Open main menu
Home
Random
Recent changes
Special pages
Community portal
Preferences
About Wikipedia
Disclaimers
Incubator escapee wiki
Search
User menu
Talk
Dark mode
Contributions
Create account
Log in
Editing
Kinetic term
Warning:
You are not logged in. Your IP address will be publicly visible if you make any edits. If you
log in
or
create an account
, your edits will be attributed to your username, along with other benefits.
Anti-spam check. Do
not
fill this in!
{{Short description|Aspect of field theory}} In [[quantum field theory]], a '''kinetic term''' is any term in the [[Lagrangian (field theory)|Lagrangian]] that is [[bilinear form|bilinear]] in the [[field (physics)|fields]] and has at least one [[derivative]]. Fields with kinetic terms are [[dynamical system|dynamical]] and together with [[mass]] terms define a [[free field|free field theory]]. Their form is primarily determined by the [[spin (physics)|spin]] of the fields along with other constraints such as [[unitarity (physics)|unitarity]] and [[Lorentz covariance|Lorentz invariance]]. Non-standard kinetic terms that break unitarity or are not [[definite quadratic form|positive-definite]] occur, such as when formulating [[ghost (physics)|ghost fields]], in some models of [[physical cosmology|cosmology]], in [[condensed matter physics|condensed matter systems]], and for non-unitary [[conformal field theory|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.<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. == 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 space]]time, the kinetic terms primarily depend on the spin of the field, with the kinetic term for a real spin-0 [[scalar field]] given by{{refn|group=nb|The [[metric signature]] used here is the mostly negative signature. Changing signatures can change the signs of the kinetic terms.}}<ref name="Peskin">{{cite book|last1=Peskin|first1=M.E.|author-link1=Michael Peskin|last2=Schroeder|first2=D.V.|author-link2=|date=1995|title=An Introduction to Quantum Field Theory|url=|doi=|location=|publisher=CRC Press|chapter=|page=|isbn=978-0201503975}}</ref>{{rp|18}} :<math> \mathcal L_{0} = \frac{1}{2}\partial_\mu \phi \partial^\mu \phi. </math> A field theory with only this term describes a real [[massless particle|massless]] [[scalar field theory|scalar field]]. The kinetic term for a [[complex number|complex]] scalar field is instead given by <math>\mathcal L_{0} = \partial_\mu \varphi^* \partial^\mu \varphi</math>, although this can be decomposed into a sum of two real kinetic terms for the real and [[imaginary number|imaginary]] components. [[Dirac fermion]] kinetic terms are given by<ref name="Schwartz"/>{{rp|1168}} :<math> \mathcal L_{1/2} = i \bar \psi \gamma^\mu \partial_\mu \psi. </math> The factor of <math>i</math> is needed to make the kinetic term hermitian, <math>\gamma^\mu</math> are the [[gamma matrices]], <math>\psi</math> is a [[Dirac spinor]], and <math>\bar \psi</math> is the [[Dirac adjoint|adjoint spinor]]. This kinetic term can be decomposed into a sum of [[chirality (physics)|left-handed and right-handed]] [[Weyl equation|Weyl fermions]] <math>\mathcal L_{1/2} = i \psi_R^\dagger \sigma^\mu \partial_\mu \psi_R + i \psi^\dagger \bar \sigma^\mu \partial_\mu \psi_L</math>, where <math>\sigma^\mu</math> and <math>\bar \sigma^\mu</math> are the [[Pauli matrices|Pauli four-vectors]]. The kinetic term for an [[abelian group|abelian]] gauge field is given in terms of the [[electromagnetic tensor|field strength tensor]] <math>F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu</math> as<ref>{{cite book|last=Fradkin|first=E.|author-link=Eduardo Fradkin|date=2021|title=Quantum Field Theory: An Integrated Approach|url=|doi=|location=|publisher=Princeton University Press|chapter=3|page=52|isbn=978-0691149080}}</ref> :<math> \mathcal L_{1} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}. </math> The [[negative number|negative]] [[sign (mathematics)|sign]] is necessary to ensure that the <math>(\partial_0 A^i)^2</math> terms are [[positive real numbers|positive]] to get positive energies. For [[non-abelian group|non-abelian]] gauge fields the field strength tensor is replaced by a [[gluon field strength tensor|non-abelian field strength tensor]] <math>F_{\mu\nu}^a = \partial_\mu A_\nu^a - \partial_\nu A_\mu^a + gf^{abc}A^b_\mu A^c_\nu</math>, where <math>f^{abc}</math> are the [[structure constants]] of the gauge group [[Lie algebra|algebra]]. These additional terms gives rise to cubic and [[quartic interaction|quartic]] interaction terms for the [[gauge boson]]s. Spin-3/2 fields, corresponding to [[gravitino]]s, have kinetic terms given by<ref>{{cite book|last1=Dall'Agata|first1=G.|author-link1=|last2=Zagermann|first2=M.|author-link2=|date=2021|title=Supergravity: From First Principles to Modern Applications|url=|doi=|location=|publisher=Springer|chapter=2|page=26|isbn=978-3662639788}}</ref> :<math> \mathcal L_{3/2} = -\frac{1}{2}\bar \psi_\mu \gamma^{\mu\nu\rho}\partial_\nu \psi_\rho. </math> A Lagrangian with only this term describes a [[massless particle|massless]] [[Rarita–Schwinger equation|Rarita–Schwinger field]]. Here <math>\gamma^{\mu\nu\rho} = \gamma^{[\mu}\gamma^\nu \gamma^{\rho]}</math> are [[antisymmetric tensor|antisymmetric]] products of gamma matrices. Spin-2 fields, corresponding to [[graviton]]s, have a unique kinetic term given by<ref name="Schwartz"/>{{rp|135}} :<math> \mathcal L_{2} = \frac{1}{4}\partial^\mu h^{\nu\rho}\partial_\mu h_{\nu\rho} - \frac{1}{2}\partial^\mu h^{\nu\rho}\partial_\nu h_{\mu\rho} + \frac{1}{2}\partial^\mu h \partial^\nu h_{\nu\mu} - \frac{1}{4}\partial^\mu h\partial_\mu h, </math> where this Lagrangian is known as the Fierz–Pauli Lagrangian. For a massless spin-2 field, this kinetic term can be uniquely extended using the fields gauge symmetry to the [[Einstein–Hilbert action|Einstein–Hilbert Lagrangian]]. One can also write down kinetic terms for fields of spin greater than two.<ref name="Schwartz"/>{{rp|138}} Kinetic terms for massless fields are only compatible with non-interacting theories. Massive [[higher-spin theory|higher-spin fields]] can form interacting [[effective field theory|effective field theories]] and are used to describe certain [[hadron]]s and some [[string (physics)|string]] [[excited state|excitation states]] in [[string theory]].<ref>{{cite journal|last1=Buchbinder|first1=I.L.|authorlink1=|last2=Reshetnyak|first2=A.|authorlink2=|date=2012|title=General Lagrangian Formulation for Higher Spin Fields with Arbitrary Index Symmetry. I. Bosonic fields|url=|journal=Nucl. Phys. B|volume=862|issue=|pages=270–326|doi=10.1016/j.nuclphysb.2012.04.016|pmid=|arxiv=1110.5044|s2cid=|access-date=}}</ref> In dimensions besides four, other kinetic terms can be written such as those for [[tensor field]]s in [[higher gauge theory|higher-form gauge theory]]. Another example is the [[Chern–Simons theory|Chern–Simons kinetic term]] in [[three-dimensional space|1+2 dimensions]], which is a kinetic term for gauge fields of the form <math>\epsilon^{\mu\nu\rho}A_\mu \partial_\nu A_\rho</math>.<ref name="Classical">{{cite book|last=Năstase |first=H.|author-link=|date=2019|title=Classical Field Theory|url=|doi=|location=|publisher=Cambridge University Press|chapter=|page=|isbn=978-1108477017}}</ref>{{rp|309}} In contrast to the regular kinetic term for gauge fields, this has a single derivative and is a [[topology|topological]] term. == Non-canonical kinetic terms == Negative-definite kinetic terms, which have the opposite sign to the canonical kinetic terms, occur in some physical systems. For example, [[Faddeev–Popov ghost]] fields occurring in gauge theories either have negative sign kinetic terms or else they have wrong [[particle statistics]], which by the [[spin-statistics theorem]] makes them unphysical.<ref name="Peskin"/>{{rp|443}} Ghost fields also occur in [[Pauli–Villars regularization]] where they cancel divergent terms in [[One-loop Feynman diagram|loop diagrams]].<ref name="Schwartz"/>{{rp|831}} In cosmology, certain scalar fields known as phantom fields also have negative kinetic terms.<ref>{{cite journal|last1=Cadwell|first1=R.R.|authorlink1=Robert R. Caldwell|date=2002|title=A Phantom menace?|url=|journal=Phys. Lett. B|volume=545|issue=|pages=23–29|doi=10.1016/S0370-2693(02)02589-3|pmid=|arxiv=astro-ph/9908168|s2cid=|access-date=}}</ref> These fields have negative kinetic energy so the dynamics drive the field up a [[potential]] towards areas of higher [[energy]]. Sometimes non-canonical kinetic terms can be converted to canonical ones through a field redefinition, although this may introduce additional interaction terms.{{refn|group=nb|For example, <math>s^{-2}\partial_\mu s \partial^\mu s</math> is equivalent to <math>\partial_\mu \phi \partial^\mu \phi</math> under the field redefinition <math>s = e^\phi</math>.}} Fields without kinetic terms are also important, with these including [[auxiliary field]]s, [[Lagrange multiplier]]s, and background fields, with all of them being non-dynamical. Auxiliary fields have numerous applications such as in [[on shell and off shell|off shell]] formulations of [[supersymmetry|supersymmetric]] theories where they are used ensure an equal number of [[boson|bosonic]] and [[fermion|fermionic]] [[degrees of freedom (physics and chemistry)|degrees of freedom]] in an off-shell [[supermultiplet]].<ref>{{cite book|last=Năstase|first=H.|author-link=|date=2024|title=Introduction to Supergravity and its Applications|url=|doi=|location=|publisher=Cambridge University Press|chapter=3|page=38–39|isbn=978-1009445597}}</ref> Lagrange multipliers are used to impose additional [[constraint (mathematics)|constraints]] or conditions on the other physical fields.<ref name="Classical"/>{{rp|327}} Background fields represent some external field that is not solved for in the field theory and so the [[action (physics)|action]] is not [[variational principle|varied]] with respect to such fields.<ref name="Srednicki"/>{{rp|327}} === Multi-field kinetic terms === In a theory with multiple fields of the same type, such as multiple scalars or multiple fermions, their kinetic and mass terms can be grouped together into kinetic and mass [[matrix (mathematics)|matrices]]. For example, for a set of real scalar fields grouped into a [[vector (mathematics and physics)|vector]] <math>\Phi = (\phi_1, \cdots, \phi_n)</math> one can write the kinetic and mass terms as :<math> \mathcal L = \frac{1}{2}\Phi_i K_{ij}\Phi_j - \frac{1}{2}\Phi_i M_{ij}\Phi_j, </math> where <math>K_{ij}</math> and <math>M_{ij}</math> must be [[hermitian matrix|hermitian]] and [[definite matrix|positive-definite]]. Similar expressions exist for fermions. The kinetic matrix can always be brought into a canonical [[diagonal matrix|diagonal]] form while also [[diagonalizable matrix|diagonalizing]] the mass matrix.<ref>{{cite journal|last1=Gedalin|first1=E.|authorlink1=|last2=Moalem|first2=A.|authorlink2=|last3=Razdolskaya|first3=L.|authorlink3=|date=2001|title=Pseudoscalar meson mixing in effective field theory|url=|journal=Phys. Rev. D|volume=64|issue=|pages=076007|doi=10.1103/PhysRevD.64.076007|pmid=|arxiv=hep-ph/0106301|s2cid=|access-date=}}</ref> This is achieved by first diagonalizing the kinetic matrix, then rescaling the fields such that all the kinetic terms are canonically normalized, making the matrix proportional to the [[identity matrix]]. The mass matrix can then be diagonalized, with this second diagonalization not affecting the kinetic matrix as it is proportional to the identity. It is not always desirable to diagonalize the kinetic and mass matrices as this may end up mixing up interactions in the full theory. For example, propagation of [[neutrino]]s is calculated in the mass [[basis (linear algebra)|basis]], which diagonalizes the kinetic and mass matrices.<ref name="Schwartz"/>{{rp|601–602}} However, the interactions that create neutrinos are written in the [[flavour (particle physics)|flavour]] basis, which instead diagonalizes the coupling of neutrinos to the [[W and Z bosons|W bosons]]. Calculations for each process are done in each respective basis. The disparity between these two basis gives rise to [[neutrino oscillation]]s. Another example occurs when one has two abelian gauge bosons, where such theories often give rise to a kinetic mixing term.<ref>{{cite book|last1=Bauer|first1=M.|author-link1=|last2=Plehn|first2=T.|author-link2=|date=2019|title=Springer Yet Another Introduction to Dark Matter: The Particle Physics Approach|url=|doi=|location=|publisher=Springer|chapter=4|page=92|isbn=978-3030162337}}</ref> This is a term of the form <math>\epsilon F_{\mu\nu}\tilde F^{\mu\nu}</math>,{{refn|group=nb|Usually <math>\epsilon</math> is taken to be a small parameter so that kinetic mixing is a perturbative interaction between the two gauge bosons.}} which has the effect of converting one gauge boson into another as it propagates. It could be eliminated by diagonalizing the kinetic terms, however this can mix up interactions. Such kinetic mixing is common in the [[phenomenology (physics)|phenomenology]] of [[dark photon]]s.<ref>{{cite journal|last1=Caputo|first1=A.|authorlink1=|last2=Millar|first2=A.J.|authorlink2=|last3=O'Hare|first3=C.A.|authorlink3=|last4=Vitagliano|first4=E.|authorlink4=|date=2021|title=Dark photon limits: A handbook|url=|journal=Phys. Rev. D|volume=104|issue=9|pages=095029|doi=|pmid=|arxiv=2105.04565|s2cid=|access-date=}}</ref> More general kinetic terms can also occur in scalar field theories in the form of [[non-linear sigma model]]s. In that case the kinetic matrix is replaced by a [[function (mathematics)|function]] of the fields themselves <math>K_{ij}\rightarrow g_{ij}(\phi)</math>.<ref>{{cite book|last1=Freedman|first1=D.Z.|author-link1=Daniel Z. Freedman|last2=Van Proeyen|first2=A.|author-link2=|date=2012|title=Supergravity|url=|doi=|location=Cambridge|publisher=Cambridge University Press|chapter=7|page=163–166|isbn=978-0521194013}}</ref> In these models, this function behaves as a [[metric tensor (general relativity)|metric]] on a [[manifold]], known as a scalar manifold, for which the scalars act as [[coordinate system|coordinates]]. A [[Taylor series|Taylor expansion]] around the [[flat manifold|flat metric]] returns the regular bilinear kinetic terms together with a series of interaction terms. === Higher-order derivatives === Higher-order derivative kinetic terms are bilinear in fields but have more than two derivatives. Such terms generally break [[perturbation theory|perturbative]] unitarity, giving rise to non-unitary theories. This is because in [[position and momentum spaces|momentum space]], unitarity requires propagators to have an [[asymptotic analysis|asymptotic]] falloff of at most <math>p^{-2}</math> in its [[momentum]], corresponding to kinetic terms with at most two derivatives in position space.<ref name="Schwartz"/>{{rp|470}} Non-unitary theories with higher-order kinetic terms are useful in a number of areas such as in condensed matter physics where unitarity is not a strict requirement. Here they have been used to study [[elasticity (physics)|elasticity]], [[phase transition]]s, and certain [[polymer]]s.<ref>{{cite journal|last1=Safari|first1=M.|authorlink1=|last2=Vacca|first2=G.P.|authorlink2=|date=2018|title=Multicritical scalar theories with higher-derivative kinetic terms: A perturbative RG approach with the $\ensuremath{\epsilon}$-expansion|url=https://link.aps.org/doi/10.1103/PhysRevD.97.041701|journal=Phys. Rev. D|volume=97|issue=4|pages=041701|doi=10.1103/PhysRevD.97.041701|pmid=|arxiv=1708.09795|s2cid=|access-date=}}</ref> These kinetic terms can also help improve the [[ultraviolet completion|ultraviolet]] behaviour of Feynman diagrams and turn [[renormalization|nonrenormalizable]] theories into renormalizable ones,<ref name="LWM">{{cite journal|last1=Anselmi|first1=D.|authorlink1=|last2=Piva|first2=M.|authorlink2=|date=2017|title=A new formulation of Lee-Wick quantum field theory|url=|journal=JHEP|volume=6|issue=|pages=066|doi=10.1007/JHEP06(2017)066|pmid=|arxiv=|s2cid=|access-date=|hdl=11568/865571|hdl-access=free}}</ref> such as for higher-derivative [[gravity]].<ref>{{cite journal|last1=Stelle|first1=K.S.|authorlink1=|date=1977|title=Renormalization of higher-derivative quantum gravity|url=https://link.aps.org/doi/10.1103/PhysRevD.16.953|journal=Phys. Rev. D|volume=16|issue=4|pages=953–969|doi=10.1103/PhysRevD.16.953|pmid=|arxiv=|s2cid=|access-date=|url-access=subscription}}</ref> A class of higher-derivative theories known as Lee–Wick models,<ref>{{cite journal|last1=Lee|first1=T.D.|authorlink1=Tsung-Dao Lee|last2=Wick|first2=G.C.|authorlink2=Gian Carlo Wick|date=1969|title=Negative metric and the unitarity of the S-matrix|url=https://www.sciencedirect.com/science/article/pii/0550321369900984|journal=Nuclear Physics B|volume=9|issue=2|pages=209–243|doi=10.1016/0550-3213(69)90098-4|pmid=|arxiv=|s2cid=|access-date=|url-access=subscription}}</ref> usually formulate at the [[S-matrix]] level, are claimed to be unitary, with them get around the aforementioned obstruction using cutting equations.<ref name="LWM"/> When higher-order derivative kinetic terms occur in a Minkowski theory and result in propagators with complex [[zeroes and poles|poles]], the theory is mathematically [[consistency|inconsistent]].<ref>{{cite journal|last1=Aglietti|first1=U.G.|authorlink1=|last2=Anselmi|first2=D.|authorlink2=|date=2017|title=Inconsistency of Minkowski higher-derivative theories|url=|journal=Eur. Phys. J. C|volume=77|issue=2|pages=84|doi=10.1140/epjc/s10052-017-4646-7|pmid=|arxiv=1612.06510|s2cid=|access-date=}}</ref> This is because these kinetic terms give rise to [[nonlocal Lagrangian|non-local]] and non-Hermitian [[ultraviolet divergence]]s that cannot be eliminated using the standard renormalization procedure. These inconsistencies do not affect higher-derivative theories which do not have propagator complex poles or purely [[Euclidean space|Euclidean]] theories. Free higher-derivative scalar field theories are solvable and do not suffer from instabilities such as [[false vacuum|vacuum decay]].<ref>{{cite journal|last1=Brust|first1=C.|authorlink1=|last2=Hinterbichler|first2=K.|authorlink2=|date=2017|title=Free \ensuremath{\square}$^{k}$ scalar conformal field theory|url=|journal=JHEP|volume=02|issue=|pages=066|doi=10.1007/JHEP02(2017)066|pmid=|arxiv=1607.07439|s2cid=|access-date=}}</ref> They can also be fully conformal. Such non-unitary conformal field theories may be useful for studying the [[dS/CFT correspondence]]. ==Notes== {{reflist|group=nb}} == References == <references /> {{DEFAULTSORT:Kinetic Term}} [[Category:Quantum field theory]]
Edit summary
(Briefly describe your changes)
By publishing changes, you agree to the
Terms of Use
, and you irrevocably agree to release your contribution under the
CC BY-SA 4.0 License
and the
GFDL
. You agree that a hyperlink or URL is sufficient attribution under the Creative Commons license.
Cancel
Editing help
(opens in new window)
Pages transcluded onto the current version of this page
(
help
)
:
Template:Cite book
(
edit
)
Template:Cite journal
(
edit
)
Template:Reflist
(
edit
)
Template:Refn
(
edit
)
Template:Rp
(
edit
)
Template:Short description
(
edit
)