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 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, two of the
fundamental forces of nature.
A physical field can be thought of as the assignment of a
physical quantity
A physical quantity is a physical property of a material or system that can be quantified by measurement. A physical quantity can be expressed as a ''value'', which is the algebraic multiplication of a ' Numerical value ' and a ' Unit '. For examp ...
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 con ...
and
time
Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, t ...
. 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.
Th ...
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 objects called
fiber bundle
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a ...
s.
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 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 element ...
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'':
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
where
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
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
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, ''M
i'', located at points, r
''i'', the gravitational field at a point r due to the masses is
If we have a continuous mass distribution ''ρ'' instead, the sum is replaced by an integral,
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 is
while in differential form it is
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 :
This is a consequence of the gravitational force F being
conservative
Conservatism is a cultural, social, and political philosophy that seeks to promote and to preserve traditional institutions, practices, and values. The central tenets of conservatism may vary in relation to the culture and civilization in ...
.
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 :
Using this and
Coulomb's law the electric field due to a single charged particle is
The electric field is
conservative
Conservatism is a cultural, social, and political philosophy that seeks to promote and to preserve traditional institutions, practices, and values. The central tenets of conservatism may vary in relation to the culture and civilization in ...
, and hence is given by the gradient of a scalar potential,
Gauss's law for electricity is in integral form
while in differential form
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
where B(r) is the
magnetic field
A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and t ...
, which is determined from ''I'' by the
Biot–Savart law:
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):
Gauss's law for magnetism in integral form is
while in differential form it is
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.
Th ...
, a set of differential equations which directly relate E and B to the electric charge density (charge per unit volume) ''ρ'' and
current density
In electromagnetism, current density is the amount of charge per unit time that flows through a unit area of a chosen cross section. The current density vector is defined as a vector whose magnitude is the electric current per cross-sectional a ...
(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
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
and the
Navier–Stokes equations
In physics, the Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician G ...
represent the conservation of momentum in the fluid, found from Newton's laws applied to the fluid,
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 a ...
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
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, 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
so substituting these into Gauss' law for each case obtains
where ''ρ
g'' 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. Mathematically ...
, ''ρ
e'' 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 in ...
, ''G'' the gravitational constant and ''k
e = 1/4πε
0'' the electric force constant.
Incidentally, this similarity arises from the similarity between
Newton's law of gravitation
Newton's law of universal gravitation is usually stated as that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distan ...
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:
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
''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
Lagrangian
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 ...
s. This is a function that, when subjected to an
action principle, 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
, a scalar called the
Lagrangian densitycan be constructed from
and its derivatives.
From this density, the action functional can be constructed by integrating over spacetime,
Where
is the volume form in curved spacetime.
Therefore, the Lagrangian itself is equal to the integral of the Lagrangian density over all space.
Then by enforcing the
action principle, the Euler–Lagrange equations are obtained
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 classica ...
.
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 o ...
'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
Magnetism is the class of physical attributes that are mediated by a magnetic field, which refers to the capacity to induce attractive and repulsive phenomena in other entities. Electric currents and the magnetic moments of elementary particles ...
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 tens ...
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
The Lagrangian
To obtain the dynamics for this field, we try and construct a scalar from the field. In the vacuum, we have
We can use
gauge field theory to get the interaction term, and this gives us
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,
Evaluating the derivative of the Lagrangian density with respect to the field components
and the derivatives of the field components
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.
Th ...
in vacuum. The source equations (Gauss' law for electricity and the Maxwell-Ampère law) are
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.
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 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 diffe ...
') caused by masses and represents the
gravitational field mathematically by a
tensor field called the
metric tensor. The
Einstein 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 ...
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 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 diffe ...
, caused by masses. The Einstein field equations,
describe how this curvature is produced by matter and radiation, where ''G
ab'' is the
Einstein tensor,
written in terms of the
Ricci tensor ''R
ab'' and
Ricci scalar , is the
stress–energy tensor 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 ...
'',
can be derived by varying the
Einstein–Hilbert action,
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 ...
of the
metric tensor ''g
ab''. Solutions of the vacuum field equations are called
vacuum solutions. An alternative interpretation, due to
Arthur Eddington, is that
is fundamental,
is merely one aspect of
, and
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 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 str ...
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 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 and Hermann Weyl.][
Further development of ]quantum field theory
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 a ...
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
In physics, the fundamental interactions, also known as fundamental forces, are the interactions that do not appear to be reducible to more basic interactions. There are four fundamental interactions known to exist: the gravitational and electr ...
, 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 Sta ...
and weak nuclear force which act on the subatomic level.
See also
* Relativistic wave equations
*Quantum field theory
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 a ...
* Classical unified field theories
* Variational methods in general relativity
* Higgs field (classical)
* Lagrangian (field theory)
* Hamiltonian field theory
* Covariant Hamiltonian field theory
Notes
References
Citations
Sources
*
External links
*
*
*
*
{{DEFAULTSORT:Classical field theory
Mathematical physics
Theoretical physics
Lagrangian mechanics
Equations