Editing Field equation

{{Short description|Partial differential equation describing physical fields}}
In [[theoretical physics]] and [[applied mathematics]], a '''field equation''' is a [[partial differential equation]] which determines the dynamics of a [[physical field]], specifically the time evolution and spatial distribution of the field. The solutions to the equation are mathematical functions which correspond directly to the field, as functions of time and space. Since the field equation is a partial differential equation, there are families of solutions which represent a variety of physical possibilities. Usually, there is not just a single equation, but a set of coupled equations which must be solved simultaneously. Field equations are not [[ordinary differential equation]]s since a field depends on space and time, which requires at least two variables.

Whereas the "[[wave equation]]", the "[[diffusion equation]]", and the "[[continuity equation]]" all have standard forms (and various special cases or generalizations), there is no single, special equation referred to as "the field equation".

The topic broadly splits into equations of [[classical field theory]] and [[quantum field theory]]. Classical field equations describe many physical properties like temperature of a substance, velocity of a fluid, stresses in an elastic material, electric and magnetic fields from a current, etc.<ref>{{cite book |last1=Fetter|first1=A. L.|last2=Walecka|first2=J. D.| title=Theoretical Mechanics of Particles and Continua|year=1980|isbn= 978-0-486-43261-8|publisher=Dover|pages=439, 471}}</ref> They also describe the fundamental forces of nature, like electromagnetism and gravity.<ref>{{cite book|first=J. D.|last=Jackson|author-link=John David Jackson (physicist)|title=Classical Electrodynamics|edition=2nd|year=1975|orig-year=1962|isbn=0-471-43132-X|publisher=[[John Wiley & Sons]]|page=[https://archive.org/details/classicalelectro00jack_0/page/218 218]|url-access=registration|url=https://archive.org/details/classicalelectro00jack_0/page/218}}</ref><ref>{{cite book|last1=Landau|first1=L.D.|author-link1=Lev Landau|last2=Lifshitz|first2=E.M.|author-link2=Evgeny Lifshitz|title=The Classical Theory of Fields|series=Course of Theoretical Physics|volume=2|edition=4th|publisher=[[Butterworth&ndash;Heinemann]]|isbn=0-7506-2768-9|year=2002|orig-year=1939|page=297}}</ref> In quantum field theory, particles or systems of "particles" like [[electron]]s and [[photon]]s are associated with fields, allowing for infinite degrees of freedom (unlike finite degrees of freedom in particle mechanics) and variable particle numbers which can be [[Matter creation|created]] or [[Annihilation|annihilated]].

==Generalities==

===Origin===

Usually, field equations are postulated (like the [[Einstein field equations]] and the [[Schrödinger equation]], which underlies all quantum field equations) or obtained from the results of experiments (like [[Maxwell's equations]]). The extent of their validity is their ability to correctly predict and agree with experimental results.

From a theoretical viewpoint, field equations can be formulated in the frameworks of [[Lagrangian field theory]], [[Hamiltonian field theory]], and field theoretic formulations of the [[principle of stationary action]].<ref>{{cite book |last1=Goldstein |first1=Herbert |author-link1=Herbert Goldstein | title=Classical Mechanics |url=https://archive.org/details/classicalmechani00gold_639 |url-access=limited |edition=2nd |year=1980|isbn= 0201029189|publisher=Addison Wesley |chapter=Chapter 12: Continuous Systems and Fields|location=San Francisco, CA |pages=[https://archive.org/details/classicalmechani00gold_639/page/n565 548], 562}}</ref> Given a suitable Lagrangian or Hamiltonian density, a function of the fields in a given system, as well as their derivatives, the principle of stationary action will obtain the field equation.

===Symmetry===

In both classical and quantum theories, field equations will satisfy the symmetry of the background physical theory. Most of the time [[Galilean symmetry]] is enough, for speeds (of propagating fields) much less than light. When particles and fields propagate at speeds close to light, [[Lorentz symmetry]] is one of the most common settings because the equation and its solutions are then consistent with special relativity.

Another symmetry arises from [[gauge freedom]], which is intrinsic to the field equations. Fields which correspond to interactions may be [[gauge field]]s, which means they can be derived from a potential, and certain values of potentials correspond to the same value of the field.

===Classification===

Field equations can be classified in many ways: classical or quantum, nonrelativistic or relativistic, according to the [[Spin (physics)|spin]] or [[mass]] of the field, and the number of components the field has and how they change under coordinate transformations (e.g. [[scalar field]]s, [[vector field]]s, [[tensor field]]s, [[spinor field]]s, [[Twistor theory|twistor field]]s etc.). They can also inherit the classification of differential equations, as [[Linear partial differential equation|linear]] or [[Nonlinear partial differential equation|nonlinear]], the order of the highest derivative, or even as [[fractional differential equation]]s. Gauge fields may be classified as in [[group theory]], as [[abelian group|abelian]] or nonabelian.

=== Waves ===

Field equations underlie wave equations, because periodically changing fields generate waves. Wave equations can be thought of as field equations, in the sense they can often be derived from field equations. Alternatively, given suitable Lagrangian or Hamiltonian densities and using the principle of stationary action, the wave equations can be obtained also.

For example, Maxwell's equations can be used to derive [[inhomogeneous electromagnetic wave equation]]s, and from the Einstein field equations one can derive equations for [[gravitational wave]]s.

=== Supplementary equations to field equations ===

Not every partial differential equation (PDE) in physics is automatically called a "field equation", even if fields are involved. They are extra equations to provide additional constraints for a given physical system.

"[[Continuity equation]]s" and "[[diffusion equation]]s" describe [[transport phenomena]], even though they may involve fields which influence the transport processes.

If a "[[constitutive equation]]" takes the form of a PDE and involves fields, it is not usually called a field equation because it does not govern the dynamical behaviour of the fields. They relate one field to another, in a given material. Constitutive equations are used along with field equations when the effects of matter need to be taken into account.

==Classical field equation==

{{main|classical mechanics|classical field theory}}

Classical field equations arise in [[continuum mechanics]] (including [[elastodynamics]] and [[fluid mechanics]]), [[heat transfer]], [[electromagnetism]], and [[gravitation]].

Fundamental classical field equations include

*[[Newton's Law of Universal Gravitation]] for nonrelativistic gravitation.
*[[Einstein field equations]] for [[General Relativity|relativistic gravitation]]
*[[Maxwell's equations]] for electromagnetism.

Important equations derived from fundamental laws include:

*[[Navier–Stokes equations]] for fluid flow.

As part of real-life [[mathematical modelling]] processes, classical field equations are accompanied by other [[equations of motion]], [[equations of state]], [[constitutive equation]]s, and continuity equations.

==Quantum field equation==
{{main|quantum mechanics|quantum field theory}}

In quantum field theory, particles are described by quantum fields which satisfy the [[Schrödinger equation]]. They are also [[creation and annihilation operators]] which satisfy [[commutation relation]]s and are subject to the [[spin–statistics theorem]].

Particular cases of [[relativistic quantum field equations]] include<ref>{{cite book|isbn=978-1-139-50432-4|first=T|last=Ohlsson|author-link= Tommy Ohlsson |title=Relativistic Quantum Physics: From Advanced Quantum Mechanics to Introductory Quantum Field Theory|publisher=Cambridge University Press|year=2011|pages=23, 42, 44|url=https://books.google.com/books?id=hRavtAW5EFcC&q=pauli-lubanski+pseudovector&pg=PA11}}</ref>

*the [[Klein–Gordon equation]] for spin-0 particles
*the [[Dirac equation]] for spin-1/2 particles
*the [[Bargmann–Wigner equations]] for particles of any spin

In quantum field equations, it is common to use [[momentum]] components of the particle instead of position coordinates of the particle's location, the fields are in [[momentum space]] and [[Fourier transform]]s relate them to the position representation.

==See also==
*[[Field strength]]
*[[Wave function]]
**[[Fundamental interaction]]
**[[Field coupling]]
**[[Field decoupling]]
**[[Coupling parameter]]
*[[Vacuum solution]]

==References==

{{reflist}}

===General===

* {{cite book| author=G. Woan| title=The Cambridge Handbook of Physics Formulas| publisher=Cambridge University Press| date=2010| isbn=978-0-521-57507-2| url-access=registration| url=https://archive.org/details/cambridgehandboo0000woan}}

===Classical field theory===

* {{Citation|first1=Charles W.|last1=Misner|author-link=Charles W. Misner|first2=Kip. S.|last2=Thorne|author2-link=Kip Thorne|first3=John A.|last3=Wheeler|author3-link=John A. Wheeler|title=[[Gravitation (book)|Gravitation]]|publisher= W. H. Freeman|date=1973|isbn=0-7167-0344-0}}
* {{Citation|last=Chadwick|first=P.|title=Continuum mechanics: Concise theory and problems|publisher= Dover (originally George Allen & Unwin Ltd.)|date=1976|bibcode=1976nyhp.book.....C |isbn=0-486-40180-4|url=https://books.google.com/books?id=1CTEAgAAQBAJ&q=chadwick+continuum+mechanics}}

===Quantum field theory===

* {{cite book| author-link = Steven Weinberg| first = S.| last = Weinberg| year = 1995| title = The Quantum Theory of Fields| volume = 1| publisher = Cambridge University Press| isbn = 0-521-55001-7| url-access = registration| url = https://archive.org/details/quantumtheoryoff00stev}} 
*{{cite book |author=[[Vladimir Berestetskii|V.B. Berestetskii]], [[Evgeny Lifshitz|E.M. Lifshitz]], [[Lev Pitaevskii|L.P. Pitaevskii]] |year=1982 |title=Quantum Electrodynamics |series=[[Course of Theoretical Physics]]|edition=2nd |volume=4 |publisher=[[Butterworth-Heinemann]] |isbn=978-0-7506-3371-0}}
*{{citation|last1=Greiner|first1=W.|author-link1=Walter Greiner|last2=Reinhardt|first2=J.|year=1996|title=Field Quantization|publisher=Springer|isbn=3-540-59179-6|url-access=registration|url=https://archive.org/details/fieldquantizatio0000grei}}
*{{cite book|first1=I.J.R.|last1=Aitchison |first2=A.J.G.|last2=Hey |title=Gauge Theories in Particle Physics: From Relativistic Quantum Mechanics to QED|volume=1|edition=3rd|publisher=IoP|year=2003|url=https://books.google.com/books?id=n7k_QS4Hb0YC&q=Gauge+Theories+in+Particle+Physics|isbn=0-7503-0864-8}}
*{{cite book|first1=I.J.R.|last1=Aitchison |first2=A.J.G.|last2=Hey |title=Gauge Theories in Particle Physics: Non-Abelian Gauge Theories: QCD and electroweak theory|volume=2|edition=3rd|publisher=IoP|year=2004|url=https://books.google.com/books?id=-mDjFSUDQsEC&q=Gauge+Theories+in+Particle+Physics+volume+2|isbn=0-7503-0950-4}}

===Classical and quantum field theory===

*{{cite book|first1=R. U.|last1=Sexl|first2=H. K.|last2=Urbantke|title=Relativity, Groups Particles. Special Relativity and Relativistic Symmetry in Field and Particle Physics|year=2001|orig-year=1992|publisher=Springer|isbn=978-3211834435|url=https://books.google.com/books?id=iyj0CAAAQBAJ&q=sexl+urbantke+relativity&pg=PR3}}

==External links==
*{{cite web|url=http://physics.illinois.edu/academics/graduates/physics-formulary.pdf|title=Physics formulary|year=1999|author=J.C.A. Wevers|access-date=27 December 2016|archive-date=27 December 2016|archive-url=https://web.archive.org/web/20161227200641/http://physics.illinois.edu/academics/graduates/physics-formulary.pdf|url-status=dead}}
*{{cite web|url=http://physics.info/equations/|title=Frequently Used Equations|year=1998|author=Glenn Elert|access-date=27 December 2016}}

[[Category:Mathematical physics]]
[[Category:Applied mathematics]]
[[Category:Partial differential equations]]
[[Category:Classical field theory]]
[[Category:Quantum field theory]]