Editing Kinetic term (section)

== 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.<ref name="Schwartz">{{cite book|first=M. D.|last=Schwartz|title=Quantum Field Theory and the Standard Model|publisher=Cambridge University Press|date=2014|chapter=|page=|isbn=9781107034730}}</ref>{{rp|30–31}} 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 [[propagator]]s used in [[Feynman diagram]]s are acquired from the kinetic and mass terms alone.<ref>{{cite book|last=Zee|first=A.|author-link=Anthony Zee|date=2003|title=Quantum Field Theory in a Nutshell|url=|doi=|location=|publisher=Princeton University Press|chapter=2|page=23|isbn=978-0691010199}}</ref>

The form of the kinetic terms is strongly restricted by the physical requirements and [[symmetry (physics)|symmetries]] that the field theory has to satisfy.<ref name="Schwartz"/>{{rp|113–118}} They have to be [[hermitian function|hermitian]] to give a [[real number|real]] Lagrangian and positive-definite to avoid [[negative energy]] modes and [[instability|instabilities]], and to preserve unitarity. Unitarity can also be broken if kinetic terms have more than two derivatives.<ref name="Schwartz"/>{{rp|133}} They must also be Lorentz invariant in [[theory of relativity|relativistic theories]]. The particular form of the kinetic term then depends on the [[representation theory of the Lorentz group|Lorentz representation]] of the fields, which in [[four-dimensional space|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.<ref name="Schwartz"/>{{rp|219}}

When the fields are [[gauge theory|gauged]], the derivatives are replaced by [[gauge covariant derivative]]s to make the kinetic terms gauge invariant.<ref name="Srednicki">{{cite book|last=Srednicki|first=M.|author-link=|date=2007|title=Quantum Field Theory|url=|doi=|location=|publisher=Cambridge University Press|chapter=|page=|isbn=978-0521864497}}</ref>{{rp|418}} When calculating Feynman diagrams, these covariant derivatives are usually expanded to get the bilinear kinetic terms together with a set of interaction terms.<ref name="Schwartz"/>{{rp|509–511}} Similarly, when a theory is elevated from flat to [[curved spacetime]], the kinetic term derivatives must be replaced by [[covariant derivative]]s.