Editing Classical field theory

{{short description|Physical theory describing classical fields}}
{{Classical mechanics}}
A '''classical field theory''' is a [[physical theory]] that predicts how one or more [[field (physics)|fields in physics]] interact with matter through '''field equations''', without considering [[Quantum mechanics|effects of quantization]]; theories that incorporate quantum mechanics are called [[quantum field theory|quantum field theories]]. In most contexts, 'classical field theory' is specifically intended to describe [[electromagnetism]] and [[gravitation]], two of the [[fundamental force]]s of nature.

A physical field can be thought of as the assignment of a [[physical quantity]] at each point of [[space]] and [[time]]. For example, in a weather forecast, the wind velocity during a day over a country is described by assigning a [[vector (mathematics and physics)|vector]] to each point in space. Each vector represents the direction of the movement of air at that point, so the set of all wind vectors in an area at a given point in time constitutes a [[vector field]]. As the day progresses, the directions in which the vectors point change as the directions of the wind change.

The first field theories, [[Newtonian gravitation]] and [[Maxwell's equations]] of electromagnetic fields were developed in classical physics before the advent of [[relativity theory]] in 1905, and had to be revised to be consistent with that theory. Consequently, classical field theories are usually categorized as ''non-relativistic'' and ''relativistic''. Modern field theories are usually expressed using the mathematics of [[tensor calculus]]. A more recent alternative mathematical formalism describes classical fields as sections of [[mathematical object]]s called [[fiber bundle]]s.

== History ==
{{Main|History of classical field theory}}
[[Michael Faraday]] coined the term "field" and lines of forces to explain electric and magnetic phenomena. [[Lord Kelvin]] in 1851 formalized the concept of field in different areas of physics.

== Non-relativistic field theories ==
Some of the simplest physical fields are vector force fields. Historically, the first time that fields were taken seriously was with [[Michael Faraday|Faraday's]] [[lines of force]] when describing the [[electric field]]. The [[gravitational field]] was then similarly described.

===Newtonian gravitation===
The first [[field theory (physics)|field theory]] of gravity was [[Newton's theory of gravitation]] in which the mutual interaction between two [[mass]]es obeys an [[inverse square law]]. This was very useful for predicting the motion of planets around the Sun.

Any massive body ''M'' has a [[gravitational field]] '''g''' which describes its influence on other massive bodies. The gravitational field of ''M'' at a point '''r''' in space is found by determining the force '''F''' that ''M'' exerts on a small [[test mass]] ''m'' located at '''r''', and then dividing by ''m'':<ref name="kleppner85">{{cite book|last1=Kleppner|first1=David|last2=Kolenkow|first2=Robert|title=An Introduction to Mechanics|page=85}}</ref>
<math display="block"> \mathbf{g}(\mathbf{r}) = \frac{\mathbf{F}(\mathbf{r})}{m}.</math>
Stipulating that ''m'' is much smaller than ''M'' ensures that the presence of ''m'' has a negligible influence on the behavior of ''M''.

According to [[Newton's law of universal gravitation]], '''F'''('''r''') is given by<ref name="kleppner85" />
<math display="block">\mathbf{F}(\mathbf{r}) = -\frac{G M m}{r^2}\hat{\mathbf{r}},</math>
where <math>\hat{\mathbf{r}}</math> is a [[unit vector]] pointing along the line from ''M'' to ''m'', and ''G'' is Newton's [[gravitational constant]]. Therefore, the gravitational field of ''M'' is<ref name="kleppner85" />
<math display="block">\mathbf{g}(\mathbf{r}) = \frac{\mathbf{F}(\mathbf{r})}{m} = -\frac{G M}{r^2}\hat{\mathbf{r}}.</math>

The experimental observation that inertial mass and gravitational mass are equal to unprecedented levels of accuracy leads to the identification of the gravitational field strength as identical to the acceleration experienced by a particle. This is the starting point of the [[equivalence principle]], which leads to [[general relativity]].

For a discrete collection of masses, ''M<sub>i</sub>'', located at points, '''r'''<sub>''i''</sub>, the gravitational field at a point '''r''' due to the masses is
<math display="block">\mathbf{g}(\mathbf{r})=-G\sum_i \frac{M_i(\mathbf{r}-\mathbf{r_i})}{|\mathbf{r}-\mathbf{r}_i|^3} \,, </math>

If we have a continuous mass distribution ''ρ'' instead, the sum is replaced by an integral,
<math display="block">\mathbf{g}(\mathbf{r})=-G \iiint_V \frac{\rho(\mathbf{x})d^3\mathbf{x}(\mathbf{r}-\mathbf{x})}{|\mathbf{r}-\mathbf{x}|^3} \, , </math>

Note that the direction of the field points from the position '''r''' to the position of the masses '''r'''<sub>''i''</sub>; this is ensured by the minus sign. In a nutshell, this means all masses attract.

In the integral form [[Gauss's law for gravity]] is
<math display="block">\iint\mathbf{g}\cdot d \mathbf{S} = -4\pi G M</math>
while in differential form it is
<math display="block">\nabla \cdot\mathbf{g} = -4\pi G\rho_m </math>

Therefore, the gravitational field '''g''' can be written in terms of the [[gradient]] of a [[gravitational potential]] {{math|''φ''('''r''')}}:
<math display="block">\mathbf{g}(\mathbf{r}) = -\nabla \phi(\mathbf{r}).</math>
This is a consequence of the gravitational force '''F''' being [[conservative field|conservative]].

=== Electromagnetism ===
==== Electrostatics ====
{{Main|Electrostatics}}

A [[test charge|charged test particle]] with charge ''q'' experiences a force '''F''' based solely on its charge. We can similarly describe the [[electric field]] '''E''' generated by the source charge ''Q'' so that {{math|1='''F''' = ''q'''''E'''}}:
<math display="block"> \mathbf{E}(\mathbf{r}) = \frac{\mathbf{F}(\mathbf{r})}{q}.</math>

Using this and [[Coulomb's law]] the electric field due to a single charged particle is
<math display="block">\mathbf{E} = \frac{1}{4\pi\varepsilon_0} \frac{Q}{r^2} \hat{\mathbf{r}} \,. </math>

The electric field is [[conservative field|conservative]], and hence is given by the gradient of a scalar potential, {{math|''V''('''r''')}}
<math display="block"> \mathbf{E}(\mathbf{r}) = -\nabla V(\mathbf{r}) \, . </math>

[[Gauss's law]] for electricity is in integral form
<math display="block">\iint\mathbf{E}\cdot d\mathbf{S} = \frac{Q}{\varepsilon_0}</math>
while in differential form
<math display="block">\nabla \cdot\mathbf{E} = \frac{\rho_e}{\varepsilon_0} \,. </math>

==== Magnetostatics ====
{{Main|Magnetostatics}}

A steady current ''I'' flowing along a path ''ℓ'' will exert a force on nearby charged particles that is quantitatively different from the electric field force described above. The force exerted by ''I'' on a nearby charge ''q'' with velocity '''v''' is
<math display="block">\mathbf{F}(\mathbf{r}) = q\mathbf{v} \times \mathbf{B}(\mathbf{r}),</math>
where '''B'''('''r''') is the [[magnetic field]], which is determined from ''I'' by the [[Biot–Savart law]]:
<math display="block">\mathbf{B}(\mathbf{r}) = \frac{\mu_0 I}{4\pi} \int \frac{d\boldsymbol{\ell} \times d\hat{\mathbf{r}}}{r^2}.</math>

The magnetic field is not conservative in general, and hence cannot usually be written in terms of a scalar potential. However, it can be written in terms of a [[magnetic vector potential|vector potential]], '''A'''('''r'''):
<math display="block"> \mathbf{B}(\mathbf{r}) = \nabla \times \mathbf{A}(\mathbf{r}) </math>

[[Gauss's law]] for magnetism in integral form is
<math display="block">\iint\mathbf{B}\cdot d\mathbf{S} = 0, </math>
while in differential form it is
<math display="block">\nabla \cdot\mathbf{B} = 0. </math>

The physical interpretation is that there are no [[magnetic monopole]]s.

==== Electrodynamics ====
{{Main|Electrodynamics}}

In general, in the presence of both a charge density ''ρ''('''r''', ''t'') and current density '''J'''('''r''', ''t''), there will be both an electric and a magnetic field, and both will vary in time. They are determined by [[Maxwell's equations]], a set of differential equations which directly relate '''E''' and '''B''' to the electric charge density (charge per unit volume) ''ρ'' and [[current density]] (electric current per unit area) '''J'''.<ref name="griffiths326">{{cite book |last=Griffiths |first=David |title=Introduction to Electrodynamics |edition=3rd |page=326 }}</ref>

Alternatively, one can describe the system in terms of its scalar and vector potentials ''V'' and '''A'''. A set of integral equations known as ''[[retarded potential]]s'' allow one to calculate ''V'' and '''A''' from ρ and '''J''',{{NoteTag|This is contingent on the correct choice of [[gauge fixing|gauge]]. ''φ'' and '''A''' are not uniquely determined by ''ρ'' and '''J'''; rather, they are only determined up to some scalar function ''f''('''r''', ''t'') known as the gauge. The retarded potential formalism requires one to choose the [[Lorenz gauge]].}} and from there the electric and magnetic fields are determined via the relations<ref name="wangsness469">{{cite book |last = Wangsness |first=Roald |title=Electromagnetic Fields |edition=2nd |page=469 }}</ref>
<math display="block"> \mathbf{E} = -\nabla V - \frac{\partial \mathbf{A}}{\partial t}</math>
<math display="block"> \mathbf{B} = \nabla \times \mathbf{A}.</math>

=== Continuum mechanics ===
==== Fluid dynamics ====
{{Main|Fluid dynamics}}

Fluid dynamics has fields of pressure, density, and flow rate that are connected by conservation laws for energy and momentum. The mass continuity equation is a continuity equation, representing the conservation of mass
<math display="block">\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf u) = 0 </math>
and the [[Navier–Stokes equations]] represent the conservation of momentum in the fluid, found from Newton's laws applied to the fluid,
<math display="block">\frac {\partial}{\partial t} (\rho \mathbf u) + \nabla \cdot (\rho \mathbf u \otimes \mathbf u + p \mathbf I) = \nabla \cdot \boldsymbol \tau + \rho \mathbf b </math>
if the density {{mvar|ρ}}, pressure {{mvar|p}}, [[deviatoric stress tensor]] {{mvar|'''τ'''}} of the fluid, as well as external body forces '''b''', are all given. The [[velocity field]] '''u''' is the vector field to solve for.

=== Other examples ===
In 1839, [[James MacCullagh]] presented field equations to describe [[reflection (physics)|reflection]] and [[refraction]] in "An essay toward a dynamical theory of crystalline reflection and refraction".<ref>[[James MacCullagh]] (1839) [https://archive.org/stream/collectedworks00maccuoft#page/144/mode/2up An essay toward a dynamical theory of crystalline reflection and refraction], ''Transactions, [[Royal Irish Academy]] 21''</ref>

== Potential theory ==
The term "[[potential theory]]" arises from the fact that, in 19th century physics, the fundamental forces of nature were believed to be derived from [[scalar potential]]s which satisfied [[Laplace's equation]]. Poisson addressed the question of the stability of the planetary [[orbit]]s, which had already been settled by Lagrange to the first degree of approximation from the perturbation forces, and derived the [[Poisson's equation]], named after him. The general form of this equation is

<math display="block">\nabla^2 \phi = \sigma </math>

where ''σ'' is a source function (as a density, a quantity per unit volume) and ø the scalar potential to solve for.

In Newtonian gravitation, masses are the sources of the field so that field lines terminate at objects that have mass. Similarly, charges are the sources and sinks of electrostatic fields: positive charges emanate electric field lines, and field lines terminate at negative charges. These field concepts are also illustrated in the general [[divergence theorem]], specifically Gauss's law's for gravity and electricity. For the cases of time-independent gravity and electromagnetism, the fields are gradients of corresponding potentials
<math display="block">\mathbf{g} = - \nabla \phi_g \,,\quad \mathbf{E} = - \nabla \phi_e </math>
so substituting these into Gauss' law for each case obtains
<math display="block">\nabla^2 \phi_g = 4\pi G \rho_g \,, \quad \nabla^2 \phi_e = 4\pi k_e \rho_e = - {\rho_e \over \varepsilon_0}</math>

where ''ρ<sub>g</sub>'' is the [[mass density]], ''ρ<sub>e</sub>'' the [[charge density]], ''G'' the gravitational constant and ''k<sub>e</sub> = 1/4πε<sub>0</sub>'' the electric force constant.

Incidentally, this similarity arises from the similarity between [[Newton's law of gravitation]] and [[Coulomb's law]].

In the case where there is no source term (e.g. vacuum, or paired charges), these potentials obey [[Laplace's equation]]:
<math display="block">\nabla^2 \phi = 0.</math>

For a distribution of mass (or charge), the potential can be expanded in a series of [[spherical harmonics]], and the ''n''th term in the series can be viewed as a potential arising from the 2<sup>''n''</sup>-moments (see [[multipole expansion]]). For many purposes only the monopole, dipole, and quadrupole terms are needed in calculations.

== Relativistic field theory ==
{{Main|Covariant classical field theory}}

Modern formulations of classical field theories generally require [[Lorentz covariance]] as this is now recognised as a fundamental aspect of nature. A field theory tends to be expressed mathematically by using [[Lagrangian (field theory)|Lagrangian]]s. This is a function that, when subjected to an [[action principle]], gives rise to the [[field equations]] and a [[Conservation law (physics)|conservation law]] for the theory. The [[action (physics)|action]] is a Lorentz scalar, from which the field equations and symmetries can be readily derived.

Throughout we use units such that the speed of light in vacuum is 1, i.e. ''c'' = 1.{{NoteTag|This is equivalent to choosing units of distance and time as light-seconds and seconds or light-years and years. Choosing ''c'' {{=}} 1 allows us to simplify the equations. For instance, ''E'' {{=}} ''mc''<sup>2</sup> reduces to ''E'' {{=}} ''m'' (since ''c''<sup>2</sup> {{=}} 1, without keeping track of units). This reduces complexity of the expressions while keeping focus on the underlying principles. This "trick" must be taken into account when performing actual numerical calculations.}}

=== Lagrangian dynamics ===
{{Main|Lagrangian (field theory)}}

Given a field tensor <math>\phi</math>, a scalar called the [[Lagrangian density]] <math display="block">\mathcal{L}(\phi,\partial\phi,\partial\partial\phi, \ldots ,x)</math> can be constructed from <math>\phi</math> and its derivatives.
From this density, the action functional can be constructed by integrating over spacetime,
<math display="block">\mathcal{S} = \int{\mathcal{L}\sqrt{-g}\, \mathrm{d}^4x}.</math>

Where <math>\sqrt{-g} \, \mathrm{d}^4x</math> is the volume form in curved spacetime. <math>(g\equiv \det(g_{\mu\nu}))</math>

Therefore, the Lagrangian itself is equal to the integral of the Lagrangian density over all space.

Then by enforcing the [[Action (physics)|action principle]], the Euler–Lagrange equations are obtained

<math display="block">\frac{\delta \mathcal{S}}{\delta\phi} = \frac{\partial\mathcal{L}}{\partial\phi} -\partial_\mu \left(\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}\right)+ \cdots +(-1)^m\partial_{\mu_1} \partial_{\mu_2} \cdots \partial_{\mu_{m-1}} \partial_{\mu_m} \left(\frac{\partial\mathcal{L}}{\partial(\partial_{\mu_1} \partial_{\mu_2}\cdots\partial_{\mu_{m-1}}\partial_{\mu_m} \phi)}\right) = 0.</math>

== Relativistic fields ==
Two of the most well-known Lorentz-covariant classical field theories are now described.

=== Electromagnetism ===
{{Main|Electromagnetic field|Electromagnetism}}

Historically, the first (classical) field theories were those describing the electric and magnetic fields (separately). After numerous experiments, it was found that these two fields were related, or, in fact, two aspects of the same field: the [[electromagnetic field]]. [[James Clerk Maxwell|Maxwell]]'s theory of [[electromagnetism]] describes the interaction of charged matter with the electromagnetic field. The first formulation of this field theory used vector fields to describe the [[electric]] and [[magnetic]] fields. With the advent of special relativity, a more complete formulation using [[tensor]] fields was found. Instead of using two vector fields describing the electric and magnetic fields, a tensor field representing these two fields together is used.

The [[electromagnetic four-potential]] is defined to be {{math|1=''A<sub>a</sub>'' = (−''φ'', '''A''')}}, and the [[four-current|electromagnetic four-current]] {{math|1=''j<sub>a</sub>'' = (−''ρ'', '''j''')}}. The electromagnetic field at any point in spacetime is described by the antisymmetric (0,2)-rank [[electromagnetic field tensor]]
<math display="block">F_{ab} = \partial_a A_b - \partial_b A_a.</math>

==== The Lagrangian ====
To obtain the dynamics for this field, we try and construct a scalar from the field. In the vacuum, we have
<math display="block">\mathcal{L} = -\frac{1}{4\mu_0}F^{ab}F_{ab}\,.</math>

We can use [[gauge field theory]] to get the interaction term, and this gives us
<math display="block">\mathcal{L} = -\frac{1}{4\mu_0}F^{ab}F_{ab} - j^aA_a\,.</math>

==== The equations ====
To obtain the field equations, the electromagnetic tensor in the Lagrangian density needs to be replaced by its definition in terms of the 4-potential ''A'', and it's this potential which enters the Euler-Lagrange equations. The EM field ''F'' is not varied in the EL equations. Therefore,
<math display="block">\partial_b\left(\frac{\partial\mathcal{L}}{\partial\left(\partial_b A_a\right)}\right)=\frac{\partial\mathcal{L}}{\partial A_a} \,.</math>

Evaluating the derivative of the Lagrangian density with respect to the field components
<math display="block">\frac{\partial\mathcal{L}}{\partial A_a} = \mu_0 j^a \,, </math>
and the derivatives of the field components
<math display="block">\frac{\partial\mathcal{L}}{\partial(\partial_b A_a)} = F^{ab} \,, </math>
obtains [[Maxwell's equations]] in vacuum. The source equations (Gauss' law for electricity and the Maxwell-Ampère law) are
<math display="block">\partial_b F^{ab}=\mu_0 j^a \, . </math>
while the other two (Gauss' law for magnetism and Faraday's law) are obtained from the fact that ''F'' is the 4-curl of ''A'', or, in other words, from the fact that the [[Bianchi identity]] holds for the electromagnetic field tensor.<ref>{{Cite web| url=http://mathworld.wolfram.com/BianchiIdentities.html|title=Bianchi Identities}}</ref>
<math display="block">6F_{[ab,c]} \, = F_{ab,c} + F_{ca,b} + F_{bc,a} = 0. </math>

where the comma indicates a [[partial derivative]].

=== Gravitation ===
{{Main|Gravitation}}
{{Further|General Relativity|Einstein field equation}}

After Newtonian gravitation was found to be inconsistent with [[special relativity]], [[Albert Einstein]] formulated a new theory of gravitation called [[general relativity]]. This treats [[gravitation]] as a geometric phenomenon ('curved [[spacetime]]') caused by masses and represents the [[gravitational field]] mathematically by a [[tensor field]] called the [[metric tensor (general relativity)|metric tensor]]. The [[Einstein field equations]] describe how this curvature is produced. [[Newtonian gravitation]] is now superseded by Einstein's theory of [[general relativity]], in which [[gravitation]] is thought of as being due to a curved [[spacetime]], caused by masses. The Einstein field equations,
<math display="block">G_{ab} = \kappa T_{ab} </math>
describe how this curvature is produced by matter and radiation, where ''G<sub>ab</sub>'' is the [[Einstein tensor]],
<math display="block">G_{ab} \, = R_{ab}-\frac{1}{2} R g_{ab}</math>
written in terms of the [[Ricci tensor]] ''R<sub>ab</sub>'' and [[Ricci scalar]] {{math|1=''R'' = ''R<sub>ab</sub>g<sup>ab</sup>''}}, {{math|''T<sub>ab</sub>''}} is the [[stress–energy tensor]] and {{math|1=''κ'' = 8''πG''/''c''<sup>4</sup>}} is a constant. In the absence of matter and radiation (including sources) the '[[vacuum field equations]]'',
<math display="block">G_{ab} = 0 </math>
can be derived by varying the [[Einstein–Hilbert action]],
<math display="block"> S = \int R \sqrt{-g} \, d^4x </math>
with respect to the metric, where ''g'' is the [[determinant]] of the [[metric tensor (general relativity)|metric tensor]] ''g<sup>ab</sup>''. Solutions of the vacuum field equations are called [[vacuum solution]]s. An alternative interpretation, due to [[Arthur Eddington]], is that <math>R</math> is fundamental, <math>T</math> is merely one aspect of <math>R</math>, and <math>\kappa</math> is forced by the choice of units.

=== Further examples ===
Further examples of Lorentz-covariant classical field theories are
* [[Klein-Gordon]] theory for real or complex scalar fields
* [[Dirac equation|Dirac]] theory for a Dirac spinor field
* [[Yang–Mills theory]] for a non-abelian gauge field

== Unification attempts ==
{{Main|Classical unified field theories}}

Attempts to create a unified field theory based on [[classical physics]] are classical unified field theories. During the years between the two World Wars, the idea of unification of [[gravity]] with [[electromagnetism]] was actively pursued by several mathematicians and physicists like [[Albert Einstein]], [[Theodor Kaluza]],<ref name=kal>{{cite journal |last=Kaluza |first=Theodor |date=1921 |title=Zum Unitätsproblem in der Physik |journal=Sitzungsber. Preuss. Akad. Wiss. Berlin. (Math. Phys.) |pages=966–972 |bibcode=1921SPAW.......966K }}</ref> [[Hermann Weyl]],<ref>{{cite journal |author=Weyl, H. |title=Gravitation und Elektrizität |journal=Sitz. Preuss. Akad. Wiss. |year=1918 |pages=147–159|doi=10.1007/978-3-663-19510-8_11 |bibcode=1918SPAW.......465W |isbn=978-3-663-19372-2 }}</ref> [[Arthur Eddington]],<ref>{{cite book |author=Eddington, A. S. |title=The Mathematical Theory of Relativity, 2nd ed. |publisher=Cambridge Univ. Press |year=1924 }}</ref> [[Gustav Mie]]<ref>{{cite journal |author=Mie, G. |title=Grundlagen einer Theorie der Materie |journal=Annalen der Physik |year=1912 |volume=37 |pages=511–534 |doi=10.1002/andp.19123420306 |issue=3|bibcode = 1912AnP...342..511M |url=https://zenodo.org/record/1424223 }}</ref> and Ernst Reichenbacher.<ref>{{cite journal |author=Reichenbächer, E. |title=Grundzüge zu einer Theorie der Elektrizität und der Gravitation |journal=Annalen der Physik |year=1917 |volume=52 |pages=134–173 |doi=10.1002/andp.19173570203 |issue=2|bibcode = 1917AnP...357..134R |url=https://zenodo.org/record/1424315 }}</ref>

Early attempts to create such theory were based on incorporation of [[electromagnetic fields]] into the geometry of [[general relativity]]. In 1918, the case for the first geometrization of the electromagnetic field was proposed in 1918 by Hermann Weyl.<ref name=Tilman>{{Citation| last = Sauer| first = Tilman| author-link = Sauer Tilman| chapter = Einstein’s Unified Field Theory Program| date = May 2014| editor1-last = Janssen| editor1-first = Michel | editor2-last = Lehner| editor2-first = Christoph | title = The Cambridge Companion to Einstein| publisher = Cambridge University Press| publication-date = May 2014| isbn = 9781139024525}}</ref>
In 1919, the idea of a five-dimensional approach was suggested by [[Theodor Kaluza]].<ref name=Tilman/> From that, a theory called [[Kaluza-Klein Theory]] was developed. It attempts to unify [[gravitation]] and [[electromagnetism]], in a five-dimensional [[space-time]].
There are several ways of extending the representational framework for a unified field theory which have been considered by Einstein and other researchers. These extensions in general are based in two options.<ref name=Tilman/> The first option is based in relaxing the conditions imposed on the original formulation, and the second is based in introducing other mathematical objects into the theory.<ref name=Tilman/> An example of the first option is relaxing the restrictions to four-dimensional space-time by considering higher-dimensional representations.<ref name=Tilman/> That is used in [[Kaluza-Klein Theory]]. For the second, the most prominent example arises from the concept of the [[affine connection]] that was introduced into [[General relativity|the theory of general relativity]] mainly through the work of [[Tullio Levi-Civita]] and [[Hermann Weyl]].<ref name=Tilman/>

Further development of [[quantum field theory]] changed the focus of searching for unified field theory from classical to quantum description. Because of that, many theoretical physicists gave up looking for a classical unified field theory.<ref name=Tilman/> Quantum field theory would include unification of two other [[Fundamental interactions|fundamental forces of nature]], the [[strong nuclear force|strong]] and [[weak nuclear force]] which act on the subatomic level.<ref>{{cite journal |last=Gadzirayi Nyambuya|first=Golden|title=Unified Field Theory – Paper I, Gravitational, Electromagnetic, Weak & the Strong Force| journal=Apeiron |date=October 2007|volume=14|issue=4|page=321|url=http://redshift.vif.com/JournalFiles/V14NO4PDF/V14N4GAD.pdf |access-date=30 December 2017}}</ref><ref>{{cite journal|last1=De Boer|first1=W.|title=Grand unified theories and supersymmetry in particle physics and cosmology|journal=Progress in Particle and Nuclear Physics|date=1994|volume=33| pages=201–301 |url=http://www-ekp.physik.uni-karlsruhe.de/~deboer/html/Lehre/Susy/deboer_review3.pdf|access-date=30 December 2017|arxiv=hep-ph/9402266|bibcode=1994PrPNP..33..201D|doi=10.1016/0146-6410(94)90045-0|s2cid=119353300}}</ref>

== See also ==
{{cols}}
*[[Relativistic wave equations]]
*[[Quantum field theory]]
*[[Classical unified field theories]]
*[[Variational methods in general relativity]]
*[[Higgs field (classical)]]
*[[Lagrangian (field theory)]]
*[[Hamiltonian field theory]]
*[[Covariant Hamiltonian field theory]]
{{colend}}

== Notes ==
{{NoteFoot}}

== References ==
=== Citations ===
{{Reflist}}

=== Sources ===
{{refbegin}}
* {{cite book
| first1 = C.
| last1 = Truesdell
| author-link = Clifford Truesdell
| first2 = R.A.
| last2 = Toupin
| author2-link = Richard Toupin
| year = 1960
| contribution = The Classical Field Theories
| title = Principles of Classical Mechanics and Field Theory/Prinzipien der Klassischen Mechanik und Feldtheorie
| editor-last = Flügge
| editor-first = Siegfried
| editor-link = Siegfried Flügge
| series = Handbuch der Physik (Encyclopedia of Physics)
| volume = III/1
| pages = 226&ndash;793
| place = Berlin&ndash;Heidelberg&ndash;New York
| publisher = Springer-Verlag
| zbl = 0118.39702
}}
{{refend}}

== External links ==
* {{cite web | last=Thidé | first=Bo | author-link=Bo Thidé | title=Electromagnetic Field Theory | url=http://www.plasma.uu.se/CED/Book/EMFT_Book.pdf | access-date=February 14, 2006 | archive-url=https://web.archive.org/web/20030917043122/http://www.plasma.uu.se/CED/Book/EMFT_Book.pdf | archive-date=September 17, 2003 | url-status=dead }}
* {{cite journal |last=Carroll |first= Sean M. | title=Lecture Notes on General Relativity | arxiv=gr-qc/9712019 |bibcode=1997gr.qc....12019C |year= 1997 }}
* {{cite web |last=Binney |first= James J. | title = Lecture Notes on Classical Fields | url=http://www-thphys.physics.ox.ac.uk/user/JamesBinney/classf.pdf| access-date=April 30, 2007 }}
* {{cite journal | author-link = Gennadi Sardanashvily | last = Sardanashvily | first = G. | title = Advanced Classical Field Theory | journal = International Journal of Geometric Methods in Modern Physics | volume = 5 | issue = 7 | pages = 1163–1189 |date=November 2008 | isbn = 978-981-283-895-7 | arxiv = 0811.0331 | doi = 10.1142/S0219887808003247 | bibcode = 2008IJGMM..05.1163S | s2cid = 13884729 }}

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