Kinetic term

Type of terms in Lagrangians

In physics, a kinetic term is the part of the Lagrangian that is bilinear in the fields (and for nonlinear sigma models, they are not even bilinear[clarification needed]), 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. Unitarity requires kinetic terms to be positive.

In mechanics, the kinetic term is

T = 1 2 x ˙ 2 = 1 2 ( ∂ x ∂ t ) 2 . {\displaystyle 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 = 1 2 ∂ μ Φ ∂ μ Φ + 1 4 g 2 F μ ν F μ ν + i ψ ¯ γ μ ∂ μ ψ . {\displaystyle 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 .}


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