Lagrangian

In classical mechanics the path chosen by a system is the path that minimises the action, which is the time integral of the lagrangian, L(q, qdot, t) = T - V. T is the kinetic energy and V the potential energy, q is some set of coordinates and qdot indicates their time derivatives. The q may be any set of coordinates, euclidian, spherical, cylindrical or anything else that seems convenient. Note that there might be time dependence in T and V.

Minimizing the action leads to a set of differential equations of motion for q, the Euler-Lagrange equations. Solving them is usually much simpler than writing down all the forces in the system, since it is possible in the Lagrangian to choose any convenient set of coordinates and to include time dependence. Also it is possible to include constraints in the equations, to keep a cart on a rollercoaster track, for instance.

In the path integral formulation of Quantum Field Theory, the Lagrangian (short for Lagrangian density) is the central object of the theory. It defines the types of fields present, describes their dynamic behavior and possible interactions, and contains all of the free parameters such as masses and coupling constants. Each different Lagrangian leads to a different theory.

Similar to Lagrangian mechanics, the Lagrangian of a field theory yields the Euler-Lagrange equations, which are the equations of motion for the fields. But the chief role of the Lagrangian in modern Quantum Field Theory is to derive the Feynman rules of the theory (usually by the way of generating functionals), which are in turn used to calculate transition amplitudes for reactions of (not necessarily elementary) particles.

The Lagrangian is a polynomial of the fields and their derivatives, with each term having a physical interpretation. The possible^† terms can be classified as follows:

Dynamic terms

are quadratic in the field derivatives and describe the variation of the fields over time and space.

Mass terms

are quadratic in the fields and define their masses.

Interaction terms

are terms of higher order that describe the possible types of interactions between the fields.

Gauge fixing terms

fix gauge fields to a particular choice of gauge, to remove infinite contributions from the path integral which are caused by integration over field configurations that are related to each other by gauge transformations. This is necessary to be able to define the propagators of the gauge fields.

Counter terms

are introduced to cancel infinities arising from loop calculations in perturbation theory. Counter terms are one possible way to implement the renormalization procedure. They are (in fact, they are required to be) of the same structure as the other terms in the Lagrangian, with the addition of Z factors, and are treated as interactions.

The symmetry properties of a Lagrangian are essential in the formulation of conservation laws, quantum numbers, and Ward identities. For example, in Quantum Electrodynamics it is the symmetry of the Lagrangian under both global as well as local gauge transformations of the fermion (electron) fields that leads to a conserved electric charge and the existence of the photon and electromagnetic interactions.

Example - Scalar φ⁴ Theory

The Lagrangian for a scalar field φ(x)

L(φ, ∂_μφ) = 1/2 (∂_μφ)(∂^μφ) - 1/2 m²φ² - g/4 φ⁴

describes a field of mass m that interacts with itself. The "strength" of the interaction is g, the coupling constant, and each interaction involves four particles. The three terms in L are

• the dynamic term 1/2 (∂_μφ)(∂^μφ),
• the mass term -1/2 m²φ²,
• and the interaction term -g/4 φ⁴.

^†Lorentz invariance and renormalizability impose some rather strict conditions on the allowed stucture of the terms.