An Introduction to Some Topics in Constructive Quantum Field Theory

Abstract

We give an elementary introduction to the spirit and some of the simpler techniques of constructive quantum field theory. In the first part of these notes we briefly review the general framework of relativistic quantum field theory, its Euclidean description and Euclidean field theory. In the second part we combine Euclidean with Lagrangean field theory and, as a result, formulate a concrete program for the construction of relativistic quantum field models. In the third part we exemplify this program by considering in some detail the famous λФ4-theory in two space-time dimensions and commenting on the construction of λФ4 in three dimensions. Finally, in part four, we discribe some recent results concerning the degeneracy of the physical vacuum (phase transitions), the spontaneous breaking of internal symmetries and the critical point in λФ4 in two and three dimensions.