Principles of Lagrangian Field Theory

Abstract

The replacement of Newton’s equation by quantum mechanical wave equations in the 1920s implied that by that time all known fundamental degrees of freedom in physics were described by fields like $\boldsymbol{A}(\boldsymbol{x},t)$ or $\Psi (\boldsymbol{x},t)$, and their dynamics was encoded in wave equations. However, all the known fundamental wave equations can be derived from a field theory version of Hamilton’s principle, i.e. the concept of the Lagrange function $L(q(t),\dot{q}(t))$ and the related action S = ∫​dt L generalizes to a Lagrange density $\mathcal{L}(\phi (\boldsymbol{x},t),\dot{\phi }(\boldsymbol{x},t),\boldsymbol{\nabla }\phi (\boldsymbol{x},t))$ with related action $S =\int \! dt\int \!d^{3}\boldsymbol{x}\,\mathcal{L}$, such that all fundamental wave equations can be derived from the variation of an action,

$$\displaystyle{\frac{\partial \mathcal{L}} {\partial \phi } - \partial _{\mu } \frac{\partial \mathcal{L}} {\partial (\partial _{\mu }\phi )} = 0.}$$

This formulation of dynamics is particularly useful for exploring the connection between symmetries and conservation laws of physical systems, and it also allows for a systematic approach to the quantization of fields, which allows us to describe creation and annihilation of particles.