Form of the manifestly covariant Lagrangian

Abstract

The preferred form for the manifestly covariant Lagrangian function of a single, charged particle in a given electromagnetic field is the subject of some disagreement in the textbooks. Some authors use a ‘‘homogeneous’’ Lagrangian and others use a ‘‘modified’’ form in which the covariant Hamiltonian function is made to be nonzero. We argue in favor of the ‘‘homogeneous’’ form. We show that the covariant Lagrangian theories can be understood only if one is careful to distinguish quantities evaluated on the varied (in the sense of the calculus of variations) world lines from quantities evaluated on the unvaried world lines. By making this distinction, we are able to derive the Hamilton–Jacobi and Klein–Gordon equations from the ‘‘homogeneous’’ Lagrangian, even though the covariant Hamiltonian function is identically zero on all world lines. The derivation of the Klein–Gordon equation in particular gives Lagrangian theoretical support to the derivations found in standard quantum texts, and is also shown to be consistent with the Feynman path-integral method. We conclude that the ‘‘homogeneous’’ Lagrangian is a completely adequate basis for covariant Lagrangian theory both in classical and quantum mechanics. The article also explores the analogy with the Fermat theorem of optics, and illustrates a simple invariant notation for the Lagrangian and other four-vector equations.