LagrangeDichte

LagrangeDichte, or Lagrangian density, is the central object in classical field theory and quantum field theory. It is a function L that depends on one or more fields φ_i(x) and their spacetime derivatives ∂μφ_i(x), and it is a scalar density under Lorentz transformations. The Lagrangian density specifies the local dynamics of the fields.

The action S is obtained by integrating the Lagrangian density over spacetime: S = ∫ d^4x L(x). Requiring the action to be stationary (δS = 0) for variations of the fields leads to the Euler–Lagrange equations for fields: ∂μ (∂L/∂(∂μφ_i)) − ∂L/∂φ_i = 0.

Common examples include the real scalar field with L = 1/2 ∂μφ ∂^μφ − 1/2 m^2 φ^2, and the electromagnetic field with L = −1/4 Fμν F^μν, where Fμν = ∂μAν − ∂νAμ. The Dirac field has L = i ψ̄ γ^μ ∂μ ψ − m ψ̄ ψ. Gauge interactions can be included by replacing ∂μ with the covariant derivative Dμ = ∂μ − i e Aμ.

Properties of the Lagrangian density include locality and Lorentz invariance. Continuous symmetries of L give conserved currents via Noether’s theorem, and translational symmetry yields the energy–momentum tensor. The Lagrangian density thus encodes both equations of motion and conserved quantities.

In quantum theory, the action appears in the path integral through the factor exp(iS/ħ). The Lagrangian density formalism therefore provides a unified framework for describing particle dynamics, their interactions, and the transition to quantum descriptions.