Classical Field Theory

Abstract

A second cornerstone of classical physics besides point-particle mechanics is field theory. Classical field theory is essentially an infinite collection of mechanical systems (one at each point in space) and hence can be viewed as an infinite-dimensional generalization of classical mechanics. More precisely, solutions of classical mechanical systems are smooth curves \(t\mapsto \gamma (t)\) from \(\mathbb {R}\) to M. In classical field theory, curves from \(\mathbb {R}\) are replaced by maps from a higher-dimensional source manifold. In this more general framework we also allow for Lagrangians with explicit time dependence. Another key feature of classical field theory is its manifest incorporation of the laws of Einstein’s theory of relativity. This chapter begins with definitions and properties of the central objects, namely fields, Lagrangians, action functionals, and the field-theoretic version of the Euler–Lagrange equations. Modern covariant field theory is customarily formulated in the language of jet bundles, which is also utilized and thus introduced here. We study symmetries and conservation laws of classical field theories in the second part of this chapter, which culminates in the field-theoretic version of Noether’s theorem. The penultimate section is devoted to a thorough presentation, from a mathematical perspective, of some prominent examples of classical field theories, such as sigma models, Yang–Mills theory, and Einstein’s theory of gravity. A key ingredient of matter-coupled Einstein gravity is the energy-momentum tensor, which is studied in detail in the final section. This chapter is largely based on Ref. [16] (Olver Applications of Lie groups to differential equations, 1993).