Theory of Fields, II: Quantum and Topological

Abstract

Quantization of classical fields is an area of fundamental importance in modern mathematical physics. Although there is no satisfactory mathematical theory of quantization of classical dynamical systems or fields, physicists have developed several methods of quantization that can be applied to specific problems. Most successful among these is QED (quantum electrodynamics), the theory of quantization of electromagnetic fields. The physical significance of electromagnetic fields is thus well understood at both the classical and the quantum level. Electromagnetic theory is the prototype of classical gauge theories. It is therefore natural to try to extend the methods of QED to the quantization of other gauge field theories. The methods of quantization may be broadly classified as non-perturbative and perturbative. The literature pertaining to each of these areas is vast. See for example, the two volumes [95, 96] edited by Deligne, et al. which contain the lectures given at the Institute for Advanced Study, Princeton, during a special year devoted to quantum fields and strings; the book by Nash [298], and [41, 354, 89]. For a collection of lectures covering various aspects of quantum field theory, see, for example, [134, 133, 376].