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Aclassical field theory is aphysical theory that predicts how one or morefields in physics interact with matter throughfield equations, without consideringeffects of quantization; theories that incorporate quantum mechanics are calledquantum field theories. In most contexts, 'classical field theory' is specifically intended to describeelectromagnetism andgravitation, two of thefundamental forces of nature.

A physical field can be thought of as the assignment of aphysical quantity at each point ofspace andtime. For example, in a weather forecast, the wind velocity during a day over a country is described by assigning avector 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 avector 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 andMaxwell's equations of electromagnetic fields were developed in classical physics before the advent ofrelativity theory in 1905, and had to be revised to be consistent with that theory. Consequently, classical field theories are usually categorized asnon-relativistic andrelativistic. Modern field theories are usually expressed using the mathematics oftensor calculus. A more recent alternativemathematical formalism describes classical fields as sections ofmathematical objects calledfiber bundles.

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.

Some of the simplest physical fields are vector force fields. Historically, the first time that fields were taken seriously was withFaraday'slines of force when describing theelectric field. Thegravitational field was then similarly described.

The firstfield theory of gravity wasNewton's theory of gravitation in which the mutual interaction between twomasses obeys aninverse square law. This was very useful for predicting the motion of planets around the Sun.

Any massive bodyM has agravitational fieldg which describes its influence on other massive bodies. The gravitational field ofM at a pointr in space is found by determining the forceF thatM exerts on a smalltest massm located atr, and then dividing bym:^[1]$${\displaystyle \mathbf{g}(\mathbf{r})=\frac{\mathbf{F}(\mathbf{r})}{m}.}$$Stipulating thatm is much smaller thanM ensures that the presence ofm has a negligible influence on the behavior ofM.

According toNewton's law of universal gravitation,F(r) is given by^[1]$${\displaystyle \mathbf{F}(\mathbf{r})=-\frac{GMm}{r^2}\hat{\mathbf{r}},}$$where${\displaystyle \hat{\mathbf{r}}}$ is aunit vector pointing along the line fromM tom, andG is Newton'sgravitational constant. Therefore, the gravitational field ofM is^[1]$${\displaystyle \mathbf{g}(\mathbf{r})=\frac{\mathbf{F}(\mathbf{r})}{m}=-\frac{GM}{r^2}\hat{\mathbf{r}}.}$$

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 theequivalence principle, which leads togeneral relativity.

For a discrete collection of masses,M_i, located at points,r_i, the gravitational field at a pointr due to the masses is$${\displaystyle \mathbf{g}(\mathbf{r})=-G\sum _i\frac{M_i(\mathbf{r}-{\mathbf{r}}_{\mathbf{i}})}{|\mathbf{r}-{\mathbf{r}}_i{|}^3},}$$

If we have a continuous mass distributionρ instead, the sum is replaced by an integral,$${\displaystyle \mathbf{g}(\mathbf{r})=-G{\iiint }_V\frac{\rho (\mathbf{x})d^3\mathbf{x}(\mathbf{r}-\mathbf{x})}{|\mathbf{r}-\mathbf{x}{|}^3},}$$

Note that the direction of the field points from the positionr to the position of the massesr_i; this is ensured by the minus sign. In a nutshell, this means all masses attract.

In the integral formGauss's law for gravity is$${\displaystyle \iint \mathbf{g}\cdot d\mathbf{S}=-4\pi GM}$$while in differential form it is$${\displaystyle \mathrm{\nabla }\cdot \mathbf{g}=-4\pi G{\rho }_m}$$

Therefore, the gravitational fieldg can be written in terms of thegradient of agravitational potentialφ(r):$${\displaystyle \mathbf{g}(\mathbf{r})=-\mathrm{\nabla }\varphi (\mathbf{r}).}$$This is a consequence of the gravitational forceF beingconservative.

Acharged test particle with chargeq experiences a forceF based solely on its charge. We can similarly describe theelectric fieldE generated by the source chargeQ so thatF =qE:$${\displaystyle \mathbf{E}(\mathbf{r})=\frac{\mathbf{F}(\mathbf{r})}{q}.}$$

Using this andCoulomb's law the electric field due to a single charged particle is$${\displaystyle \mathbf{E}=\frac{1}{4\pi {\epsilon }_0}\frac{Q}{r^2}\hat{\mathbf{r}}.}$$

The electric field isconservative, and hence is given by the gradient of a scalar potential,V(r)$${\displaystyle \mathbf{E}(\mathbf{r})=-\mathrm{\nabla }V(\mathbf{r}).}$$

Gauss's law for electricity is in integral form$${\displaystyle \iint \mathbf{E}\cdot d\mathbf{S}=\frac{Q}{{\epsilon }_0}}$$while in differential form$${\displaystyle \mathrm{\nabla }\cdot \mathbf{E}=\frac{{\rho }_e}{{\epsilon }_0}.}$$

A steady currentI flowing along a pathℓ will exert a force on nearby charged particles that is quantitatively different from the electric field force described above. The force exerted byI on a nearby chargeq with velocityv is$${\displaystyle \mathbf{F}(\mathbf{r})=q\mathbf{v}\times \mathbf{B}(\mathbf{r}),}$$whereB(r) is themagnetic field, which is determined fromI by theBiot–Savart law:$${\displaystyle \mathbf{B}(\mathbf{r})=\frac{{\mu }_{0}I}{4\pi }\int \frac{d\mathit{\ell }\times d\hat{\mathbf{r}}}{r^2}.}$$

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 avector potential,A(r):$${\displaystyle \mathbf{B}(\mathbf{r})=\mathrm{\nabla }\times \mathbf{A}(\mathbf{r})}$$

Gauss's law for magnetism in integral form is$${\displaystyle \iint \mathbf{B}\cdot d\mathbf{S}=0,}$$while in differential form it is$${\displaystyle \mathrm{\nabla }\cdot \mathbf{B}=0.}$$

The physical interpretation is that there are nomagnetic monopoles.

In general, in the presence of both a charge densityρ(r,t) and current densityJ(r,t), there will be both an electric and a magnetic field, and both will vary in time. They are determined byMaxwell's equations, a set of differential equations which directly relateE andB to the electric charge density (charge per unit volume)ρ andcurrent density (electric current per unit area)J.^[2]

Alternatively, one can describe the system in terms of its scalar and vector potentialsV andA. A set of integral equations known asretarded potentials allow one to calculateV andA from ρ andJ,^{[note 1]} and from there the electric and magnetic fields are determined via the relations^[3]$${\displaystyle \mathbf{E}=-\mathrm{\nabla }V-\frac{\mathrm{\partial }\mathbf{A}}{\mathrm{\partial }t}}$$$${\displaystyle \mathbf{B}=\mathrm{\nabla }\times \mathbf{A}.}$$

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$${\displaystyle \frac{\mathrm{\partial }\rho }{\mathrm{\partial }t}+\mathrm{\nabla }\cdot (\rho \mathbf{u})=0}$$and theNavier–Stokes equations represent the conservation of momentum in the fluid, found from Newton's laws applied to the fluid,$${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }t}(\rho \mathbf{u})+\mathrm{\nabla }\cdot (\rho \mathbf{u}\otimes \mathbf{u}+p\mathbf{I})=\mathrm{\nabla }\cdot \mathit{\tau }+\rho \mathbf{b}}$$if the densityρ, pressurep,deviatoric stress tensorτ of the fluid, as well as external body forcesb, are all given. Thevelocity fieldu is the vector field to solve for.

In 1839,James MacCullagh presented field equations to describereflection andrefraction in "An essay toward a dynamical theory of crystalline reflection and refraction".^[4]

The term "potential theory" arises from the fact that, in 19th century physics, the fundamental forces of nature were believed to be derived fromscalar potentials which satisfiedLaplace's equation. Poisson addressed the question of the stability of the planetaryorbits, which had already been settled by Lagrange to the first degree of approximation from the perturbation forces, and derived thePoisson's equation, named after him. The general form of this equation is

$${\displaystyle {\mathrm{\nabla }}^2\varphi =\sigma }$$

whereσ is a source function (as a density, a quantity per unit volume) and φ 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 generaldivergence 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$${\displaystyle \mathbf{g}=-\mathrm{\nabla }{\varphi }_g,\mathbf{E}=-\mathrm{\nabla }{\varphi }_e}$$so substituting these into Gauss' law for each case obtains$${\displaystyle {\mathrm{\nabla }}^2{\varphi }_g=4\pi G{\rho }_g,{\mathrm{\nabla }}^2{\varphi }_e=4\pi k_e{\rho }_e=-\frac{{\rho }_e}{{\epsilon }_0}}$$

whereρ_g is themass density,ρ_e thecharge density,G the gravitational constant andk_e = 1/4πε₀ the electric force constant.

Incidentally, this similarity arises from the similarity betweenNewton's law of gravitation andCoulomb's law.

In the case where there is no source term (e.g. vacuum, or paired charges), these potentials obeyLaplace's equation:$${\displaystyle {\mathrm{\nabla }}^2\varphi =0.}$$

For a distribution of mass (or charge), the potential can be expanded in a series ofspherical harmonics, and thenth term in the series can be viewed as a potential arising from the 2ⁿ-moments (seemultipole expansion). For many purposes only the monopole, dipole, and quadrupole terms are needed in calculations.

Modern formulations of classical field theories generally requireLorentz covariance as this is now recognised as a fundamental aspect of nature. A field theory tends to be expressed mathematically by usingLagrangians. This is a function that, when subjected to anaction principle, gives rise to thefield equations and aconservation law for the theory. Theaction 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.^{[note 2]}

Given a field tensor${\displaystyle \varphi }$, a scalar called theLagrangian density$${\displaystyle \mathcal{L}(\varphi ,\mathrm{\partial }\varphi ,\mathrm{\partial }\mathrm{\partial }\varphi ,\dots ,x)}$$ can be constructed from${\displaystyle \varphi }$ and its derivatives.From this density, the action functional can be constructed by integrating over spacetime,$${\displaystyle \mathcal{S}=\int \mathcal{L}\sqrt{-g}{\mathrm{d}}^{4}x.}$$

Where${\displaystyle \sqrt{-g}{\mathrm{d}}^{4}x}$ is the volume form in curved spacetime.${\displaystyle (g\equiv det(g_{\mu \nu }))}$

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

Then by enforcing theaction principle, the Euler–Lagrange equations are obtained

$${\displaystyle \frac{\delta \mathcal{S}}{\delta \varphi }=\frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }\varphi }-{\mathrm{\partial }}_{\mu }\left(\frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }({\mathrm{\partial }}_{\mu }\varphi )}\right)+\cdots +(-1{)}^m{\mathrm{\partial }}_{{\mu }_1}{\mathrm{\partial }}_{{\mu }_2}\cdots {\mathrm{\partial }}_{{\mu }_{m-1}}{\mathrm{\partial }}_{{\mu }_m}\left(\frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }({\mathrm{\partial }}_{{\mu }_1}{\mathrm{\partial }}_{{\mu }_2}\cdots {\mathrm{\partial }}_{{\mu }_{m-1}}{\mathrm{\partial }}_{{\mu }_m}\varphi )}\right)=0.}$$

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

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: theelectromagnetic field.Maxwell's theory ofelectromagnetism describes the interaction of charged matter with the electromagnetic field. The first formulation of this field theory used vector fields to describe theelectric andmagnetic fields. With the advent of special relativity, a more complete formulation usingtensor 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.

Theelectromagnetic four-potential is defined to beA_a = (−φ,A), and theelectromagnetic four-currentj_a = (−ρ,j). The electromagnetic field at any point in spacetime is described by the antisymmetric (0,2)-rankelectromagnetic field tensor$${\displaystyle F_{ab}={\mathrm{\partial }}_a{A}_b-{\mathrm{\partial }}_b{A}_a.}$$

To obtain the dynamics for this field, we try and construct a scalar from the field. In the vacuum, we have$${\displaystyle \mathcal{L}=-\frac{1}{4{\mu }_0}{F}^{ab}{F}_{ab}.}$$

We can usegauge field theory to get the interaction term, and this gives us$${\displaystyle \mathcal{L}=-\frac{1}{4{\mu }_0}{F}^{ab}{F}_{ab}-j^a{A}_a.}$$

To obtain the field equations, the electromagnetic tensor in the Lagrangian density needs to be replaced by its definition in terms of the 4-potentialA, and it's this potential which enters the Euler-Lagrange equations. The EM fieldF is not varied in the EL equations. Therefore,$${\displaystyle {\mathrm{\partial }}_b\left(\frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }\left({\mathrm{\partial }}_b{A}_a\right)}\right)=\frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }{A}_a}.}$$

Evaluating the derivative of the Lagrangian density with respect to the field components$${\displaystyle \frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }{A}_a}={\mu }_0{j}^a,}$$and the derivatives of the field components$${\displaystyle \frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }({\mathrm{\partial }}_b{A}_a)}=F^{ab},}$$obtainsMaxwell's equations in vacuum. The source equations (Gauss' law for electricity and the Maxwell-Ampère law) are$${\displaystyle {\mathrm{\partial }}_b{F}^{ab}={\mu }_0{j}^a.}$$while the other two (Gauss' law for magnetism and Faraday's law) are obtained from the fact thatF is the 4-curl ofA, or, in other words, from the fact that theBianchi identity holds for the electromagnetic field tensor.^[5]$${\displaystyle 6F_{[ab,c]}=F_{ab,c}+F_{ca,b}+F_{bc,a}=0.}$$

where the comma indicates apartial derivative.

After Newtonian gravitation was found to be inconsistent withspecial relativity,Albert Einstein formulated a new theory of gravitation calledgeneral relativity. This treatsgravitation as a geometric phenomenon ('curvedspacetime') caused by masses and represents thegravitational field mathematically by atensor field called themetric tensor. TheEinstein field equations describe how this curvature is produced.Newtonian gravitation is now superseded by Einstein's theory ofgeneral relativity, in whichgravitation is thought of as being due to a curvedspacetime, caused by masses. The Einstein field equations,$${\displaystyle G_{ab}=\kappa T_{ab}}$$describe how this curvature is produced by matter and radiation, whereG_ab is theEinstein tensor,$${\displaystyle G_{ab}=R_{ab}-\frac{1}{2}R{g}_{ab}}$$written in terms of theRicci tensorR_ab andRicci scalarR =R_abg^ab,T_ab is thestress–energy tensor andκ = 8πG/c⁴ is a constant. In the absence of matter and radiation (including sources) thevacuum field equations,$${\displaystyle G_{ab}=0}$$can be derived by varying theEinstein–Hilbert action,$${\displaystyle S=\int R\sqrt{-g}{d}^{4}x}$$with respect to the metric, whereg is thedeterminant of themetric tensorg^ab. Solutions of the vacuum field equations are calledvacuum solutions. An alternative interpretation, due toArthur Eddington, is that${\displaystyle R}$ is fundamental,${\displaystyle T}$ is merely one aspect of${\displaystyle R}$, and${\displaystyle \kappa }$ is forced by the choice of units.

Further examples of Lorentz-covariant classical field theories are

• Klein-Gordon theory for real or complex scalar fields
• Dirac theory for a Dirac spinor field
• Yang–Mills theory for a non-abelian gauge field

Attempts to create a unified field theory based onclassical physics are classical unified field theories. During the years between the two World Wars, the idea of unification ofgravity withelectromagnetism was actively pursued by several mathematicians and physicists likeAlbert Einstein,Theodor Kaluza,^[6]Hermann Weyl,^[7]Arthur Eddington,^[8]Gustav Mie^[9] and Ernst Reichenbacher.^[10]

Early attempts to create such a theory were based on incorporation ofelectromagnetic fields into the geometry ofgeneral relativity. In 1918, the case for the first geometrization of the electromagnetic field was proposed in 1918 by Hermann Weyl.^[11]In 1919, the idea of a five-dimensional approach was suggested byTheodor Kaluza.^[11] From that, a theory calledKaluza-Klein Theory was developed. It attempts to unifygravitation andelectromagnetism, in a five-dimensionalspace-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.^[11] 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.^[11] An example of the first option is relaxing the restrictions to four-dimensional space-time by considering higher-dimensional representations.^[11] That is used inKaluza-Klein Theory. For the second, the most prominent example arises from the concept of theaffine connection that was introduced intothe theory of general relativity mainly through the work ofTullio Levi-Civita andHermann Weyl.^[11]

Further development ofquantum 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.^[11] Quantum field theory would include unification of two otherfundamental forces of nature, thestrong andweak nuclear force which act on the subatomic level.^[12][13]

• Thidé, Bo."Electromagnetic Field Theory"(PDF). Archived fromthe original(PDF) on September 17, 2003. RetrievedFebruary 14, 2006.
• Carroll, Sean M. (1997). "Lecture Notes on General Relativity".arXiv:gr-qc/9712019.
• Binney, James J."Lecture Notes on Classical Fields"(PDF). RetrievedApril 30, 2007.
• Sardanashvily, G. (November 2008). "Advanced Classical Field Theory".International Journal of Geometric Methods in Modern Physics.5 (7):1163–1189.arXiv:0811.0331.Bibcode:2008IJGMM..05.1163S.doi:10.1142/S0219887808003247.ISBN 978-981-283-895-7.S2CID 13884729.

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