Lagrangians and Lagrangian Densities in Field Theory
The time integral of the Lagrangian is called the action denoted by S. In field theory, a distinction is occasionally made between the Lagrangian L, of which the action is the time integral:
and the Lagrangian density, which one integrates over all space-time to get the action:
The Lagrangian is then the spatial integral of the Lagrangian density. However, is also frequently simply called the Lagrangian, especially in modern use; it is far more useful in relativistic theories since it is a locally defined, Lorentz scalar field. Both definitions of the Lagrangian can be seen as special cases of the general form, depending on whether the spatial variable is incorporated into the index i or the parameters s in φi(s). Quantum field theories in particle physics, such as quantum electrodynamics, are usually described in terms of, and the terms in this form of the Lagrangian translate quickly to the rules used in evaluating Feynman diagrams.
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