the author is the interpretation of the massive fermion as a soliton of the gauge field (Skyrmion). He also proposes that the internal symmetry algebra of hadrons is generated from the conformal reflection group (an idea originally due to Budinich) and claims even that, in the context of conformal supersymmetry, the only four possible internal symmetry algebras compatible with CP invariance correspond exactly to the four fundamental interactions. In the final chapter he discusses the connection of his model with non-commutative geometry and string theory. The main weakness of the book is the unevenness of style. Whereas a few sections are indeed textbook-like (mainly the chapters on electroweak theory and the Skyrme model), the larger part of it (even the first chapter entitled ``Theory of spinors'') is written in the technical, sometimes even obscure, style of a research paper, whose reading is further complicated by numerous misprints. In the first and last chapters many advanced mathematical concepts are invoked but not explained, on the other hand the discussion of the Skyrme model presupposes acquaintance with elementary particle phenomenology. I regret that the author has not succeeded in communicating his approach in a more transparent manner.
\(\text{SL}(2,\mathbb{C})\) gauge fields
scientific article; zbMATH DE number 1999045