In quantum field theory, a kinetic term is any term in the Lagrangian that is bilinear in the fields and has at least one derivative. Fields with kinetic terms are dynamical and together with mass terms define a free field theory. Their form is primarily determined by the spin of the fields along with other constraints such as unitarity and Lorentz invariance. Non-standard kinetic terms that break unitarity or are not positive-definite occur, such as when formulating ghost fields, in some models of cosmology, in condensed matter systems, and for non-unitary conformal field theories.
Overview In a Lagrangian, bilinear field terms are split into two types: those without derivatives and those with derivatives. The former give fields mass and are known as mass terms. The latter, those which have at least one derivative, are known as kinetic terms and these make fields dynamical. A field theory with only bilinear terms is a free field theory. Interacting theories must have additional interacting terms, which have three or more fields per term. In a field theory, the propagators used in Feynman diagrams are acquired from the kinetic and mass terms alone. The form of the kinetic terms is strongly restricted by the physical requirements and symmetries that the field theory has to satisfy. They have to be hermitian to give a real Lagrangian and positive-definite to avoid negative energy modes and instabilities, and to preserve unitarity. Unitarity can also be broken if kinetic terms have more than two derivatives. They must also be Lorentz invariant in relativistic theories. The particular form of the kinetic term then depends on the Lorentz representation of the fields, which in four dimensions is primarily fixed by the spin. Integer spin fields having two derivatives in their kinetic terms while half-integer spin fields having only one derivative. When the fields are gauged, the derivatives are replaced by gauge covariant derivatives to make the kinetic terms gauge invariant. When calculating Feynman diagrams, these covariant derivatives are usually expanded to get the bilinear kinetic terms together with a set of interaction terms. Similarly, when a theory is elevated from flat to curved spacetime, the kinetic term derivatives must be replaced by covariant derivatives.
Canonical kinetic terms by spin The kinetic terms in unitary Lorentz invariant field theories are often expressed in certain canonical forms. In four-dimensional Minkowski spacetime, the kinetic terms primarily depend on the spin of the field, with the kinetic term for a real spin-0 scalar field given by
L 0 = 1 2 ∂ μ ϕ ∂ μ ϕ . {\displaystyle {\mathcal {L}}_{0}={\frac {1}{2}}\partial _{\mu }\phi \partial ^{\mu }\phi .}
A field theory with only this term describes a real massless scalar field. The kinetic term for a complex scalar field is instead given by L 0 = ∂ μ φ ∗ ∂ μ φ {\displaystyle {\mathcal {L}}_{0}=\partial _{\mu }\varphi ^{*}\partial ^{\mu }\varphi } , although this can be decomposed into a sum of two real kinetic terms for the real and imaginary components. Dirac fermion kinetic terms are given by
L 1 / 2 = i ψ ¯ γ μ ∂ μ ψ . {\displaystyle {\mathcal {L}}_{1/2}=i{\bar {\psi }}\gamma ^{\mu }\partial _{\mu }\psi .}
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