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This monograph deals with the geometrical and topological aspects related to quantum field theory with special reference to the electroweak theory and skyrmions. This book is unique in its emphasis on the topological aspects of a fermion manifested through chiral anomaly which is responsible for the generation of mass. This has its relevance in electroweak theory where it is observed that weak interaction gauge bosons attain mass topologically. These geometrical and topological features help us to consider a massive fermion as a skyrmion and for a composite state we can realise the internal symmetry of hadrons from reflection group. Also, an overview of noncommutative geometry has been presented and it is observed that the manifold M 4 x Z2 has its relevance in the description of a massive fermion as skyrmion when the discrete space is considered as the internal space and the symmetry breaking gives rise to chiral anomaly leading to topological features. TOC:From the Contents: 1: Theory of Spinors.- 2: Fermions and Topology.- 3: Electroweak Theory.- 4: Skyrme Model.- 5: Geometrical Aspects of a Skyrmion.- 6: Noncommutative Geometry.
Table des matières
"1: Theory of Spinors. 1.1. Spinors and Spin Structure. 1.2. Spinors in Different Dimensions. 1.3. Supersymmetry and Superspace.
2: Fermions and Topology. 2.1. Fermi Field and Nonlinear Sigma Model. 2.2. Quantization and Anomaly. 2.3. Anomaly and Topology.
3: Electroweak Theory. 3.1. Weinberg-Salam Theory. 3.2. Topological Features in Field Theory. 3.3. Topological Origin of Mass.
4: Skyrme Model. 4.1. Nonlinear Sigma Model. 4.2. Skyrme Model for Nucleons. 4.3. Baryons as Three Flavor Solitons.
5: Geometrical Aspects of a Skyrmion. 5.1. Microlocal Space Time and Fermions. 5.2. Internal Symmetry of Hadrons. 5.3. Supersymmetry and Internal Symmetry.
6: Noncommutative Geometry. 6.1. Quantum Space Time. 6.2. Noncommutative Geometry and Particle Physics. 6.3. Discrete Space as the Internal Space."
Résumé
This is a monograph on geometrical and topological features which arise in quantum field theory. It is well known that when a chiral fermion interacts with a gauge field we have chiral anomaly which corresponds to the fact that divergence of the axial vector current does not vanish. It is observed that this is related to certain topological features associated with the fermion and leads to the realization of the topological origin of fermion number as well as the Berry phase. The role of gauge fields in the quantization procedure has its implications in these topological features of a fermion and helps us to consider a massive fermion as a soliton (skyrrnion). In this formalism chiral anomaly is found to be responsible for mass generation. This has its relevance in electroweak theory where it is observed that weak interaction gauge bosons attain mass topologically. The geometrical feature of a skyrmion also helps us to realize the internal symmetry of hadrons from reflection group. Finally it has been shown that noncommutative geometry where the space time manifold is taken to be X = M x Zz has its relevance in the description of a massive 4 fermion as a skyrmion when the discrete space is considered as the internal space and the symmetry breaking leads to chiral anomaly. In chap. l preliminary mathematical formulations related to the spinor structure have been discussed. In chap.
