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
(section)
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!
== 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.
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)