A kinetic term contains derivatives of a field and determines its free propagation. Its quadratic differential operator becomes the inverse quantum field theory propagator.
A gauge-covariant kinetic term replaces ordinary derivatives by covariant derivatives. Expanding it generates interactions between the matter field and one or more gauge fields, including contact vertices with two gauge bosons.
A mass term is a quadratic term without derivatives whose coefficient sets the mass scale of a free field.
The parameter multiplying a scalar quadratic mass term. A negative value describes an instability of the expansion point, rather than a negative value of the physical mass of a stable excitation. Nonnegative quartic terms do not stabilize a negative quadratic term along an exactly flat quartic ray.
A field interaction term is nonlinear in the fields and produces the interaction vertices of perturbation theory.
An off shell field configuration is not required to satisfy the Euler-Lagrange field equation. An external four-momentum is off shell if it is not constrained to its physical mass shell. On shell identities can therefore fail for general untransformed correlation functions, even when the associated S-matrix is unchanged by a field redefinition.
Adding to the cross-coupled scalar Lagrangian density produces independent equations and . For the convention , their frequencies are respectively and . Above the threshold they oscillate with decaying or growing envelopes; below it they have nonoscillatory real temporal exponents. At equality the repeated temporal root supplies . At each equation also has a constant solution. The full system is generically unstable because of the amplified second field, even though it arises from an autonomous variational system with conserved, indefinite total energy.
For an action , vary the field with compact support, or fix both and its first derivatives on the boundary. One integration by parts on the first-derivative term and two on the second-derivative term give the displayed Euler-Lagrange field equation. With ordered repeated indices, define the second-derivative coefficient symmetrically so that . This convention avoids an unintended factor of two in mixed derivative terms. Higher spatial derivatives alone do not require higher time derivatives.
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In the case of field however, we can expand the Lagrangian out further, to also integrate over the space coordinates and their derivatives.
Since we are now working with something that gets integrated over space to obtain the total action, much like density would be integrated over space to obtain a total mass, the name "Lagrangian density" is fitting.
Of course, if we were to write it like that all the time we would go mad, so we can just write a much more condensed vectorized version using the gradient with :
And in the context of special relativity, people condense that even further by adding to the spacetime Four-vector as well, so you don't even need to write that separate pesky .
The main point of talking about the Lagrangian density instead of a Lagrangian for fields is likely that it treats space and time in a more uniform way, which is a basic requirement of special relativity: we have to be able to mix them up somehow to do Lorentz transformations. Notably, this is a key ingredient in a/the formulation of quantum field theory.