Kinetic term

In physics, a kinetic term is the part of the Lagrangian that is bilinear in the fields (this does not include the mass term!) (and for nonlinear sigma models, they are not even bilinear), and usually contains two derivatives with respect to time (or space); in the case of fermions, the kinetic term usually has one derivative only. The equation of motion derived from such a Lagrangian contains differential operators which are generated by the kinetic term.

In mechanics, the kinetic term is

$T = \frac{1}{2}\dot x^2 = \frac{1}{2}\left( \frac{\partial x}{\partial t}\right)^2 .$

In quantum field theory, the kinetic terms for real scalar fields, electromagnetic field and Dirac field are

$T = \frac{1}{2}\partial_\mu \Phi \partial^\mu \Phi + \frac{1}{4g^2}F_{\mu\nu}F^{\mu\nu} + i \bar \psi \gamma^\mu \partial_\mu \psi .$