Quantum Fibre Bundles. An Introduction

Abstract

The description of space time at the Planck scale is one of thefundamental problems of theoretical physics. There is an approachto this subject based on deformation quantization and quantumgroups. In this context we must indicate that the generalizationof the notion of fibre bundles and corresponding gauge theory forquantum groups is necessary. Such a generalization has been givenrecently by S. Majid and the author. It has been expressed interms of Hopf algebras and their actions and coactions onalgebras. The present paper is a nice review of the concept ofquantum fibre bundles and quantum gauge theory. The essentialidea in this subject is based on the dualization of the classicalconcept of the theory of fibre bundles and gauge theories and onthe fact that the algebra of regular functions on a group is aHopf algebra. The paper contains a short introduction to thenotion of Hopf algebras and quantum generalization of the theorypincipal fibre bundles, associated bundles, differentialstructures, connections, etc. It is interesting that the notionof a principal quantum bundle is related to the theory ofalgebraic extensions. Note that the presentation of the quantumprincipal bundles with general differential structures is notcomplete, but some references to this subject are given. Thequantum group generalization of gauge fields, cross sections andmatter fields is also described. A few illustrative examples aregiven. The quantum analogue of the Hopf fibration is consideredamong them. Some open problems are discussed. The paper couldalso be interesting in general for anyone who is not a specialistworking in the area of quantum groups but who just has aninterest in the foundations of theoretical physics and also forbeginning scientists.\par {For the entire collection see MR98b:58006.}