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A classical field theory is a physical theory that predicts how one or more physical fields interact with matter through field equations, without considering effects of quantization; theories that incorporate quantum mechanics are called
quantum field theories In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles ...
. In most contexts, 'classical field theory' is specifically intended to describe
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
and
gravitation In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stron ...
, two of the fundamental forces of nature. A physical field can be thought of as the assignment of a physical quantity at each point of
space Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consi ...
and
time Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, through the present, into the future. It is a component quantity of various me ...
. For example, in a weather forecast, the wind velocity during a day over a country is described by assigning a 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 Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits ...
of electromagnetic fields were developed in classical physics before the advent of
relativity theory The theory of relativity usually encompasses two interrelated theories by Albert Einstein: special relativity and general relativity, proposed and published in 1905 and 1915, respectively. Special relativity applies to all physical phenomena ...
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 objects called fiber bundles.


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 Faraday's lines of force when describing the electric field. The
gravitational field In physics, a gravitational field is a model used to explain the influences that a massive body extends into the space around itself, producing a force on another massive body. Thus, a gravitational field is used to explain gravitational pheno ...
was then similarly described.


Newtonian gravitation

The first field theory of gravity was Newton's theory of gravitation in which the mutual interaction between two
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different ele ...
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 In physics, a gravitational field is a model used to explain the influences that a massive body extends into the space around itself, producing a force on another massive body. Thus, a gravitational field is used to explain gravitational pheno ...
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'': \mathbf(\mathbf) = \frac. 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 \mathbf(\mathbf) = -\frac\hat, where \hat is a
unit vector In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction v ...
pointing along the line from ''M'' to ''m'', and ''G'' is Newton's
gravitational constant The gravitational constant (also known as the universal gravitational constant, the Newtonian constant of gravitation, or the Cavendish gravitational constant), denoted by the capital letter , is an empirical physical constant involved in ...
. Therefore, the gravitational field of ''M'' is \mathbf(\mathbf) = \frac = -\frac\hat. 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 In the theory of general relativity, the equivalence principle is the equivalence of gravitational and inertial mass, and Albert Einstein's observation that the gravitational "force" as experienced locally while standing on a massive body (su ...
, which leads to
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
. For a discrete collection of masses, ''Mi'', located at points, r''i'', the gravitational field at a point r due to the masses is \mathbf(\mathbf)=-G\sum_i \frac \,, If we have a continuous mass distribution ''ρ'' instead, the sum is replaced by an integral, \mathbf(\mathbf)=-G \iiint_V \frac \, , Note that the direction of the field points from the position r to the position of the masses r''i''; this is ensured by the minus sign. In a nutshell, this means all masses attract. In the integral form
Gauss's law for gravity In physics, Gauss's law for gravity, also known as Gauss's flux theorem for gravity, is a law of physics that is equivalent to Newton's law of universal gravitation. It is named after Carl Friedrich Gauss. It states that the flux (surface int ...
is \iint\mathbf\cdot d \mathbf = -4\pi G M while in differential form it is \nabla \cdot\mathbf = -4\pi G\rho_m Therefore, the gravitational field g can be written in terms of the
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
of a gravitational potential : \mathbf(\mathbf) = -\nabla \phi(\mathbf). This is a consequence of the gravitational force F being conservative.


Electromagnetism


Electrostatics

A 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 : \mathbf(\mathbf) = \frac. Using this and
Coulomb's law Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is convention ...
the electric field due to a single charged particle is \mathbf = \frac \frac \hat \,. The electric field is conservative, and hence is given by the gradient of a scalar potential, \mathbf(\mathbf) = -\nabla V(\mathbf) \, .
Gauss's law In physics and electromagnetism, Gauss's law, also known as Gauss's flux theorem, (or sometimes simply called Gauss's theorem) is a law relating the distribution of electric charge to the resulting electric field. In its integral form, it st ...
for electricity is in integral form \iint\mathbf\cdot d\mathbf = \frac while in differential form \nabla \cdot\mathbf = \frac \,.


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 \mathbf(\mathbf) = q\mathbf \times \mathbf(\mathbf), where B(r) is the magnetic field, which is determined from ''I'' by the Biot–Savart law: \mathbf(\mathbf) = \frac \int \frac. 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 vector potential, A(r): \mathbf(\mathbf) = \nabla \times \mathbf(\mathbf)
Gauss's law In physics and electromagnetism, Gauss's law, also known as Gauss's flux theorem, (or sometimes simply called Gauss's theorem) is a law relating the distribution of electric charge to the resulting electric field. In its integral form, it st ...
for magnetism in integral form is \iint\mathbf\cdot d\mathbf = 0, while in differential form it is \nabla \cdot\mathbf = 0. The physical interpretation is that there are no magnetic monopoles.


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 Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits ...
, 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. 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 potentials'' allow one to calculate ''V'' and A from ρ and J, and from there the electric and magnetic fields are determined via the relations \mathbf = -\nabla V - \frac \mathbf = \nabla \times \mathbf.


Continuum mechanics


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 \frac + \nabla \cdot (\rho \mathbf u) = 0 and the Navier–Stokes equations represent the conservation of momentum in the fluid, found from Newton's laws applied to the fluid, \frac (\rho \mathbf u) + \nabla \cdot (\rho \mathbf u \otimes \mathbf u + p \mathbf I) = \nabla \cdot \boldsymbol \tau + \rho \mathbf b if the density , pressure , deviatoric stress tensor 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 and
refraction In physics, refraction is the redirection of a wave as it passes from one medium to another. The redirection can be caused by the wave's change in speed or by a change in the medium. Refraction of light is the most commonly observed phenomen ...
in "An essay toward a dynamical theory of crystalline reflection and refraction".


Potential theory

The term "
potential theory In mathematics and mathematical physics, potential theory is the study of harmonic functions. The term "potential theory" was coined in 19th-century physics when it was realized that two fundamental forces of nature known at the time, namely gra ...
" arises from the fact that, in 19th century physics, the fundamental forces of nature were believed to be derived from scalar potentials which satisfied Laplace's equation. Poisson addressed the question of the stability of the planetary
orbit In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as ...
s, which had already been settled by Lagrange to the first degree of approximation from the perturbation forces, and derived the
Poisson's equation Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with ...
, named after him. The general form of this equation is \nabla^2 \phi = \sigma where ''σ'' 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 general
divergence theorem In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the '' flux'' of a vector field through a closed surface to the ''divergence'' of the field in the ...
, 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 \mathbf = - \nabla \phi_g \,,\quad \mathbf = - \nabla \phi_e so substituting these into Gauss' law for each case obtains \nabla^2 \phi_g = 4\pi G \rho_g \,, \quad \nabla^2 \phi_e = 4\pi k_e \rho_e = - where ''ρg'' is the
mass density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
, ''ρe'' the
charge density In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system i ...
, ''G'' the gravitational constant and ''ke = 1/4πε0'' the electric force constant. Incidentally, this similarity arises from the similarity between Newton's law of gravitation and
Coulomb's law Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is convention ...
. In the case where there is no source term (e.g. vacuum, or paired charges), these potentials obey Laplace's equation: \nabla^2 \phi = 0. For a distribution of mass (or charge), the potential can be expanded in a series of
spherical harmonics In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. Since the spherical harmonics form ...
, and the ''n''th term in the series can be viewed as a potential arising from the 2''n''-moments (see multipole expansion). For many purposes only the monopole, dipole, and quadrupole terms are needed in calculations.


Relativistic 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 Lagrangians. This is a function that, when subjected to an
action principle In physics, action is a scalar quantity describing how a physical system has changed over time. Action is significant because the equations of motion of the system can be derived through the principle of stationary action. In the simple case ...
, gives rise to the field equations and a conservation law for the theory. The 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.


Lagrangian dynamics

Given a field tensor \phi, a scalar called the Lagrangian density\mathcal(\phi,\partial\phi,\partial\partial\phi, \ldots ,x)can be constructed from \phi and its derivatives. From this density, the action functional can be constructed by integrating over spacetime, \mathcal = \int. Where \sqrt \, \mathrm^4x is the volume form in curved spacetime. (g\equiv \det(g_)) Therefore, the Lagrangian itself is equal to the integral of the Lagrangian density over all space. Then by enforcing the
action principle In physics, action is a scalar quantity describing how a physical system has changed over time. Action is significant because the equations of motion of the system can be derived through the principle of stationary action. In the simple case ...
, the Euler–Lagrange equations are obtained \frac = \frac -\partial_\mu \left(\frac\right)+ \cdots +(-1)^m\partial_ \partial_ \cdots \partial_ \partial_ \left(\frac\right) = 0.


Relativistic fields

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


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 An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical ...
.
Maxwell Maxwell may refer to: People * Maxwell (surname), including a list of people and fictional characters with the name ** James Clerk Maxwell, mathematician and physicist * Justice Maxwell (disambiguation) * Maxwell baronets, in the Baronetage of ...
's theory of
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
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 In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other 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 , and the electromagnetic four-current . The electromagnetic field at any point in spacetime is described by the antisymmetric (0,2)-rank electromagnetic field tensor F_ = \partial_a A_b - \partial_b A_a.


The Lagrangian

To obtain the dynamics for this field, we try and construct a scalar from the field. In the vacuum, we have \mathcal = -\fracF^F_\,. We can use
gauge field theory In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups ...
to get the interaction term, and this gives us \mathcal = -\fracF^F_ - j^aA_a\,.


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, \partial_b\left(\frac\right)=\frac \,. Evaluating the derivative of the Lagrangian density with respect to the field components \frac = \mu_0 j^a \,, and the derivatives of the field components \frac = F^ \,, obtains
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits ...
in vacuum. The source equations (Gauss' law for electricity and the Maxwell-Ampère law) are \partial_b F^=\mu_0 j^a \, . 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. 6F_ \, = F_ + F_ + F_ = 0. where the comma indicates a partial derivative.


Gravitation

After Newtonian gravitation was found to be inconsistent with
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The law ...
,
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theor ...
formulated a new theory of gravitation called
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
. This treats
gravitation In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stron ...
as a geometric phenomenon ('curved
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
') caused by masses and represents the
gravitational field In physics, a gravitational field is a model used to explain the influences that a massive body extends into the space around itself, producing a force on another massive body. Thus, a gravitational field is used to explain gravitational pheno ...
mathematically by a tensor field called the metric tensor. The Einstein field equations describe how this curvature is produced. Newtonian gravitation is now superseded by Einstein's theory of
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, in which
gravitation In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stron ...
is thought of as being due to a curved
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
, caused by masses. The Einstein field equations, G_ = \kappa T_ describe how this curvature is produced by matter and radiation, where ''Gab'' is the
Einstein tensor In differential geometry, the Einstein tensor (named after Albert Einstein; also known as the trace-reversed Ricci tensor) is used to express the curvature of a pseudo-Riemannian manifold. In general relativity, it occurs in the Einstein fie ...
, G_ \, = R_-\frac R g_ written in terms of the
Ricci tensor In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure ...
''Rab'' and
Ricci scalar In the mathematical field of Riemannian geometry, the scalar curvature (or the Ricci scalar) is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the geome ...
, is the
stress–energy tensor The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor physical quantity that describes the density and flux of energy and momentum in spacetime, generalizing the str ...
and is a constant. In the absence of matter and radiation (including sources) the '
vacuum field equations In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it. The equations were published by Einstein in 1915 in the form ...
'', G_ = 0 can be derived by varying the
Einstein–Hilbert action The Einstein–Hilbert action (also referred to as Hilbert action) in general relativity is the action that yields the Einstein field equations through the stationary-action principle. With the metric signature, the gravitational part of the a ...
, S = \int R \sqrt \, d^4x with respect to the metric, where ''g'' is the
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
of the metric tensor ''gab''. Solutions of the vacuum field equations are called vacuum solutions. An alternative interpretation, due to Arthur Eddington, is that R is fundamental, T is merely one aspect of R, and \kappa 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 theory for a Dirac spinor field *
Yang–Mills theory In mathematical physics, Yang–Mills theory is a gauge theory based on a special unitary group SU(''N''), or more generally any compact, reductive Lie algebra. Yang–Mills theory seeks to describe the behavior of elementary particles using t ...
for a non-abelian gauge field


Unification attempts

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 In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
with
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
was actively pursued by several mathematicians and physicists like
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theor ...
, Theodor Kaluza, Hermann Weyl, Arthur Eddington, Gustav Mie and Ernst Reichenbacher. Early attempts to create such theory were based on incorporation of electromagnetic fields into the geometry of
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
. In 1918, the case for the first geometrization of the electromagnetic field was proposed in 1918 by Hermann Weyl. In 1919, the idea of a five-dimensional approach was suggested by Theodor Kaluza. From that, a theory called Kaluza-Klein Theory was developed. It attempts to unify
gravitation In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stron ...
and
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
, 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. 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. An example of the first option is relaxing the restrictions to four-dimensional space-time by considering higher-dimensional representations. 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 the theory of general relativity mainly through the work of
Tullio Levi-Civita Tullio Levi-Civita, (, ; 29 March 1873 – 29 December 1941) was an Italian mathematician, most famous for his work on absolute differential calculus ( tensor calculus) and its applications to the theory of relativity, but who also made signi ...
and Hermann Weyl. 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. Quantum field theory would include unification of two other fundamental forces of nature, the
strong Strong may refer to: Education * The Strong, an educational institution in Rochester, New York, United States * Strong Hall (Lawrence, Kansas), an administrative hall of the University of Kansas * Strong School, New Haven, Connecticut, United S ...
and
weak nuclear force In nuclear physics and particle physics, the weak interaction, which is also often called the weak force or weak nuclear force, is one of the four known fundamental interactions, with the others being electromagnetism, the strong interacti ...
which act on the subatomic level.


See also

* Relativistic wave equations * Quantum field theory *
Classical unified field theories Since the 19th century, some physicists, notably Albert Einstein, have attempted to develop a single theoretical framework that can account for all the fundamental forces of nature – a unified field theory. Classical unified field theories are ...
*
Variational methods in general relativity Variational methods in general relativity refers to various mathematical techniques that employ the use of variational calculus in Einstein's theory of general relativity. The most commonly used tools are Lagrangians and Hamiltonians and are used ...
*
Higgs field (classical) Spontaneous symmetry breaking, a vacuum Higgs field, and its associated fundamental particle the Higgs boson are quantum phenomena. A vacuum Higgs field is responsible for spontaneous symmetry breaking the gauge symmetries of fundamental interact ...
*
Lagrangian (field theory) Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
*
Hamiltonian field theory In theoretical physics, Hamiltonian field theory is the field-theoretic analogue to classical Hamiltonian mechanics. It is a formalism in classical field theory alongside Lagrangian field theory. It also has applications in quantum field theory ...
*
Covariant Hamiltonian field theory In theoretical physics, Hamiltonian field theory is the field-theoretic analogue to classical Hamiltonian mechanics. It is a formalism in classical field theory alongside Lagrangian field theory. It also has applications in quantum field theory. ...


Notes


References


Citations


Sources

*


External links

* * * * {{DEFAULTSORT:Classical field theory Mathematical physics Theoretical physics Lagrangian mechanics Equations