PREFACE: WHAT IS GEOMETRY IN QUANTUM THEORY

Abstract

In this scientific preface to the first issue of International Journal of Geometric Methods in Modern Physics1, we briefly survey some peculiarities of geometric techniques in quantummodels. Contemporary quantum theory meets an explosion of different types of quantization. Some of them (geometric quantization, deformation quantization, noncommutative geometry, topological field theory etc.) speak the language of geometry, algebraic and differential topology. We do not pretend for any comprehensive analysis of these quantization techniques, but aims to formulate and illustrate their main peculiarities. As in any survey, a selection of topics has to be done, and we apologize in advance if some relevant works are omitted. Geometry of classical mechanics and field theory is mainly differential geometry of finitedimensional smooth manifolds, fiber bundles and Lie groups. The key point why geometry plays a prominent role in classical field theory lies in the fact that it enables one to deal with invariantly defined objects. Gauge theory has shown clearly that this is a basic physical principle. At first, a pseudo-Riemannian metric has been identified to a gravitational field in the framework of Einstein’s General Relativity. In 60-70th, one has observed that connections on a principal bundle provide the mathematical model of classical gauge potentials [1-3]. Furthermore, since the characteristic classes of principal bundles are expressed in terms of the gauge strengths, one can also describe the topological phenomena in classical gauge models [4]. Spontaneous symmetry breaking and Higgs fields have been explained in terms of reduced G-structures [5]. A gravitational field seen as a pseudo-Riemannian metric exemplifies such a Higgs field [6]. In a general setting, differential geometry of smooth fiber bundles gives the adequate mathematical formulation of classical field theory, where fields are represented by sections of fiber bundles and their dynamics is phrased in terms of jet manifolds [7]. Autonomous classical mechanics speaks the geometric language of symplectic and Poisson Web: http://www.worldscinet.com/ijgmmp/ijgmmp.shtml