Classical field theory with derivative corrections

Abstract

A reformulation of classical field theory in arbitrary spacetime dimensions is presented in the general case where the Lagrangian density depends on an arbitrary number of derivatives of the field. From the appropriate action S = ∫ d^Dx L(φ, ∂_μφ, ∂_μνφ, ∂_μνρφ, ...) one derives the correct classical field equation. This turns out to be a generalization of the Euler-Lagrange equation fully and correctly accounting for the role of higher-order field derivatives in the classical field theory. Thus, a field theory based on a Lagrangian involving higher derivatives of the field must use these generalized Euler-Lagrange equations and not just the usual Euler-Lagrange equations, (∂L/∂φ) − ∂_μ[∂L/∂(∂_μφ)] = 0.
The generalization of the appropriate stress tensor T_μν, which summarizes the conserved quantities of the theory, is also sought now that the Lagrangian is allowed to depend on higher derivatives of the field.