The Schwinger Action Principle and Effective Action

Abstract

This book is an introduction to the Schwinger action principle in quan- tum mechanics and quantum field theory, with applications to a variety of different models, not only those of interest to particle physics. The book begins with a brief review of the action principle in classical mechan- ics and classical field theory. It then moves on to quantum field theory, focusing on the effective action method. This is introduced as simply as possible by using the zero-point energy of the simple harmonic oscilla- tor as the starting point. This allows the utility of the method, and the process of regularization and renormalization of quantum field theory, to be demonstrated with a minimum of formal development. The book concludes with a more complete definition of the effective action, and demonstrates how the provisional definition used earlier is the first term in the systematic loop expansion. Several applications of the Schwinger action principle are given, including Bose–Einstein condensation, the Casimir effect, and trapped Fermi gases. The renormalization of interacting scalar field theory is presented to two-loop order. This book will interest graduate students and researchers in theoretical physics who are familiar with quantum mechanics.