Field (physics)
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In
science Science is a systematic discipline that builds and organises knowledge in the form of testable hypotheses and predictions about the universe. Modern science is typically divided into twoor threemajor branches: the natural sciences, which stu ...
, a field is a
physical quantity A physical quantity (or simply quantity) is a 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 ''nu ...
, represented by a scalar,
vector Vector most often refers to: * Euclidean vector, a quantity with a magnitude and a direction * Disease vector, an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematics a ...
, or
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ...
, that has a value for each point in space and time. An example of a
scalar field In mathematics and physics, a scalar field is a function associating a single number to each point in a region of space – possibly physical space. The scalar may either be a pure mathematical number ( dimensionless) or a scalar physical ...
is a weather map, with the surface
temperature Temperature is a physical quantity that quantitatively expresses the attribute of hotness or coldness. Temperature is measurement, measured with a thermometer. It reflects the average kinetic energy of the vibrating and colliding atoms making ...
described by assigning a
number A number is a mathematical object used to count, measure, and label. The most basic examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can ...
to each point on the map. A surface wind map, assigning an arrow to each point on a map that describes the wind speed and direction 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 space, most commonly Euclidean space \mathbb^n. A vector field on a plane can be visualized as a collection of arrows with given magnitudes and dire ...
, 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 called E-field) is a field (physics), physical field that surrounds electrically charged particles such as electrons. In classical electromagnetism, the electric field of a single charge (or group of charges) descri ...
is another rank-1 tensor field, while electrodynamics 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 field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
, 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) is a physical field, varying in space and time, that represents the electric and magnetic influences generated by and acting upon electric charges. The field at any point in space and time can be regarde ...
s to be a physical entity, making the field concept a supporting
paradigm In science and philosophy, a paradigm ( ) is a distinct set of concepts or thought patterns, including theories, research methods, postulates, and standards for what constitute legitimate contributions to a field. The word ''paradigm'' is Ancient ...
of the edifice of modern physics.
Richard Feynman Richard Phillips Feynman (; May 11, 1918 – February 15, 1988) was an American theoretical physicist. He is best known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, the physics of t ...
said, "The fact that the electromagnetic field can possess momentum and energy makes it very real, and ..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 or the
electrostatic field An electric field (sometimes called E-field) is a physical field that surrounds electrically charged particles such as electrons. In classical electromagnetism, the electric field of a single charge (or group of charges) describes their capac ...
in classical electromagnetism, is inversely proportional to the square of the distance from the source (i.e. they follow Gauss's law). A field can be classified as a scalar field, a vector field, a spinor field or a tensor field according to whether the represented physical quantity is a scalar, a
vector Vector most often refers to: * Euclidean vector, a quantity with a magnitude and a direction * Disease vector, an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematics a ...
, a
spinor In geometry and physics, spinors (pronounced "spinner" IPA ) are elements of a complex numbers, complex vector space that can be associated with Euclidean space. A spinor transforms linearly when the Euclidean space is subjected to a slight (infi ...
, or a
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ...
, 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 Newtonian
gravitational field In physics, a gravitational field or gravitational acceleration field is a vector field used to explain the influences that a body extends into the space around itself. A gravitational field is used to explain gravitational phenomena, such as ...
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, for instance a
boson In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0, 1, 2, ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have half odd-intege ...
.


History

To
Isaac Newton Sir Isaac Newton () was an English polymath active as a mathematician, physicist, astronomer, alchemist, theologian, and author. Newton was a key figure in the Scientific Revolution and the Age of Enlightenment, Enlightenment that followed ...
, his law of universal gravitation simply expressed the gravitational
force In physics, a force is an influence that can cause an Physical object, object to change its velocity unless counterbalanced by other forces. In mechanics, force makes ideas like 'pushing' or 'pulling' mathematically precise. Because the Magnitu ...
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 "Sola ...
, 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, a gravitational field or gravitational acceleration field is a vector field used to explain the influences that a body extends into the space around itself. A gravitational field is used to explain gravitational phenomena, such as ...
, 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. His idea in ''
Opticks ''Opticks: or, A Treatise of the Reflexions, Refractions, Inflexions and Colours of Light'' is a collection of three books by Isaac Newton that was published in English language, English in 1704 (a scholarly Latin translation appeared in 1706). ...
'' that optical reflection and
refraction In physics, refraction is the redirection of a wave as it passes from one transmission medium, 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 commo ...
arise from interactions across the entire surface is arguably the beginning of the field theory of electric force. The development of the independent concept of a field truly began in the nineteenth century with the development of the theory of
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interacti ...
. In the early stages, André-Marie Ampère 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 electrostatic force of att ...
could manage with Newton-style laws that expressed the forces between pairs of
electric charge Electric charge (symbol ''q'', sometimes ''Q'') is a physical property of matter that causes it to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative''. Like charges repel each other and ...
s or
electric current An electric current is a flow of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is defined as the net rate of flow of electric charge through a surface. The moving particles are called charge c ...
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 physical phenomena associated with the presence and motion of matter possessing an electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described by Maxwel ...
and
magnetic field A magnetic field (sometimes called B-field) is a physical 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 ...
s; in 1845
Michael Faraday Michael Faraday (; 22 September 1791 – 25 August 1867) was an English chemist and physicist who contributed to the study of electrochemistry and electromagnetism. His main discoveries include the principles underlying electromagnetic inducti ...
became the first to coin the term "magnetic field". And Lord Kelvin provided a formal definition for a field in 1851. 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 physicist and mathematician who was responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism an ...
's discovery that waves in these fields, called
electromagnetic waves In physics, electromagnetic radiation (EMR) is a self-propagating wave of the electromagnetic field that carries momentum and radiant energy through space. It encompasses a broad spectrum, classified by frequency or its inverse, wavelength, ran ...
, 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) is a physical field, varying in space and time, that represents the electric and magnetic influences generated by and acting upon electric charges. The field at any point in space and time can be regarde ...
expressed the deformation of some underlying medium—the
luminiferous aether Luminiferous aether or ether (''luminiferous'' meaning 'light-bearing') was the postulated Transmission medium, medium for the propagation of light. It was invoked to explain the ability of the apparently wave-based light to propagate through empt ...
—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 187918 April 1955) was a German-born theoretical physicist who is best known for developing the theory of relativity. Einstein also made important contributions to quantum mechanics. His mass–energy equivalence f ...
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 the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
were first applied to the electromagnetic field. In 1927,
Paul Dirac Paul Adrien Maurice Dirac ( ; 8 August 1902 – 20 October 1984) was an English mathematician and Theoretical physics, theoretical physicist who is considered to be one of the founders of quantum mechanics. Dirac laid the foundations for bot ...
used quantum fields to successfully explain how the decay of an
atom Atoms are the basic particles of the chemical elements. An atom consists of a atomic nucleus, nucleus of protons and generally neutrons, surrounded by an electromagnetically bound swarm of electrons. The chemical elements are distinguished fr ...
to a lower
quantum state In quantum physics, a quantum state is a mathematical entity that embodies the knowledge of a quantum system. Quantum mechanics specifies the construction, evolution, and measurement of a quantum state. The result is a prediction for the system ...
led to the
spontaneous emission Spontaneous emission is the process in which a Quantum mechanics, quantum mechanical system (such as a molecule, an atom or a subatomic particle) transits from an excited state, excited energy state to a lower energy state (e.g., its ground state ...
of a
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless particles that can ...
, 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 who made significant contributions to quantum mechanics and quantum field theory. He contributed much to the mathematical form of matri ...
,
Eugene Wigner Eugene Paul Wigner (, ; November 17, 1902 – January 1, 1995) was a Hungarian-American theoretical physicist who also contributed to mathematical physics. He received the Nobel Prize in Physics in 1963 "for his contributions to the theory of th ...
,
Werner Heisenberg Werner Karl Heisenberg (; ; 5 December 1901 – 1 February 1976) was a German theoretical physicist, one of the main pioneers of the theory of quantum mechanics and a principal scientist in the German nuclear program during World War II. He pub ...
, and
Wolfgang Pauli Wolfgang Ernst Pauli ( ; ; 25 April 1900 – 15 December 1958) was an Austrian theoretical physicist and a pioneer of quantum mechanics. In 1945, after having been nominated by Albert Einstein, Pauli received the Nobel Prize in Physics "for the ...
) that all particles, including
electron The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary charge, elementary electric charge. It is a fundamental particle that comprises the ordinary matter that makes up the universe, along with up qua ...
s and
proton A proton is a stable subatomic particle, symbol , Hydron (chemistry), H+, or 1H+ with a positive electric charge of +1 ''e'' (elementary charge). Its mass is slightly less than the mass of a neutron and approximately times the mass of an e ...
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. He is best known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, the physics of t ...
seriously considered Newton's pre-field concept of
action at a distance Action at a distance is the concept in physics that an object's motion (physics), motion can be affected by another object without the two being in Contact mechanics, physical contact; that is, it is the concept of the non-local interaction of ob ...
(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, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
and
quantum electrodynamics In particle physics, quantum electrodynamics (QED) is the Theory of relativity, relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quant ...
).


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 of materials,
fluid dynamics In physics, physical chemistry and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids – liquids and gases. It has several subdisciplines, including (the study of air and other gases in motion ...
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, Electrical network, electr ...
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 Faraday's lines of force when describing the
electric field An electric field (sometimes called E-field) is a field (physics), physical field that surrounds electrically charged particles such as electrons. In classical electromagnetism, the electric field of a single charge (or group of charges) descri ...
. The
gravitational field In physics, a gravitational field or gravitational acceleration field is a vector field used to explain the influences that a body extends into the space around itself. A gravitational field is used to explain gravitational phenomena, such as ...
was then similarly described.


Newtonian gravitation

A classical field theory describing gravity is Newtonian gravitation, which describes the gravitational force as a mutual interaction between two
mass Mass is an Intrinsic and extrinsic properties, intrinsic property of a physical body, body. It was traditionally believed to be related to the physical quantity, quantity of matter in a body, until the discovery of the atom and particle physi ...
es. Any body with mass ''M'' is associated with a
gravitational field In physics, a gravitational field or gravitational acceleration field is a vector field used to explain the influences that a body extends into the space around itself. A gravitational field is used to explain gravitational phenomena, such as ...
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 mass ''m'' located at r and the test mass itself: : \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 Newton's law of universal gravitation describes gravity as a force by stating that every particle attracts every other particle in the universe with a force that is Proportionality (mathematics)#Direct proportionality, proportional to the product ...
, 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 (mathematics and physics), vector (often a vector (geometry), spatial vector) of Norm (mathematics), length 1. A unit vector is often denoted by a lowercase letter with a circumfle ...
lying along the line joining ''M'' and ''m'' and pointing from ''M'' to ''m''. Therefore, the gravitational field of M is :\mathbf(\mathbf) = \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 The equivalence principle is the hypothesis that the observed equivalence of gravitational and inertial mass is a consequence of nature. The weak form, known for centuries, relates to masses of any composition in free fall taking the same t ...
, which leads to
general relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
. Because the gravitational force F is
conservative Conservatism is a cultural, social, and political philosophy and ideology that seeks to promote and preserve traditional institutions, customs, and values. The central tenets of conservatism may vary in relation to the culture and civiliza ...
, the gravitational field g can be rewritten in terms of the
gradient In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p gives the direction and the rate of fastest increase. The g ...
of a scalar function, the
gravitational potential In classical mechanics, the gravitational potential is a scalar potential associating with each point in space the work (energy transferred) per unit mass that would be needed to move an object to that point from a fixed reference point in the ...
Φ(r): :\mathbf(\mathbf) = -\nabla \Phi(\mathbf).


Electromagnetism

Michael Faraday Michael Faraday (; 22 September 1791 – 25 August 1867) was an English chemist and physicist who contributed to the study of electrochemistry and electromagnetism. His main discoveries include the principles underlying electromagnetic inducti ...
first realized the importance of a field as a physical quantity, during his investigations into
magnetism Magnetism is the class of physical attributes that occur through a magnetic field, which allows objects to attract or repel each other. Because both electric currents and magnetic moments of elementary particles give rise to a magnetic field, ...
. He realized that
electric Electricity is the set of physical phenomena associated with the presence and motion of matter possessing an electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described by Maxwel ...
and magnetic 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 physicist and mathematician who was responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism an ...
, of the first unified field theory in physics with the introduction of equations for the
electromagnetic field An electromagnetic field (also EM field) is a physical field, varying in space and time, that represents the electric and magnetic influences generated by and acting upon electric charges. The field at any point in space and time can be regarde ...
. The modern versions of these equations are called
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, Electrical network, electr ...
.


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 called E-field) is a field (physics), physical field that surrounds electrically charged particles such as electrons. In classical electromagnetism, the electric field of a single charge (or group of charges) descri ...
E so that . Using this and
Coulomb's law Coulomb's inverse-square law, or simply Coulomb's law, is an experimental scientific law, law of physics that calculates the amount of force (physics), force between two electric charge, electrically charged particles at rest. This electric for ...
tells us that the electric field due to a single charged particle is : \mathbf = \frac\frac\hat. The electric field is
conservative Conservatism is a cultural, social, and political philosophy and ideology that seeks to promote and preserve traditional institutions, customs, and values. The central tenets of conservatism may vary in relation to the culture and civiliza ...
, and hence can be described by a scalar potential, ''V''(r): : \mathbf(\mathbf) = -\nabla V(\mathbf).


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(\mathbf) = q\mathbf \times \mathbf(\mathbf), where B(r) is the
magnetic field A magnetic field (sometimes called B-field) is a physical 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 ...
, 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(\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) = \boldsymbol \times \mathbf(\mathbf)


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, Electrical network, electr ...
, 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 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 = -\boldsymbol V - \frac : \mathbf = \boldsymbol \times \mathbf. At the end of the 19th century, the
electromagnetic field An electromagnetic field (also EM field) is a physical field, varying in space and time, that represents the electric and magnetic influences generated by and acting upon electric charges. The field at any point in space and time can be regarde ...
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

Einstein's theory of gravity, called
general relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
, is another example of a field theory. Here the principal field is the
metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
, a symmetric 2nd-rank tensor field in
spacetime In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualiz ...
. This replaces
Newton's law of universal gravitation Newton's law of universal gravitation describes gravity as a force by stating that every particle attracts every other particle in the universe with a force that is Proportionality (mathematics)#Direct proportionality, proportional to the product ...
.


Waves as fields

Wave In physics, mathematics, engineering, and related fields, a wave is a propagating dynamic disturbance (change from List of types of equilibrium, equilibrium) of one or more quantities. ''Periodic waves'' oscillate repeatedly about an equilibrium ...
s can be constructed as physical fields, due to their finite propagation speed and causal nature when a simplified physical model of an 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 limits for diffraction. In practice though, the field theories of optics are superseded by the electromagnetic field theory of Maxwell Gravity waves are waves in the surface of water, defined by a height field.


Fluid dynamics

Fluid dynamics has fields of
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and eve ...
,
density Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be u ...
, 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 The Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances. They were named after French engineer and physicist Claude-Louis Navier and the Irish physicist and mathematician Georg ...
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
flow velocity In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the f ...
u is the vector field to solve for.


Elasticity

Linear elasticity Linear elasticity is a mathematical model of how solid objects deform and become internally stressed by prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mechani ...
is defined in terms of
constitutive equations In physics and engineering, a constitutive equation or constitutive relation is a relation between two or more physical quantities (especially kinetic quantities as related to kinematic quantities) that is specific to a material or substance o ...
between tensor fields, : \sigma_ = L_ \varepsilon_ where \sigma_ are the components of the 3x3 Cauchy stress tensor, \varepsilon_ the components of the 3x3 infinitesimal strain and L_ is the
elasticity tensor The elasticity tensor is a fourth-rank tensor describing the stress-strain relation in a linear elastic material. Other names are elastic modulus tensor and stiffness tensor. Common symbols include \mathbf and \mathbf. The defining equation can ...
, a fourth-rank tensor with 81 components (usually 21 independent components).


Thermodynamics and transport equations

Assuming that the temperature ''T'' is an intensive quantity, i.e., a single-valued,
differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
function of three-dimensional space (a
scalar field In mathematics and physics, a scalar field is a function associating a single number to each point in a region of space – possibly physical space. The scalar may either be a pure mathematical number ( dimensionless) or a scalar physical ...
), i.e., that T=T(\mathbf), then the
temperature gradient A temperature gradient is a physical quantity that describes in which direction and at what rate the temperature changes the most rapidly around a particular location. The temperature spatial gradient is a vector quantity with Dimensional analysis, ...
is a vector field defined as \nabla T. In
thermal conduction Thermal conduction is the diffusion of thermal energy (heat) within one material or between materials in contact. The higher temperature object has molecules with more kinetic energy; collisions between molecules distributes this kinetic energy ...
, the temperature field appears in Fourier's law, : \mathbf = -k \nabla T where q is the
heat flux In physics and engineering, heat flux or thermal flux, sometimes also referred to as heat flux density, heat-flow density or heat-flow rate intensity, is a flow of energy per unit area per unit time (physics), time. Its SI units are watts per sq ...
field and ''k'' the thermal conductivity. Temperature and
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and eve ...
gradients are also important for meteorology.


Quantum fields

It is now believed that
quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
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 In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
. For example, quantizing classical electrodynamics gives
quantum electrodynamics In particle physics, quantum electrodynamics (QED) is the Theory of relativity, relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quant ...
. Quantum electrodynamics is arguably the most successful scientific theory;
experiment An experiment is a procedure carried out to support or refute a hypothesis, or determine the efficacy or likelihood of something previously untried. Experiments provide insight into cause-and-effect by demonstrating what outcome occurs whe ...
al
data Data ( , ) are a collection of discrete or continuous values that convey information, describing the quantity, quality, fact, statistics, other basic units of meaning, or simply sequences of symbols that may be further interpreted for ...
confirm its predictions to a higher precision (to more
significant digit Significant figures, also referred to as significant digits, are specific Numerical digit, digits within a number that is written in positional notation that carry both reliability and necessity in conveying a particular quantity. When presen ...
s) than any other theory. The two other fundamental quantum field theories are
quantum chromodynamics In theoretical physics, quantum chromodynamics (QCD) is the study of the strong interaction between quarks mediated by gluons. Quarks are fundamental particles that make up composite hadrons such as the proton, neutron and pion. QCD is a type of ...
and the electroweak theory. In quantum chromodynamics, the color field lines are coupled at short distances by
gluon A gluon ( ) is a type of Massless particle, massless elementary particle that mediates the strong interaction between quarks, acting as the exchange particle for the interaction. Gluons are massless vector bosons, thereby having a Spin (physi ...
s, which are polarized by the field and line up with it. This effect increases within a short distance (around 1 fm from the vicinity of the quarks) making the color force increase within a short distance, confining the quarks within
hadron In particle physics, a hadron is a composite subatomic particle made of two or more quarks held together by the strong nuclear force. Pronounced , the name is derived . They are analogous to molecules, which are held together by the electri ...
s. 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 The Standard Model of particle physics is the Scientific theory, theory describing three of the four known fundamental forces (electromagnetism, electromagnetic, weak interaction, weak and strong interactions – excluding gravity) in the unive ...
of
particle physics Particle physics or high-energy physics is the study of Elementary particle, fundamental particles and fundamental interaction, forces that constitute matter and radiation. The field also studies combinations of elementary particles up to the s ...
.
General relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
, 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 theory one deals with odd fields, e.g.
Faddeev–Popov ghost In physics, Faddeev–Popov ghosts (also called Faddeev–Popov gauge ghosts or Faddeev–Popov ghost fields) are extraneous fields which are introduced into gauge quantum field theories to maintain the consistency of the path integral form ...
s. 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, Dirac, 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 In geometry and physics, spinors (pronounced "spinner" IPA ) are elements of a complex numbers, complex vector space that can be associated with Euclidean space. A spinor transforms linearly when the Euclidean space is subjected to a slight (infi ...
are not tensors, so may need calculus for spinor fields), but these in theory can still be subjected to analytical methods given appropriate 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 or a
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
of the field, and treating it as a classical or quantum mechanical system with an infinite number of
degrees of freedom In many scientific fields, the degrees of freedom of a system is the number of parameters of the system that may vary independently. For example, a point in the plane has two degrees of freedom for translation: its two coordinates; a non-infinite ...
. 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 density in terms of the field components; the dynamics can be obtained by using the action principle. It is possible to construct simple fields without any prior knowledge of physics using only mathematics from
multivariable calculus Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving multiple variables ('' mult ...
,
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 the two fundamental forces of nature known at the time, namely g ...
and
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to ho ...
s (PDEs). For example, scalar PDEs might consider quantities such as amplitude, density and pressure fields for the wave equation and
fluid dynamics In physics, physical chemistry and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids – liquids and gases. It has several subdisciplines, including (the study of air and other gases in motion ...
; temperature/concentration fields for the
heat In thermodynamics, heat is energy in transfer between a thermodynamic system and its surroundings by such mechanisms as thermal conduction, electromagnetic radiation, and friction, which are microscopic in nature, involving sub-atomic, ato ...
/ 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 Continuum mechanics is a branch of mechanics that deals with the deformation of and transmission of forces through materials modeled as a ''continuous medium'' (also called a ''continuum'') rather than as discrete particles. Continuum mec ...
may involve for example, directional elasticity (from which comes the term ''tensor'', derived from the
Latin Latin ( or ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken by the Latins (Italic tribe), Latins in Latium (now known as Lazio), the lower Tiber area aroun ...
word for stretch),
complex fluid Complex fluids are mixtures that have a coexistence between two Phase (matter) , phases: solid–liquid (Suspension (chemistry) , suspensions or solutions of macromolecules such as polymers), solid–gas (Granular material, granular), liquid–gas ...
flows or
anisotropic diffusion In image processing and computer vision, anisotropic diffusion, also called Perona–Malik diffusion, is a technique aiming at reducing image noise without removing significant parts of the image content, typically edges, lines or other details t ...
, which are framed as matrix-tensor PDEs, and then require matrices or tensor fields, hence
matrix Matrix (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the m ...
or tensor calculus. The scalars (and hence the vectors, matrices and tensors) can be real or complex as both are fields in the abstract-algebraic/ 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 manifolds ( covariant classical field theory).Giachetta, G., Mangiarotti, L., Sardanashvily, G. (2009) ''Advanced Classical Field Theory''. Singapore: World Scientific, () In
modern physics Modern physics is a branch of physics that developed in the early 20th century and onward or branches greatly influenced by early 20th century physics. Notable branches of modern physics include quantum mechanics, special relativity, and genera ...
, the most often studied fields are those that model the four
fundamental forces In physics, the fundamental interactions or fundamental forces are interactions in nature that appear not to be reducible to more basic interactions. There are four fundamental interactions known to exist: * gravity * electromagnetism * weak int ...
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 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, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualiz ...
. The terms used in this classification are: *
scalar field In mathematics and physics, a scalar field is a function associating a single number to each point in a region of space – possibly physical space. The scalar may either be a pure mathematical number ( dimensionless) or a scalar physical ...
s (such as
temperature Temperature is a physical quantity that quantitatively expresses the attribute of hotness or coldness. Temperature is measurement, measured with a thermometer. It reflects the average kinetic energy of the vibrating and colliding atoms making ...
) 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 space, most commonly Euclidean space \mathbb^n. A vector field on a plane can be visualized as a collection of arrows with given magnitudes and dire ...
s (such as the magnitude and direction of the
force In physics, a force is an influence that can cause an Physical object, object to change its velocity unless counterbalanced by other forces. In mechanics, force makes ideas like 'pushing' or 'pulling' mathematically precise. Because the Magnitu ...
at each point in a
magnetic field A magnetic field (sometimes called B-field) is a physical 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 ...
) which are specified by attaching a vector to each point of space. The components of this vector transform between themselves 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 fields, (such as the 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 fields (such as the Dirac spinor) arise in
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
to describe particles with spin which transform like vectors except for 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 Particle physics or high-energy physics is the study of Elementary particle, fundamental particles and fundamental interaction, forces that constitute matter and radiation. The field also studies combinations of elementary particles up to the s ...
, the
color Color (or colour in English in the Commonwealth of Nations, Commonwealth English; American and British English spelling differences#-our, -or, see spelling differences) is the visual perception based on the electromagnetic spectrum. Though co ...
symmetry of the interaction of
quark A quark () is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nucleus, atomic nuclei ...
s is an example of an internal symmetry, that of the
strong interaction In nuclear physics and particle physics, the strong interaction, also called the strong force or strong nuclear force, is one of the four known fundamental interaction, fundamental interactions. It confines Quark, quarks into proton, protons, n ...
. Other examples are
isospin In nuclear physics and particle physics, isospin (''I'') is a quantum number related to the up- and down quark content of the particle. Isospin is also known as isobaric spin or isotopic spin. Isospin symmetry is a subset of the flavour symmetr ...
, weak isospin, strangeness and any other 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 and philosophy, a paradigm ( ) is a distinct set of concepts or thought patterns, including theories, research methods, postulates, and standards for what constitute legitimate contributions to a field. The word ''paradigm'' is Ancient ...
toward many-body systems and
statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applicati ...
. 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) is a physical field, varying in space and time, that represents the electric and magnetic influences generated by and acting upon electric charges. The field at any point in space and time can be regarde ...
, 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 thermally fluctuating classical fields are nowhere differentiable. Random fields are indexed sets of
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
s; 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 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 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 In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that p ...
from a space of functions into the
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s.


See also

* Conformal field theory * Covariant Hamiltonian field theory *
Field strength In physics, field strength refers to a value in a vector-valued field (e.g., in volts per meter, V/m, for an electric field ''E''). For example, an electromagnetic field has both electric field strength and magnetic field strength. Field str ...
* Lagrangian and Eulerian specification of a field *
Scalar field theory In theoretical physics, scalar field theory can refer to a relativistically invariant classical or quantum theory of scalar fields. A scalar field is invariant under any Lorentz transformation. The only fundamental scalar quantum field that has ...
* Velocity field


Notes


References


Further reading

* * Landau, Lev D. and Lifshitz, Evgeny M. (1971). ''Classical Theory of Fields'' (3rd ed.). London: Pergamon. . Vol. 2 of the Course of Theoretical Physics. *


External links


Particle and Polymer Field Theories
{{DEFAULTSORT:Field (Physics) Mathematical physics Physical quantities