Classical Field Theory and Lagrangian Density

Core Idea

Classical field theory replaces discrete particle coordinates with continuous fields phi(x,t) as dynamical variables. The Lagrangian becomes a Lagrangian density L integrated over space, and the Euler-Lagrange equations generalize to field equations governing how fields evolve in spacetime.

Explainer

In classical particle mechanics, you specify a system by its Lagrangian L(q, dq/dt) and derive the equations of motion from the Euler-Lagrange equations. The transition to field theory replaces the discrete coordinates q_i(t) with a continuous field phi(x, t) -- or in relativistic notation, phi(x^mu). The field assigns a number (or a set of numbers, for vector or spinor fields) to every point in spacetime. Instead of a finite number of degrees of freedom, you now have infinitely many: one for each spatial point.

The Lagrangian of particle mechanics becomes a Lagrangian density L(phi, partial_mu phi), and the total Lagrangian is L = integral L d^3x. The action is S = integral L d^4x = integral L d^3x dt, integrated over all of spacetime. The field-theoretic Euler-Lagrange equation follows from demanding that the action is stationary under variations of phi: partial_mu (partial L / partial (partial_mu phi)) - partial L / partial phi = 0. This single equation, together with a choice of L, generates all the classical field equations you already know. Maxwell's equations, the Klein-Gordon equation, and the Dirac equation all arise from specific choices of Lagrangian density.

The power of this formulation is that symmetries of the Lagrangian density directly constrain the physics. A Lagrangian density that is a Lorentz scalar automatically produces Lorentz-covariant field equations. Internal symmetries (like phase rotations of a complex field) lead to conserved currents. The requirement that L contain no explicit spacetime dependence guarantees energy-momentum conservation. Rather than postulating field equations and then checking their properties, you build L from symmetry principles and derive everything else. This is the starting point for quantization: once you have the classical Lagrangian density, you can quantize the field using canonical or path-integral methods.

The simplest example is the free real scalar field, with L = (1/2)(partial_mu phi)(partial^mu phi) - (1/2)m² phi². The first term is the kinetic energy density (the relativistic generalization of (1/2)(dphi/dt)², including spatial gradient terms), and the second is a mass term. The Euler-Lagrange equation gives the Klein-Gordon equation. More complex Lagrangians include interaction terms (like lambda phi⁴ / 4! for self-interacting scalars), coupling to other fields, and gauge field terms. The entire Standard Model of particle physics is specified by a single Lagrangian density, and every prediction of the theory follows from it.