Field (physics)

TheInfoList

In physics, a field is a
physical quantity A physical quantity is a physical property of a material or system that can be Quantification (science), quantified by measurement. A physical quantity can be expressed as a ''value'', which is the algebraic multiplication of a ''numerical value'' ...
, represented by a number or another
tensor In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

, that has a value for each
point Point or points may refer to: Places * Point, LewisImage:Point Western Isles NASA World Wind.png, Satellite image of Point Point ( gd, An Rubha), also known as the Eye Peninsula, is a peninsula some 11 km long in the Outer Hebrides (or Western I ...
in
space and time In physics, spacetime is any mathematical model which fuses the three-dimensional space, three dimensions of space and the one dimension of time into a single four-dimensional manifold. The fabric of space-time is a conceptual model combining the ...
. For example, on a weather map, the surface
temperature Temperature is a physical quantity that expresses hot and cold. It is the manifestation of thermal energy, present in all matter, which is the source of the occurrence of heat, a flow of energy, when a body is in contact with another that is ...

is described by assigning a
number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deduct ...
to each point on the map; the temperature can be considered at a certain point in time or over some interval of time, to study the dynamics of temperature change. A surface wind map, assigning an
arrow s and nock. An arrow is a fin-stabilized projectile launched by a bow and arrow, bow. A typical arrow usually consists of a long, stiff, straight ''shaft'' with a weighty (and usually sharp and pointed) ''arrowhead'' attached to the front end, m ...
to each point on a map that describes the wind at that point, is an example of a
vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. For instance, a vector field in the plane can be visualised as a collection of arrows with a given magnitude and direction, each att ...

, i.e. a 1-dimensional (rank-1) tensor field. Field theories, mathematical descriptions of how field values change in space and time, are ubiquitous in physics. For instance, the
electric field An electric field (sometimes E-field) is the physical field that surrounds electrically-charged particle In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' ' ...

is another rank-1 tensor field, while
electrodynamics Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The electromagnetic force is carried by electromagneti ...
can be formulated in terms of two interacting vector fields at each point in spacetime, or as a single-rank 2-tensor field. In the modern framework of the quantum theory of fields, even without referring to a test particle, a field occupies space, contains energy, and its presence precludes a classical "true vacuum". This has led physicists to consider
electromagnetic field An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the in ...
s to be a physical entity, making the field concept a supporting
paradigm In science Science (from the Latin word ''scientia'', meaning "knowledge") is a systematic enterprise that Scientific method, builds and Taxonomy (general), organizes knowledge in the form of Testability, testable explanations and prediction ...
of the edifice of modern physics. "The fact that the electromagnetic field can possess momentum and energy makes it very real ... a particle makes a field, and a field acts on another particle, and the field has such familiar properties as energy content and momentum, just as particles can have." In practice, the strength of most fields diminishes with distance, eventually becoming undetectable. For instance the strength of many relevant classical fields, such as the gravitational field in
Newton's theory of gravity 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 Proportionality (mathematics)#Direct proportionality, directly proportional to the product of their m ...
or the
electrostatic field An electric field (sometimes E-field) is the field (physics), physical field that surrounds each electric charge and exerts force on all other charges in the field, either attracting or repelling them. Electric fields originate from electric cha ...
in classical electromagnetism, is inversely proportional to the square of the distance from the source (i.e., they follow
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 state ...
). A field can be classified as a
scalar field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

, a
vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. For instance, a vector field in the plane can be visualised as a collection of arrows with a given magnitude and direction, each att ...

, a
spinor fieldIn differential geometry Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The Differential g ...
or a
tensor field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
according to whether the represented physical quantity is a
scalar Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers *Scalar (physics), a physical quantity that can be described by a single element of a number field such as ...
, a
vector Vector may refer to: Biology *Vector (epidemiology) In epidemiology Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and risk factor, determinants of health and disease conditions in defined pop ...
, a
spinor In geometry and physics, spinors are elements of a complex numbers, complex vector space that can be associated with Euclidean space. Like Euclidean vector, geometric vectors and more general tensors, spinors linear transformation, transform line ...
, or a
tensor In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

, respectively. A field has a consistent tensorial character wherever it is defined: i.e. a field cannot be a scalar field somewhere and a vector field somewhere else. For example, the
NewtonianNewtonian refers to the work of Isaac Newton, in particular: * Newtonian mechanics, i.e. classical mechanics * Newtonian telescope, a type of reflecting telescope * Newtonian cosmology * Newtonian dynamics * Newtonianism, the philosophical principle ...
gravitational field In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "P ...

is a vector field: specifying its value at a point in spacetime requires three numbers, the components of the gravitational field vector at that point. Moreover, within each category (scalar, vector, tensor), a field can be either a ''classical field'' or a ''quantum field'', depending on whether it is characterized by numbers or quantum operators respectively. In this theory an equivalent representation of field is a
field particle In quantum field theory In theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict List of natural phenomena, natural p ...
, for instance a
boson In quantum mechanics, a boson (, ) is a particle that follows Bose–Einstein statistics. Bosons make up one of two classes of elementary particles, the other being fermions. The name boson was coined by Paul Dirac to commemorate the contributio ...

.

# History

To
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March Old Style and New Style dates, 1726/27) was an English mathematician, physicist, astronomer, theologian, and author (described in his time as a "natural philosophy, natural philosopher") ...

, his
law of universal gravitation Newton's law of universal gravitation is usually stated as that every particle In the Outline of physical science, physical sciences, a particle (or corpuscule in older texts) is a small wikt:local, localized physical body, object to which can ...
simply expressed the gravitational
force In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force ...

that acted between any pair of massive objects. When looking at the motion of many bodies all interacting with each other, such as the planets in the
Solar System The Solar SystemCapitalization of the name varies. The International Astronomical Union, the authoritative body regarding astronomical nomenclature, specifies capitalizing the names of all individual astronomical objects but uses mixed "Solar ...

, dealing with the force between each pair of bodies separately rapidly becomes computationally inconvenient. In the eighteenth century, a new quantity was devised to simplify the bookkeeping of all these gravitational forces. This quantity, the
gravitational field In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "P ...

, gave at each point in space the total gravitational acceleration which would be felt by a small object at that point. This did not change the physics in any way: it did not matter if all the gravitational forces on an object were calculated individually and then added together, or if all the contributions were first added together as a gravitational field and then applied to an object. The development of the independent concept of a field truly began in the nineteenth century with the development of the theory of
electromagnetism Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The electromagnetic force is carried by electromagnet ...

. In the early stages,
André-Marie Ampère André-Marie Ampère (, ; ; 20 January 177510 June 1836) was a French physicist A physicist is a scientist A scientist is a person who conducts Scientific method, scientific research to advance knowledge in an Branches of science, area of ...
and
Charles-Augustin de Coulomb Charles-Augustin de Coulomb (; ; 14 June 1736 – 23 August 1806) was a French officer, engineer, and physicist. He is best known as the eponymous discoverer of what is now called Coulomb's law, the description of the electrostatics, electrostatic ...
could manage with Newton-style laws that expressed the forces between pairs of
electric charge Electric charge is the physical property of matter that causes it to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respectively). Like ch ...
s or
electric current An electric current is a stream of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is measured as the net rate of flow of electric charge through a surface or into a control volume. The moving part ...
s. However, it became much more natural to take the field approach and express these laws in terms of
electric Electricity is the set of physics, physical Phenomenon, phenomena associated with the presence and motion of matter that has a property of electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnet ...

and
magnetic field A magnetic field is a vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. For instance, a vector field in the plane can be visualised as a collection of arrows with ...

s; in 1849
Michael Faraday Michael Faraday (; 22 September 1791 – 25 August 1867) was an English scientist A scientist is a person who conducts Scientific method, scientific research to advance knowledge in an Branches of science, area of interest. In clas ...

became the first to coin the term "field". The independent nature of the field became more apparent with
James Clerk Maxwell James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish scientist A scientist is a person who conducts Scientific method, scientific research to advance knowledge in an Branches of science, area of interest. In classica ...

's discovery that propagated at a finite speed. Consequently, the forces on charges and currents no longer just depended on the positions and velocities of other charges and currents at the same time, but also on their positions and velocities in the past. Maxwell, at first, did not adopt the modern concept of a field as a fundamental quantity that could independently exist. Instead, he supposed that the
electromagnetic field An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the in ...
expressed the deformation of some underlying medium—the
luminiferous aether upright=1.25, The luminiferous aether: it was hypothesised that the Earth moves through a "medium" of aether that carries light Luminiferous aether or ether ("luminiferous", meaning "light-bearing") was the postulated medium for the propagation of ...
—much like the tension in a rubber membrane. If that were the case, the observed velocity of the electromagnetic waves should depend upon the velocity of the observer with respect to the aether. Despite much effort, no experimental evidence of such an effect was ever found; the situation was resolved by the introduction of the special theory of relativity by
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest physicists of all time. Einstein is known for developing the theory of relativity The theory ...

in 1905. This theory changed the way the viewpoints of moving observers were related to each other. They became related to each other in such a way that velocity of electromagnetic waves in Maxwell's theory would be the same for all observers. By doing away with the need for a background medium, this development opened the way for physicists to start thinking about fields as truly independent entities. In the late 1920s, the new rules of
quantum mechanics Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quan ...
were first applied to the electromagnetic field. In 1927,
Paul Dirac Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century. Dirac made fundamental contributions to the early developme ...

used
quantum field In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity and quantum mechanics Quantum mechanics is a fundamental Scientific theory, theory in physics that prov ...
s to successfully explain how the decay of an
atom An atom is the smallest unit of ordinary matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atom ...

to a lower
quantum state In quantum physics Quantum mechanics is a fundamental theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is of ...
led to the
spontaneous emission Spontaneous emission is the process in which a quantum mechanical system (such as a molecule File:Pentacene on Ni(111) STM.jpg, A scanning tunneling microscopy image of pentacene molecules, which consist of linear chains of five carbon rings. ...
of a
photon The photon ( el, φῶς, phōs, light) is a type of elementary particle In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. Particles currently thought to be eleme ...

, the quantum of the electromagnetic field. This was soon followed by the realization (following the work of
Pascual Jordan Ernst Pascual Jordan (; 18 October 1902 – 31 July 1980) was a German theoretical and mathematical physicist A physicist is a scientist A scientist is a person who conducts Scientific method, scientific research to advance knowledge in an ...
,
Eugene Wigner Eugene Paul "E. P." Wigner ( hu, Wigner Jenő Pál, ; November 17, 1902 – January 1, 1995) was a Hungarian-American theoretical physicist and also contributed to mathematical physics. He received the Nobel Prize in Physics in 1963 "for his cont ...
,
Werner Heisenberg Werner Karl Heisenberg (; ; 5 December 1901 – 1 February 1976) was a German theoretical physicist and one of the key pioneers of quantum mechanics Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a de ...
, and
Wolfgang Pauli Wolfgang Ernst Pauli (; ; 25 April 1900 – 15 December 1958) was an Austrian theoretical physicist and one of the pioneers of quantum physics Quantum mechanics is a fundamental theory in physics that provides a description of the physica ...

) that all particles, including
electron The electron is a subatomic particle In physical sciences, subatomic particles are smaller than atom An atom is the smallest unit of ordinary matter In classical physics and general chemistry, matter is any substance that has ma ...

s and
proton A proton is a subatomic particle, symbol or , with a positive electric charge of +1''e'' elementary charge and a mass slightly less than that of a neutron. Protons and neutrons, each with masses of approximately one atomic mass unit, are collecti ...

s, could be understood as the quanta of some quantum field, elevating fields to the status of the most fundamental objects in nature. That said, John Wheeler and
Richard Feynman Richard Phillips Feynman (; May 11, 1918 – February 15, 1988) was an American theoretical physicist, known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, the physics of the superflui ...

seriously considered Newton's pre-field concept of
action at a distance In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through S ...
(although they set it aside because of the ongoing utility of the field concept for research in
general relativity General relativity, also known as the general theory of relativity, is the differential geometry, geometric scientific theory, theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern ph ...
and
quantum electrodynamics In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity and quantum mecha ...
).

# Classical fields

There are several examples of classical fields. Classical field theories remain useful wherever quantum properties do not arise, and can be active areas of research.
Elasticity Elasticity often refers to: *Elasticity (physics), continuum mechanics of bodies that deform reversibly under stress Elasticity may also refer to: Information technology * Elasticity (data store), the flexibility of the data model and the clu ...
of materials,
fluid dynamics In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. " ...
and
Maxwell's equations Maxwell's equations are a set of coupled partial differential equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), s ...
are cases in point. Some of the simplest physical fields are vector force fields. Historically, the first time that fields were taken seriously was with
lines of force A line of force in Faraday Michael Faraday (; 22 September 1791 – 25 August 1867) was an English scientist A scientist is a person who conducts Scientific method, scientific research to advance knowledge in an Branches of science, ...
when describing the
electric field An electric field (sometimes E-field) is the physical field that surrounds electrically-charged particle In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' ' ...

. The
gravitational field In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "P ...

was then similarly described.

## Newtonian gravitation

A classical field theory describing gravity is
Newtonian gravitationNewtonian refers to the work of Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March Old Style and New Style dates, 1726/27) was an English mathematician, physicist, astronomer, theologian, and author (described in his time as a ...

, which describes the gravitational force as a mutual interaction between two
mass Mass is the physical quantity, quantity of ''matter'' in a physical body. It is also a measure (mathematics), measure of the body's ''inertia'', the resistance to acceleration (change of velocity) when a net force is applied. An object's mass ...
es. Any body with mass ''M'' is associated with a
gravitational field In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "P ...

g which describes its influence on other bodies with mass. The gravitational field of ''M'' at a point r in space corresponds to the ratio between force F that ''M'' exerts on a small or negligible
test massIn physical theories, a test particle, or test charge, is an idealized model of an object whose physical properties (usually mass Mass is both a property Property (''latin: Res Privata'') in the Abstract and concrete, abstract is what belo ...
''m'' located at r and the test mass itself: : $\mathbf\left(\mathbf\right) = \frac.$ 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 Newton's law of universal gravitation is usually stated as that every particle In the Outline of physical science, physical sciences, a particle (or corpuscule in older texts) is a small wikt:local, localized physical body, object to which can ...
, F(r) is given by :$\mathbf\left(\mathbf\right) = -\frac\hat,$ where $\hat$ is a
unit vector In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
lying along the line joining ''M'' and ''m'' and pointing from ''M'' to ''m''. Therefore, the gravitational field of M is :$\mathbf\left(\mathbf\right) = \frac = -\frac\hat.$ The experimental observation that inertial mass and gravitational mass are equal to an unprecedented level of accuracy leads to the identity that gravitational field strength is identical to the acceleration experienced by a particle. This is the starting point of the
equivalence principle In the theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational ...
general relativity General relativity, also known as the general theory of relativity, is the differential geometry, geometric scientific theory, theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern ph ...
. Because the gravitational force F is
conservative Conservatism is an aesthetic Aesthetics, or esthetics (), is a branch of philosophy that deals with the nature of beauty and taste (sociology), taste, as well as the philosophy of art (its own area of philosophy that comes out of aest ...
, the gravitational field g can be rewritten in terms of the
gradient In vector calculus, the gradient of a scalar-valued function, scalar-valued differentiable function of Function of several variables, several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the Vec ...

of a scalar function, the
gravitational potential In classical mechanics, the gravitational potential at a location is equal to the work Work may refer to: * Work (human activity), intentional activity people perform to support themselves, others, or the community ** Manual labour, physical wo ...

Φ(r): :$\mathbf\left(\mathbf\right) = -\nabla \Phi\left(\mathbf\right).$

## Electromagnetism

Michael Faraday Michael Faraday (; 22 September 1791 – 25 August 1867) was an English scientist A scientist is a person who conducts Scientific method, scientific research to advance knowledge in an Branches of science, area of interest. In clas ...

first realized the importance of a field as a physical quantity, during his investigations into
magnetism Magnetism is a class of physical attributes that are mediated by 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 i ...

. He realized that
electric Electricity is the set of physics, physical Phenomenon, phenomena associated with the presence and motion of matter that has a property of electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnet ...

and
magnetic Magnetism is a class of physical phenomena that are mediated by 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 ...

fields are not only fields of force which dictate the motion of particles, but also have an independent physical reality because they carry energy. These ideas eventually led to the creation, by
James Clerk Maxwell James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish scientist A scientist is a person who conducts Scientific method, scientific research to advance knowledge in an Branches of science, area of interest. In classica ...

, of the first unified field theory in physics with the introduction of equations for the
electromagnetic field An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the in ...
. The modern version of these equations is called
Maxwell's equations Maxwell's equations are a set of coupled partial differential equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), s ...
.

### Electrostatics

A charged test particle with charge ''q'' experiences a force F based solely on its charge. We can similarly describe the
electric field An electric field (sometimes E-field) is the physical field that surrounds electrically-charged particle In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' ' ...

E so that . Using this and
Coulomb's law between two point charge A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an idealization of particle In the Outline of physical science, physical sciences, a particle (or corpuscule in older te ...
tells us that the electric field due to a single charged particle is : $\mathbf = \frac\frac\hat.$ The electric field is
conservative Conservatism is an aesthetic Aesthetics, or esthetics (), is a branch of philosophy that deals with the nature of beauty and taste (sociology), taste, as well as the philosophy of art (its own area of philosophy that comes out of aest ...
, and hence can be described by a scalar potential, ''V''(r): : $\mathbf\left(\mathbf\right) = -\nabla V\left(\mathbf\right).$

### Magnetostatics

A steady current ''I'' flowing along a path ''ℓ'' will create a field B, that exerts a force on nearby moving 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 : $\mathbf\left(\mathbf\right) = q\mathbf \times \mathbf\left(\mathbf\right),$ where B(r) is the
magnetic field A magnetic field is a vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. For instance, a vector field in the plane can be visualised as a collection of arrows with ...

, which is determined from ''I'' by the
Biot–Savart law In physics, specifically electromagnetism, the Biot–Savart law ( or ) is an equation describing the magnetic field generated by a constant electric current. It relates the magnetic field to the magnitude, direction, length, and proximity of the ...
: :$\mathbf\left(\mathbf\right) = \frac \int \frac.$ 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 In vector calculus, a vector potential is a vector field whose Curl (mathematics), curl is a given vector field. This is analogous to a ''scalar potential'', which is a scalar field whose gradient is a given vector field. Formally, given a vector ...
, A(r): : $\mathbf\left(\mathbf\right) = \boldsymbol \times \mathbf\left(\mathbf\right)$

### 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 are a set of coupled partial differential equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), s ...
, a set of differential equations which directly relate E and B to ρ and 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 potential In electrodynamics, the retarded potentials are the electromagnetic potentials for the electromagnetic field An electromagnetic field (also EM field) is a classical (i.e. non-quantum) field produced by accelerating electric charge Electric ...
s'' allow one to calculate ''V'' and A from ρ and J, and from there the electric and magnetic fields are determined via the relations : $\mathbf = -\boldsymbol V - \frac$ : $\mathbf = \boldsymbol \times \mathbf.$ At the end of the 19th century, the
electromagnetic field An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the in ...
was understood as a collection of two vector fields in space. Nowadays, one recognizes this as a single antisymmetric 2nd-rank tensor field in spacetime.

## Gravitation in general relativity

File:Relativistic gravity field (physics).svg, 350px, left, In
general relativity General relativity, also known as the general theory of relativity, is the differential geometry, geometric scientific theory, theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern ph ...
, mass-energy warps space time (Einstein tensor G), and rotating asymmetric mass-energy distributions with angular momentum J generate Gravitoelectromagnetism, GEM fields H Einstein's theory of gravity, called
general relativity General relativity, also known as the general theory of relativity, is the differential geometry, geometric scientific theory, theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern ph ...
, is another example of a field theory. Here the principal field is the
metric tensor In the mathematics, mathematical field of differential geometry, one definition of a metric tensor is a type of function which takes as input a pair of tangent vectors and at a point of a surface (or higher dimensional differentiable manifold) an ...
, a symmetric 2nd-rank tensor field in
spacetime In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "P ...
. This replaces
Newton's law of universal gravitation Newton's law of universal gravitation is usually stated as that every particle In the Outline of physical science, physical sciences, a particle (or corpuscule in older texts) is a small wikt:local, localized physical body, object to which can ...
.

## Waves as fields

Wave In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. " ...

s can be constructed as physical fields, due to their finite propagation speed and causal nature when a simplified
physical model , Beaconsfield, UK City Centre Image:Models of battle at australian war memorial museum.jpg, Model of a war scene — Australian War Memorial, Canberra A physical model (most commonly referred to simply as a model but in this context distinguished ...
of an Physical system#The concept of closed systems in physics, isolated closed system is set . They are also subject to the inverse-square law. For electromagnetic waves, there are optical fields, and terms such as Near and far field, near- and far-field limits for diffraction. In practice though, the field theories of optics are superseded by the electromagnetic field theory of Maxwell.

# Quantum fields

It is now believed that
quantum mechanics Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quan ...
should underlie all physical phenomena, so that a classical field theory should, at least in principle, permit a recasting in quantum mechanical terms; success yields the corresponding quantum field theory. For example, Quantization (physics), quantizing classical electrodynamics gives
quantum electrodynamics In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity and quantum mecha ...
. Quantum electrodynamics is arguably the most successful scientific theory; experimental data confirm its predictions to a higher Accuracy and precision, precision (to more significant digits) than any other theory. The two other fundamental quantum field theories are quantum chromodynamics and the electroweak theory. In quantum chromodynamics, the color field lines are coupled at short distances by gluons, which are polarized by the field and line up with it. This effect increases within a short distance (around 1 femtometre, fm from the vicinity of the quarks) making the color force increase within a short distance, Color confinement, confining the quarks within hadrons. As the field lines are pulled together tightly by gluons, they do not "bow" outwards as much as an electric field between electric charges. These three quantum field theories can all be derived as special cases of the so-called standard model of particle physics. General relativity, the Einsteinian field theory of gravity, has yet to be successfully quantized. However an extension, thermal field theory, deals with quantum field theory at ''finite temperatures'', something seldom considered in quantum field theory. In BRST formalism, BRST theory one deals with odd fields, e.g. Faddeev–Popov ghosts. There are different descriptions of odd classical fields both on graded manifolds and supermanifolds. As above with classical fields, it is possible to approach their quantum counterparts from a purely mathematical view using similar techniques as before. The equations governing the quantum fields are in fact PDEs (specifically, relativistic wave equations (RWEs)). Thus one can speak of Yang–Mills field, Yang–Mills, Dirac field, Dirac, Klein–Gordon field, Klein–Gordon and Schrödinger fields as being solutions to their respective equations. A possible problem is that these RWEs can deal with complicated mathematical objects with exotic algebraic properties (e.g. spinors are not tensors, so may need calculus for
spinor fieldIn differential geometry Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The Differential g ...
s), but these in theory can still be subjected to analytical methods given appropriate Generalization (mathematics), mathematical generalization.

# Field theory

Field theory usually refers to a construction of the dynamics of a field, i.e. a specification of how a field changes with time or with respect to other independent physical variables on which the field depends. Usually this is done by writing a Lagrangian (field theory), Lagrangian or a Hamiltonian mechanics, Hamiltonian of the field, and treating it as a classical mechanics, classical or quantum mechanics, quantum mechanical system with an infinite number of degrees of freedom (physics and chemistry), degrees of freedom. The resulting field theories are referred to as classical or quantum field theories. The dynamics of a classical field are usually specified by the Lagrangian (field theory), Lagrangian density in terms of the field components; the dynamics can be obtained by using the Action (physics), action principle. It is possible to construct simple fields without any prior knowledge of physics using only mathematics from multivariable calculus, several variable calculus, potential theory and partial differential equations (PDEs). For example, scalar PDEs might consider quantities such as amplitude, density and pressure fields for the wave equation and
fluid dynamics In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. " ...
; temperature/concentration fields for the heat equation, heat/diffusion equations. Outside of physics proper (e.g., radiometry and computer graphics), there are even light fields. All these previous examples are scalar fields. Similarly for vectors, there are vector PDEs for displacement, velocity and vorticity fields in (applied mathematical) fluid dynamics, but vector calculus may now be needed in addition, being calculus for vector fields (as are these three quantities, and those for vector PDEs in general). More generally problems in continuum mechanics may involve for example, directional elasticity tensor, elasticity (from which comes the term ''tensor'', derived from the Latin word for stretch), complex fluid flows or anisotropic diffusion, which are framed as matrix-tensor PDEs, and then require matrices or tensor fields, hence matrix calculus, matrix or tensor calculus. The scalars (and hence the vectors, matrices and tensors) can be real or complex as both are field (algebra), fields in the abstract-algebraic/ring theory, ring-theoretic sense. In a general setting, classical fields are described by sections of fiber bundles and their dynamics is formulated in the terms of jet bundle, jet manifolds (covariant classical field theory).Giachetta, G., Mangiarotti, L., Gennadi Sardanashvily, Sardanashvily, G. (2009) ''Advanced Classical Field Theory''. Singapore: World Scientific, () In modern physics, the most often studied fields are those that model the four fundamental forces which one day may lead to the Unified Field Theory.

## Symmetries of fields

A convenient way of classifying a field (classical or quantum) is by the Symmetry in physics, symmetries it possesses. Physical symmetries are usually of two types:

### Spacetime symmetries

Fields are often classified by their behaviour under transformations of
spacetime In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "P ...
. The terms used in this classification are: *
scalar field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s (such as
temperature Temperature is a physical quantity that expresses hot and cold. It is the manifestation of thermal energy, present in all matter, which is the source of the occurrence of heat, a flow of energy, when a body is in contact with another that is ...

) whose values are given by a single variable at each point of space. This value does not change under transformations of space. *
vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. For instance, a vector field in the plane can be visualised as a collection of arrows with a given magnitude and direction, each att ...

s (such as the magnitude and direction of the force (physics), force at each point in a
magnetic field A magnetic field is a vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. For instance, a vector field in the plane can be visualised as a collection of arrows with ...

) which are specified by attaching a vector to each point of space. The components of this vector transform between themselves covariance and contravariance of vectors, contravariantly under rotations in space. Similarly, a dual (or co-) vector field attaches a dual vector to each point of space, and the components of each dual vector transform covariantly. *
tensor field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s, (such as the Stress (physics), stress tensor of a crystal) specified by a tensor at each point of space. Under rotations in space, the components of the tensor transform in a more general way which depends on the number of covariant indices and contravariant indices. *
spinor fieldIn differential geometry Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The Differential g ...
s (such as the Dirac spinor) arise in quantum field theory to describe particles with spin (physics), spin which transform like vectors except for the one of their components; in other words, when one rotates a vector field 360 degrees around a specific axis, the vector field turns to itself; however, spinors would turn to their negatives in the same case.

### Internal symmetries

Fields may have internal symmetries in addition to spacetime symmetries. In many situations, one needs fields which are a list of spacetime scalars: (φ1, φ2, ... φ''N''). For example, in weather prediction these may be temperature, pressure, humidity, etc. In particle physics, the color charge, color symmetry of the interaction of quarks is an example of an internal symmetry, that of the strong interaction. Other examples are isospin, weak isospin, strangeness and any other flavour (particle physics), flavour symmetry. If there is a symmetry of the problem, not involving spacetime, under which these components transform into each other, then this set of symmetries is called an ''internal symmetry''. One may also make a classification of the charges of the fields under internal symmetries.

## Statistical field theory

Statistical field theory attempts to extend the field-theoretic
paradigm In science Science (from the Latin word ''scientia'', meaning "knowledge") is a systematic enterprise that Scientific method, builds and Taxonomy (general), organizes knowledge in the form of Testability, testable explanations and prediction ...
toward many-body systems and statistical mechanics. As above, it can be approached by the usual infinite number of degrees of freedom argument. Much like statistical mechanics has some overlap between quantum and classical mechanics, statistical field theory has links to both quantum and classical field theories, especially the former with which it shares many methods. One important example is mean field theory.

## Continuous random fields

Classical fields as above, such as the
electromagnetic field An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the in ...
, are usually infinitely differentiable functions, but they are in any case almost always twice differentiable. In contrast, generalized functions are not continuous. When dealing carefully with classical fields at finite temperature, the mathematical methods of continuous random fields are used, because Thermal fluctuations, thermally fluctuating classical fields are nowhere differentiable. Random fields are indexed sets of random variables; a continuous random field is a random field that has a set of functions as its index set. In particular, it is often mathematically convenient to take a continuous random field to have a Schwartz space of functions as its index set, in which case the continuous random field is a Distribution (mathematics), tempered distribution. We can think about a continuous random field, in a (very) rough way, as an ordinary function that is $\pm\infty$ almost everywhere, but such that when we take a weighted average of all the infinity, infinities over any finite region, we get a finite result. The infinities are not well-defined; but the finite values can be associated with the functions used as the weight functions to get the finite values, and that can be well-defined. We can define a continuous random field well enough as a linear map from a space of functions into the real numbers.

* Conformal field theory * Covariant Hamiltonian field theory * Field strength * History of the philosophy of field theory * Lagrangian and Eulerian specification of the flow field, Lagrangian and Eulerian specification of a field * Scalar field theory * Velocity field