Inscience, afield is aphysical quantity, represented by ascalar,vector, ortensor, that has a value for eachpoint inspace and time.^[1][2][3] An example of ascalar field is a weather map, with the surfacetemperature described by assigning anumber to each point on the map. A surface wind map,^[4] assigning an arrow to each point on a map that describes the windspeed and direction at that point, is an example of avector field, 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, theelectric field is another rank-1 tensor field, whileelectrodynamics can be formulated in terms oftwo interacting vector fields at each point in spacetime, or as asingle-rank 2-tensor field.^[5][6][7]

In the modern framework of thequantum field theory, even without referring to a test particle, a field occupies space, contains energy, and its presence precludes a classical "true vacuum".^[8] This has led physicists to considerelectromagnetic fields to be a physical entity, making the field concept a supportingparadigm of the edifice of modern physics.Richard Feynman 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."^[9] 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 inNewton's theory of gravity or theelectrostatic field in classical electromagnetism, is inversely proportional to the square of the distance from the source (i.e. they followGauss's law).

A field can be classified as a scalar field, a vector field, aspinor field or atensor field according to whether the represented physical quantity is ascalar, avector, aspinor, or atensor, 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, theNewtoniangravitational field 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 aclassical field or aquantum field, depending on whether it is characterized by numbers orquantum operators respectively. In this theory an equivalent representation of field is afield particle, for instance aboson.^[10]

ToIsaac Newton, hislaw of universal gravitation simply expressed the gravitationalforce 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 theSolar System, 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, thegravitational field, 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.^[11] His idea inOpticks that opticalreflection andrefraction arise from interactions across the entire surface is arguably the beginning of the field theory of electric force.^[12]

The development of the independent concept of a field truly began in the nineteenth century with the development of the theory ofelectromagnetism. In the early stages,André-Marie Ampère andCharles-Augustin de Coulomb could manage with Newton-style laws that expressed the forces between pairs ofelectric charges orelectric currents. However, it became much more natural to take the field approach and express these laws in terms ofelectric andmagnetic fields; in 1845Michael Faraday became the first to coin the term "magnetic field".^[13] And Lord Kelvin provided a formal definition for a field in 1851.^[14]

The independent nature of the field became more apparent withJames Clerk Maxwell's discovery that waves in these fields, calledelectromagnetic waves, 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.^[11]

Maxwell, at first, did not adopt the modern concept of a field as a fundamental quantity that could independently exist. Instead, he supposed that theelectromagnetic field expressed the deformation of some underlying medium—theluminiferous aether—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 thespecial theory of relativity byAlbert Einstein 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.^[11]

In the late 1920s, the new rules ofquantum mechanics were first applied to the electromagnetic field. In 1927,Paul Dirac usedquantum fields to successfully explain how the decay of anatom to a lowerquantum state led to thespontaneous emission of aphoton, the quantum of the electromagnetic field. This was soon followed by the realization (following the work ofPascual Jordan,Eugene Wigner,Werner Heisenberg, andWolfgang Pauli) that all particles, includingelectrons andprotons, could be understood as the quanta of some quantum field, elevating fields to the status of the most fundamental objects in nature.^[11] That said,John Wheeler andRichard Feynman seriously considered Newton's pre-field concept ofaction at a distance (although they set it aside because of the ongoing utility of the field concept for research ingeneral relativity andquantum electrodynamics).

There are several examples ofclassical fields. Classical field theories remain useful wherever quantum properties do not arise, and can be active areas of research.Elasticity of materials,fluid dynamics andMaxwell's equations are cases in point.

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

A classical field theory describing gravity isNewtonian gravitation, which describes the gravitational force as a mutual interaction between twomasses.

Any body with massM is associated with agravitational fieldg which describes its influence on other bodies with mass. The gravitational field ofM at a pointr in space corresponds to the ratio between forceF thatM exerts on a small or negligibletest massm located atr and the test mass itself:^[15]

${\displaystyle \mathbf{g}(\mathbf{r})=\frac{\mathbf{F}(\mathbf{r})}{m}.}$ 

Stipulating thatm is much smaller thanM ensures that the presence ofm has a negligible influence on the behavior ofM.

According toNewton's law of universal gravitation,F(r) is given by^[15]

${\displaystyle \mathbf{F}(\mathbf{r})=-\frac{GMm}{r^2}\hat{\mathbf{r}},}$ 

where${\displaystyle \hat{\mathbf{r}}}$  is aunit vector lying along the line joiningM andm and pointing fromM tom. Therefore, the gravitational field ofM is^[15]

${\displaystyle \mathbf{g}(\mathbf{r})=\frac{\mathbf{F}(\mathbf{r})}{m}=-\frac{GM}{r^2}\hat{\mathbf{r}}.}$ 

The experimental observation that inertial mass and gravitational mass are equalto 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 theequivalence principle, which leads togeneral relativity.

Because the gravitational forceF isconservative, the gravitational fieldg can be rewritten in terms of thegradient of a scalar function, thegravitational potential Φ(r):

${\displaystyle \mathbf{g}(\mathbf{r})=-\mathrm{\nabla }\mathrm{\Phi }(\mathbf{r}).}$ 

Michael Faraday first realized the importance of a field as a physical quantity, during his investigations intomagnetism. He realized thatelectric andmagnetic 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, byJames Clerk Maxwell, of the first unified field theory in physics with the introduction of equations for theelectromagnetic field. The modern versions of these equations are calledMaxwell's equations.

Acharged test particle with chargeq experiences a forceF based solely on its charge. We can similarly describe theelectric fieldE so thatF =qE. Using this andCoulomb's law tells us that the electric field due to a single charged particle is

${\displaystyle \mathbf{E}=\frac{1}{4\pi {ϵ}_0}\frac{q}{r^2}\hat{\mathbf{r}}.}$ 

The electric field isconservative, and hence can be described by a scalar potential,V(r):

${\displaystyle \mathbf{E}(\mathbf{r})=-\mathrm{\nabla }V(\mathbf{r}).}$ 

A steady currentI flowing along a pathℓ 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 byI on a nearby chargeq with velocityv is

${\displaystyle \mathbf{F}(\mathbf{r})=q\mathbf{v}\times \mathbf{B}(\mathbf{r}),}$ 

whereB(r) is themagnetic field, which is determined fromI by theBiot–Savart law:

${\displaystyle \mathbf{B}(\mathbf{r})=\frac{{\mu }_0}{4\pi }\int \frac{Id\mathit{\ell }\times \hat{\mathbf{r}}}{r^2}.}$ 

The magnetic field is not conservative in general, and hence cannot usually be written in terms of a scalar potential. However, it can be written in terms of avector potential,A(r):

${\displaystyle \mathbf{B}(\mathbf{r})=\mathbf{\nabla }\times \mathbf{A}(\mathbf{r})}$ 

In general, in the presence of both a charge density ρ(r,t) and current densityJ(r,t), there will be both an electric and a magnetic field, and both will vary in time. They are determined byMaxwell's equations, a set of differential equations which directly relateE andB to ρ andJ.^[18]

Alternatively, one can describe the system in terms of its scalar and vector potentialsV andA. A set of integral equations known asretarded potentials allow one to calculateV andA from ρ andJ,^{[note 1]} and from there the electric and magnetic fields are determined via the relations^[19]

${\displaystyle \mathbf{E}=-\mathbf{\nabla }V-\frac{\mathrm{\partial }\mathbf{A}}{\mathrm{\partial }t}}$ 

${\displaystyle \mathbf{B}=\mathbf{\nabla }\times \mathbf{A}.}$ 

At the end of the 19th century, theelectromagnetic field 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.

Einstein's theory of gravity, calledgeneral relativity, is another example of a field theory. Here the principal field is themetric tensor, a symmetric 2nd-rank tensor field inspacetime. This replacesNewton's law of universal gravitation.

Waves can be constructed as physical fields, due to theirfinite propagation speed andcausal nature when a simplifiedphysical model of anisolated closed system is set^{[clarification needed]}. They are also subject to theinverse-square law.

For electromagnetic waves, there areoptical fields, and terms such asnear- 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 has fields ofpressure,density, and flow rate that are connected by conservation laws for energy and momentum. The mass continuity equation is acontinuity equation, representing the conservation of mass$${\displaystyle \frac{\mathrm{\partial }\rho }{\mathrm{\partial }t}+\mathrm{\nabla }\cdot (\rho \mathbf{u})=0}$$ and theNavier–Stokes equations represent the conservation of momentum in the fluid, found from Newton's laws applied to the fluid,$${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }t}(\rho \mathbf{u})+\mathrm{\nabla }\cdot (\rho \mathbf{u}\otimes \mathbf{u}+p\mathbf{I})=\mathrm{\nabla }\cdot \mathit{\tau }+\rho \mathbf{b}}$$ if the densityρ, pressurep,deviatoric stress tensorτ of the fluid, as well as external body forcesb, are all given. Theflow velocityu is the vector field to solve for.

Linear elasticity is defined in terms ofconstitutive equations between tensor fields,

${\displaystyle {\sigma }_{ij}=L_{ijkl}{\epsilon }_{kl}}$ 

where${\displaystyle {\sigma }_{ij}}$  are the components of the 3x3Cauchy stress tensor,${\displaystyle {\epsilon }_{ij}}$  the components of the 3x3infinitesimal strain and${\displaystyle L_{ijkl}}$  is theelasticity tensor, a fourth-rank tensor with 81 components (usually 21 independent components).

Assuming that the temperatureT is anintensive quantity, i.e., a single-valued,continuous anddifferentiablefunction of three-dimensional space (ascalar field), i.e., that${\displaystyle T=T(\mathbf{r})}$ , then thetemperature gradient is a vector field defined as${\displaystyle \mathrm{\nabla }T}$ . Inthermal conduction, the temperature field appears in Fourier's law,

${\displaystyle \mathbf{q}=-k\mathrm{\nabla }T}$ 

whereq is theheat flux field andk thethermal conductivity.

Temperature andpressure gradients are also important for meteorology.

It is now believed thatquantum mechanics 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 correspondingquantum field theory. For example,quantizingclassical electrodynamics givesquantum electrodynamics. Quantum electrodynamics is arguably the most successful scientific theory;experimentaldata confirm its predictions to a higherprecision (to moresignificant digits) than any other theory.^[22] The two other fundamental quantum field theories arequantum chromodynamics and theelectroweak theory.

In quantum chromodynamics, the color field lines are coupled at short distances bygluons, which are polarized by the field and line up with it. This effect increases within a short distance (around 1fm from the vicinity of the quarks) making the color force increase within a short distance,confining the quarks withinhadrons. As the field lines are pulled together tightly by gluons, they do not "bow" outwards as much as an electric field between electric charges.^[23]

These three quantum field theories can all be derived as special cases of the so-calledstandard model ofparticle 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 atfinite temperatures, something seldom considered in quantum field theory.

InBRST theory one deals with odd fields, e.g.Faddeev–Popov ghosts. There are different descriptions of odd classical fields both ongraded manifolds andsupermanifolds.

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 ofYang–Mills,Dirac,Klein–Gordon andSchrödinger fields as being solutions to their respective equations. A possible problem is that these RWEs can deal with complicatedmathematical objects with exotic algebraic properties (e.g.spinors are nottensors, so may need calculus forspinor fields), but these in theory can still be subjected to analytical methods given appropriatemathematical generalization.

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 aLagrangian or aHamiltonian of the field, and treating it as aclassical orquantum mechanical system with an infinite number ofdegrees 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 theLagrangian density in terms of the field components; the dynamics can be obtained by using theaction principle.

It is possible to construct simple fields without any prior knowledge of physics using only mathematics frommultivariable calculus,potential theory andpartial differential equations (PDEs). For example, scalar PDEs might consider quantities such as amplitude, density and pressure fields for the wave equation andfluid dynamics; temperature/concentration fields for theheat/diffusion equations. Outside of physics proper (e.g., radiometry and computer graphics), there are evenlight fields. All these previous examples arescalar 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 forvector fields (as are these three quantities, and those for vector PDEs in general). More generally problems incontinuum mechanics may involve for example, directionalelasticity (from which comes the termtensor, derived from theLatin word for stretch),complex fluid flows oranisotropic diffusion, which are framed as matrix-tensor PDEs, and then require matrices or tensor fields, hencematrix ortensor calculus. The scalars (and hence the vectors, matrices and tensors) can be real or complex as both arefields in the abstract-algebraic/ring-theoretic sense.

In a general setting, classical fields are described by sections offiber bundles and their dynamics is formulated in the terms ofjet manifolds (covariant classical field theory).^[24]

Inmodern physics, the most often studied fields are those that model the fourfundamental forces which one day may lead to theUnified Field Theory.

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

Fields are often classified by their behaviour under transformations ofspacetime. The terms used in this classification are:

• scalar fields (such astemperature) whose values are given by a single variable at each point of space. This value does not change under transformations of space.
• vector fields (such as the magnitude and direction of theforce at each point in amagnetic field) which are specified by attaching a vector to each point of space. The components of this vector transform between themselvescontravariantly 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 thestress 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 theDirac spinor) arise inquantum field theory to describe particles withspin 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.

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. Inparticle physics, thecolor symmetry of the interaction ofquarks is an example of an internal symmetry, that of thestrong interaction. Other examples areisospin,weak isospin,strangeness and any otherflavour 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 aninternal symmetry. One may also make a classification of the charges of the fields under internal symmetries.

Statistical field theory attempts to extend the field-theoreticparadigm toward many-body systems andstatistical 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 ismean field theory.

Classical fields as above, such as theelectromagnetic field, 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, becausethermally fluctuating classical fields arenowhere differentiable.Random fields are indexed sets ofrandom 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 aSchwartz space of functions as its index set, in which case the continuous random field is atempered distribution.

We can think about a continuous random field, in a (very) rough way, as an ordinary function that is${\displaystyle \pm \mathrm{\infty }}$  almost everywhere, but such that when we take aweighted average of all theinfinities 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 alinear map from a space of functions into thereal numbers.

• "Fields".Principles of Physical Science. Vol. 25 (15th ed.). 1994. p. 815 – viaEncyclopædia Britannica (Macropaedia).
• Landau, Lev D. andLifshitz, Evgeny M. (1971).Classical Theory of Fields (3rd ed.). London: Pergamon.ISBN 0-08-016019-0. Vol. 2 of theCourse of Theoretical Physics.
• Jepsen, Kathryn (July 18, 2013)."Real talk: Everything is made of fields"(PDF).Symmetry Magazine. Archived fromthe original(PDF) on March 4, 2016. RetrievedJune 9, 2015.

• Particle and Polymer Field Theories