Geometry, topology and quantum field theory.
type: Item
instance of: scholarly article
title: Geometry, topology and quantum field theory.
zbMATH Open document ID: 1034.81002
author: Pratul Bandyopadhyay
published in: Fundamental Theories of Physics
publication date: 2003-10-30T00:00:00Z
review text: 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 exac…
Mathematics Subject Classification ID: 81-02
Mathematics Subject Classification ID: 81Txx
Mathematics Subject Classification ID: 53C80
Mathematics Subject Classification ID: 81V22
Mathematics Subject Classification ID: 81V35
Mathematics Subject Classification ID: 81V17
Mathematics Subject Classification ID: 81R60
zbMATH DE Number: 1999045
zbMATH Keywords: chiral anomaly
zbMATH Keywords: vortex line
zbMATH Keywords: conformal spinors
zbMATH Keywords: noncommutative geometry
zbMATH Keywords: stochastic quantization
zbMATH Keywords: complexified space-time
zbMATH Keywords: \(\text{SL}(2,\mathbb{C})\) gauge fields
zbMATH Keywords: gauge theory of gravitation
zbMATH Keywords: field of a magnetic monopole
zbMATH Keywords: Skyrmion
reviewed by: Helmut Rumpf
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1999045