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Field (physics)From Wikipedia, the free encyclopedia

The magnitude and direction of a two-dimensional electric field surrounding two equally charged (repelling) particles.

Brightness represents magnitude and hue represents direction.

Oppositely charged (attracting) particles.

A field is aphysical quantitythat has a value for each point inspaceandtime.[1]

For example, in a weather

forecast, the wind velocity during a day over a country is described by assigning a vector to each point in

space. Each vector represents the direction of the movement of air at that point. As the day progresses, the

directions in which the vectors point change as the directions of the wind change.

A field can be classified as ascalar field, avector field, aspinor field, or atensor fieldaccording to whether the

value of the field at each point is ascalar, avector, aspinor(e.g., a Dirac electron) or, more generally, atensor,

respectively. For example, theNewtoniangravitational fieldis 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 fieldor a quantum field, depending

on whether it is characterized by numbers orquantum operatorsrespectively.

A field may be thought of as extending throughout the whole of space. In practice, the strength of every known

field has been found to diminish with distance to the point of being undetectable. For instance, inNewton'stheory of gravity, the gravitational field strength is inversely proportional to the square of the distance from the

gravitating object. Therefore the Earth's gravitational field quickly becomes undetectable (oncosmicscales).

Defining the field as "numbers in space" shouldn't detract from the idea that it hasphysicalreality.It occupies

space. It contains energy. Its presence eliminates a true vacuum.[2]The field creates a "condition in

space"[3]

such that when we put a particle in it, the particle "feels" a force.

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If an electrical charge is accelerated, the effects on another charge do not appear instantaneously. The first

charge feels areactionforce, picking upmomentum, but the second charge feels nothing until the influence,

traveling at thespeed of light, reaches it and gives it the momentum. Where is the momentum before the

second charge moves? By the law ofconservation of momentumit must be somewhere. Physicists have found

it of "great utility for the analysis of forces"[3]to think of it as being in the field.

This utility leads to physicists believing thatelectromagnetic fieldsactually exist, making the field concept a

supportingparadigmof the entire edifice of modern physics. That said,John WheelerandRichard

Feynmanhave entertained Newton's pre-field concept ofaction at a distance(although they put it on the back

burner because of the ongoing utility of the field concept for research ingeneral relativityandquantum

electrodynamics).

"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".[3]

Contents

[hide]

1 History

2 Classical fields

o 2.1 Newtonian gravitation

o 2.2 Electromagnetism

2.2.1 Electrostatics

2.2.2 Magnetostatics

2.2.3 Electrodynamics

o 2.3 Gravitation in general relativity

o 2.4 Waves as fields

3 Quantum fields

4 Field theory

o 4.1 Symmetries of fields

4.1.1 Spacetime symmetries

4.1.2 Internal symmetries

o 4.2 Statistical field theory

o 4.3 Continuous random fields

o 4.4 Mathematics of fields

https://en.wikipedia.org/wiki/Reaction_(physics)https://en.wikipedia.org/wiki/Reaction_(physics)https://en.wikipedia.org/wiki/Reaction_(physics)https://en.wikipedia.org/wiki/Momentumhttps://en.wikipedia.org/wiki/Momentumhttps://en.wikipedia.org/wiki/Momentumhttps://en.wikipedia.org/wiki/Speed_of_lighthttps://en.wikipedia.org/wiki/Speed_of_lighthttps://en.wikipedia.org/wiki/Speed_of_lighthttps://en.wikipedia.org/wiki/Conservation_of_momentumhttps://en.wikipedia.org/wiki/Conservation_of_momentumhttps://en.wikipedia.org/wiki/Conservation_of_momentumhttps://en.wikipedia.org/wiki/Field_(physics)#cite_note-Feynman-3https://en.wikipedia.org/wiki/Field_(physics)#cite_note-Feynman-3https://en.wikipedia.org/wiki/Field_(physics)#cite_note-Feynman-3https://en.wikipedia.org/wiki/Electromagnetic_fieldhttps://en.wikipedia.org/wiki/Electromagnetic_fieldhttps://en.wikipedia.org/wiki/Electromagnetic_fieldhttps://en.wikipedia.org/wiki/Paradigmhttps://en.wikipedia.org/wiki/Paradigmhttps://en.wikipedia.org/wiki/Paradigmhttps://en.wikipedia.org/wiki/John_Archibald_Wheelerhttps://en.wikipedia.org/wiki/John_Archibald_Wheelerhttps://en.wikipedia.org/wiki/John_Archibald_Wheelerhttps://en.wikipedia.org/wiki/Richard_Feynmanhttps://en.wikipedia.org/wiki/Richard_Feynmanhttps://en.wikipedia.org/wiki/Richard_Feynmanhttps://en.wikipedia.org/wiki/Richard_Feynmanhttps://en.wikipedia.org/wiki/Action_at_a_distance_(physics)https://en.wikipedia.org/wiki/Action_at_a_distance_(physics)https://en.wikipedia.org/wiki/Action_at_a_distance_(physics)https://en.wikipedia.org/wiki/General_relativityhttps://en.wikipedia.org/wiki/General_relativityhttps://en.wikipedia.org/wiki/General_relativityhttps://en.wikipedia.org/wiki/Quantum_electrodynamicshttps://en.wikipedia.org/wiki/Quantum_electrodynamicshttps://en.wikipedia.org/wiki/Quantum_electrodynamicshttps://en.wikipedia.org/wiki/Quantum_electrodynamicshttps://en.wikipedia.org/wiki/Field_(physics)#cite_note-Feynman-3https://en.wikipedia.org/wiki/Field_(physics)#cite_note-Feynman-3https://en.wikipedia.org/wiki/Field_(physics)#cite_note-Feynman-3https://en.wikipedia.org/wiki/Field_(physics)https://en.wikipedia.org/wiki/Field_(physics)https://en.wikipedia.org/wiki/Field_(physics)https://en.wikipedia.org/wiki/Field_(physics)#Historyhttps://en.wikipedia.org/wiki/Field_(physics)#Historyhttps://en.wikipedia.org/wiki/Field_(physics)#Classical_fieldshttps://en.wikipedia.org/wiki/Field_(physics)#Classical_fieldshttps://en.wikipedia.org/wiki/Field_(physics)#Newtonian_gravitationhttps://en.wikipedia.org/wiki/Field_(physics)#Newtonian_gravitationhttps://en.wikipedia.org/wiki/Field_(physics)#Electromagnetismhttps://en.wikipedia.org/wiki/Field_(physics)#Electromagnetismhttps://en.wikipedia.org/wiki/Field_(physics)#Electrostaticshttps://en.wikipedia.org/wiki/Field_(physics)#Electrostaticshttps://en.wikipedia.org/wiki/Field_(physics)#Magnetostaticshttps://en.wikipedia.org/wiki/Field_(physics)#Magnetostaticshttps://en.wikipedia.org/wiki/Field_(physics)#Electrodynamicshttps://en.wikipedia.org/wiki/Field_(physics)#Electrodynamicshttps://en.wikipedia.org/wiki/Field_(physics)#Gravitation_in_general_relativityhttps://en.wikipedia.org/wiki/Field_(physics)#Gravitation_in_general_relativityhttps://en.wikipedia.org/wiki/Field_(physics)#Waves_as_fieldshttps://en.wikipedia.org/wiki/Field_(physics)#Waves_as_fieldshttps://en.wikipedia.org/wiki/Field_(physics)#Quantum_fieldshttps://en.wikipedia.org/wiki/Field_(physics)#Quantum_fieldshttps://en.wikipedia.org/wiki/Field_(physics)#Field_theoryhttps://en.wikipedia.org/wiki/Field_(physics)#Field_theoryhttps://en.wikipedia.org/wiki/Field_(physics)#Symmetries_of_fieldshttps://en.wikipedia.org/wiki/Field_(physics)#Symmetries_of_fieldshttps://en.wikipedia.org/wiki/Field_(physics)#Spacetime_symmetrieshttps://en.wikipedia.org/wiki/Field_(physics)#Spacetime_symmetrieshttps://en.wikipedia.org/wiki/Field_(physics)#Internal_symmetrieshttps://en.wikipedia.org/wiki/Field_(physics)#Internal_symmetrieshttps://en.wikipedia.org/wiki/Field_(physics)#Statistical_field_theoryhttps://en.wikipedia.org/wiki/Field_(physics)#Statistical_field_theoryhttps://en.wikipedia.org/wiki/Field_(physics)#Continuous_random_fieldshttps://en.wikipedia.org/wiki/Field_(physics)#Continuous_random_fieldshttps://en.wikipedia.org/wiki/Field_(physics)#Mathematics_of_fieldshttps://en.wikipedia.org/wiki/Field_(physics)#Mathematics_of_fieldshttps://en.wikipedia.org/wiki/Field_(physics)#Mathematics_of_fieldshttps://en.wikipedia.org/wiki/Field_(physics)#Continuous_random_fieldshttps://en.wikipedia.org/wiki/Field_(physics)#Statistical_field_theoryhttps://en.wikipedia.org/wiki/Field_(physics)#Internal_symmetrieshttps://en.wikipedia.org/wiki/Field_(physics)#Spacetime_symmetrieshttps://en.wikipedia.org/wiki/Field_(physics)#Symmetries_of_fieldshttps://en.wikipedia.org/wiki/Field_(physics)#Field_theoryhttps://en.wikipedia.org/wiki/Field_(physics)#Quantum_fieldshttps://en.wikipedia.org/wiki/Field_(physics)#Waves_as_fieldshttps://en.wikipedia.org/wiki/Field_(physics)#Gravitation_in_general_relativityhttps://en.wikipedia.org/wiki/Field_(physics)#Electrodynamicshttps://en.wikipedia.org/wiki/Field_(physics)#Magnetostaticshttps://en.wikipedia.org/wiki/Field_(physics)#Electrostaticshttps://en.wikipedia.org/wiki/Field_(physics)#Electromagnetismhttps://en.wikipedia.org/wiki/Field_(physics)#Newtonian_gravitationhttps://en.wikipedia.org/wiki/Field_(physics)#Classical_fieldshttps://en.wikipedia.org/wiki/Field_(physics)#Historyhttps://en.wikipedia.org/wiki/Field_(physics)https://en.wikipedia.org/wiki/Field_(physics)#cite_note-Feynman-3https://en.wikipedia.org/wiki/Quantum_electrodynamicshttps://en.wikipedia.org/wiki/Quantum_electrodynamicshttps://en.wikipedia.org/wiki/General_relativityhttps://en.wikipedia.org/wiki/Action_at_a_distance_(physics)https://en.wikipedia.org/wiki/Richard_Feynmanhttps://en.wikipedia.org/wiki/Richard_Feynmanhttps://en.wikipedia.org/wiki/John_Archibald_Wheelerhttps://en.wikipedia.org/wiki/Paradigmhttps://en.wikipedia.org/wiki/Electromagnetic_fieldhttps://en.wikipedia.org/wiki/Field_(physics)#cite_note-Feynman-3https://en.wikipedia.org/wiki/Conservation_of_momentumhttps://en.wikipedia.org/wiki/Speed_of_lighthttps://en.wikipedia.org/wiki/Momentumhttps://en.wikipedia.org/wiki/Reaction_(physics)

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5 See also

6 Notes

7 References

8 Further reading

9 External links

History [edit]

The first field to appear in physics was thegravitational field. ToIsaac Newtonhislaw of universal

gravitationsimply expressed the gravitationalforcethat acted between any pair of massive objects. In the

eighteenth century, a new entity was devised to simplify the bookkeeping of all these gravitational forces. This

entity, the gravitational field, gave at each point in space the total gravitational force on an object with unit mass

at that point. This did not change the physics in any way: it did not matter if you calculated all the gravitational

forces on an object individually and then added them together, or if you first added all the contributions together

as a gravitational field and then applied it to an object.[4]

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,Andr-Marie AmpreandCharles-

Augustin de Coulombcould manage with Newton-style laws that expressed the forces between pairs ofelectric

chargesorelectric currents. However, it became much more natural to take the field approach and express

these laws in terms ofelectricandmagnetic fields; in 1849Michael Faradaybecame the first to coin the term

"field".[4]

The independent nature of the field became more apparent withJames Clerk Maxwell's discovery thatwaves in

these fieldspropagated 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.[4]

Maxwell, at first, did not adopt the modern concept of a field as fundamental entity that could independently

exist. Instead he supposed that theelectromagnetic fieldexpressed the deformation of some underlying

mediumtheluminiferous aethermuch like the tension in a rubber membrane. A direct consequence of this

hypothesis was that the observed velocity of the electromagnetic waves should depend on 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 thetheory of special relativitybyAlbert Einsteinin 1905.

This theory changed the way the viewpoints of moving observers should be 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.[4]

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.org/wiki/Michael_Faradayhttps://en.wikipedia.org/wiki/Magnetic_fieldhttps://en.wikipedia.org/wiki/Electric_fieldhttps://en.wikipedia.org/wiki/Electric_currenthttps://en.wikipedia.org/wiki/Electric_chargehttps://en.wikipedia.org/wiki/Electric_chargehttps://en.wikipedia.org/wiki/Charles-Augustin_de_Coulombhttps://en.wikipedia.org/wiki/Charles-Augustin_de_Coulombhttps://en.wikipedia.org/wiki/Andr%C3%A9-Marie_Amp%C3%A8rehttps://en.wikipedia.org/wiki/Electromagnetismhttps://en.wikipedia.org/wiki/Field_(physics)#cite_note-Weinberg1977-4https://en.wikipedia.org/wiki/Forcehttps://en.wikipedia.org/wiki/Law_of_universal_gravitationhttps://en.wikipedia.org/wiki/Law_of_universal_gravitationhttps://en.wikipedia.org/wiki/Isaac_Newtonhttps://en.wikipedia.org/wiki/Gravitational_fieldhttps://en.wikipedia.org/w/index.php?title=Field_(physics)&action=edit&section=1https://en.wikipedia.org/wiki/Field_(physics)#External_linkshttps://en.wikipedia.org/wiki/Field_(physics)#Further_readinghttps://en.wikipedia.org/wiki/Field_(physics)#Referenceshttps://en.wikipedia.org/wiki/Field_(physics)#Noteshttps://en.wikipedia.org/wiki/Field_(physics)#See_also

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In the late 1920s, the new rules ofquantum mechanicswere first applied to the electromagnetic fields. In

1927,Paul Diracusedquantum fieldsto successfully explain how the decay of anatomto lowerquantum

statelead to thespontaneous emissionof 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 includingelectronsandprotonscould be understood as the quanta of

some quantum field, elevating fields to the most fundamental objects in nature.[4]

Classical fields [edit]

Main article:Classical field theory

There are several examples ofclassical fields. Classical field theories remain useful wherever quantum

properties do not arise, and can be active areas of research.Elasticityof materials,fluid

dynamicsandMaxwell's equationsare 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 forcewhen describing theelectric field. Thegravitational fieldwas then

similarly described.

Newtonian gravitation [edit]

Inclassical gravitation, mass is the source of an attractivegravitational fieldg.

A classical field theory describing gravity isNewtonian gravitation, which describes the gravitational force as a

mutual interaction between twomasses.

Any massive body Mhas agravitational fieldg which describes its influence on other massive bodies. The

gravitational field ofMat a point rin space is found by determining the force F that Mexerts on a smalltest

massm located at r, and then dividing by m:[5]

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Stipulating that m is much smaller than Mensures that the presence ofm has a negligible influence the

behavior ofM.

According toNewton's law of gravitation,F(r) is given by[5]

where is aunit vectorlying along the line joining Mand m and pointing from m to M. Therefore, the

gravitational field ofM is[5]

The experimental observation that inertial mass and gravitational mass are equal

tounprecedented levels of accuracyleads to the identification of the gravitational field strength as

identical to the acceleration experienced by a particle. This is the starting point of theequivalenceprinciple, which leads togeneral relativity.

Because the gravitational force F isconservative, the gravitational field g can be rewritten in

terms of thegradientof agravitational potential(r):

Electromagnetism [edit]

Main article:Electromagnetism

Michael Faradayfirst realized the importance of a field as a physical object, during his

investigations intomagnetism. He realized thatelectricandmagneticfields 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

version of these equations is calledMaxwell's equations.

Electrostatics [edit]

Main article:Electrostatics

Acharged test particlewith charge q experiences a force F based solely on its charge. We

can similarly describe theelectric fieldE so that F = qE. Using this andCoulomb's lawtells us

that the electric field due to a single charged particle as

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

Magnetostatics [edit]

Main article:Magnetostatics

A steady current Iflowing along a path will exert a force on nearby charged

particles that is quantitatively different from the electric field force described above.

The force exerted by Ion a nearby charge q with velocity v is

where B(r) is themagnetic field, which is determined from Iby theBiot-Savart

law:

The magnetic field is not conservative in general, and hence cannot

usually be written in terms of a scalar potential. However, it can be written

in terms of avector potential,A(r):

E fields due to stationary electric charges and B fields due to

stationarymagnetic charges. In motion (velocityv), an electriccharge induces

a Bfield while a magneticcharge induces an E field.Conventional currentis

used.

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Top:E field due to anelectric dipole momentd. Bottom left:B field due to

a mathematicalmagnetic dipolem formed by two magnetic

monopoles. Bottom right:B field due to a puremagnetic dipole

momentm found in ordinary matter (notfrom monopoles).

TheE fieldsandB fieldsdue toelectric charges(black/white) andmagnetic

poles(red/blue).[6][7]

Electrodynamics [edit]

Main article:Electrodynamics

In general, in the presence of both a charge density (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 byMaxwell's equations, a

set of differential equations which directly relate E and Bto andJ.[8]

Alternatively, one can describe the system in terms of its scalar and

vector potentials Vand A. A set of integral equations known

asretarded potentialsallow one to calculate VandAfrom andJ,[note

1]and from there the electric and magnetic fields are determined via

the relations[9]

At the end of the 19th century, theelectromagnetic fieldwas

understood as a collection of two vector fields in space.

Nowadays, one recognizes this as a single antisymmetric

2nd-rank tensor field in spacetime.

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Gravitation in general relativity [edit]

Ingeneral relativity, mass-energy warps space time (Einstein

tensorG),[10]and rotating asymmetric mass-energy distributions

withangular momentumJ generateGEM fieldsH[11]

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 as fields [edit]

Wavescan be constructed as physical fields, due to

theirfinite propagation speedandcausal naturewhen a

simplifiedphysical modelof anisolated closed systemis

set[clarification needed]

. They are also subject to theinverse-square

law.

For electromagnetic waves, there areoptical fields, and

terms such asnear- and far-fieldlimits for diffraction. In

practice, though the field theories of optics are superseded

by the electromagnetic field theory of Maxwell.

Quantum fields [edit]

Main article:Quantum field theory

Further information:Wavefunction collapse

It is now believed thatquantum mechanicsshould underlie

all physical phenomena, so that a classical field theory

should, at least in principle, permit a recasting in quantum

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mechanical terms; success yields the

correspondingquantum field theory. For

example,quantizingclassical

electrodynamicsgivesquantum electrodynamics. Quantum

electrodynamics is arguably the most successful scientific

theory;experimentaldataconfirm its predictions to a

higherprecision(to moresignificant digits) than any other

theory.[12]The two other fundamental quantum field theories

arequantum chromodynamicsand theelectroweak theory.

Fields due tocolor charges, like inquarks(G is thegluon field

strength tensor). These are "colorless" combinations. Top: Color

charge has "ternary neutral states" as well as binary neutrality

(analogous toelectric charge). Bottom: The quark/antiquark

combinations.[6][7]

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 1fmfrom the vicinity of the quarks)

making the color force increase within a short

distance,confining the quarkswithinhadrons. As the field

lines are pulled together tightly by gluons, they do not "bow"

outwards as much as an electric field between electric

charges.[13]

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These three quantum field theories can all be derived as

special cases of the so-calledstandard modelofparticle

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.

InBRST theoryone deals with odd fields, e.g.ghosts. There

are different descriptions of odd classical fields both

ongraded manifoldsandsupermanifolds.

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 (more precisely,relativistic

wave equations(RWEs)). Thus one can speak ofYang-

Mills,Dirac,Klein-GordonandSchroedinger fieldsas being

solutions to their respective equations. A possible problem is

that these RWEs can deal with complicatedmathematical

objectswith exotic algebraic properties (e.g.spinorsare

nottensors, so may need calculus overspinor fields), but

these in theory can still be subjected to analytical methods

given appropriatemathematical generalization.

Some theories, such as theBatalinVilkovisky formalism,

contains both fields and antifields.

Field theory [edit]

A field theory is aphysical theorythat describes how one or

more physical fields interact with matter.

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

aLagrangianor aHamiltonianof the field, and treating it as

theclassical mechanics(orquantum mechanics) of a system

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kipedia.org/wiki/Supermanifoldhttps://en.wikipedia.org/wiki/Graded_manifoldhttps://en.wikipedia.org/wiki/Faddeev%E2%80%93Popov_ghosthttps://en.wikipedia.org/wiki/BRST_formalismhttps://en.wikipedia.org/wiki/Thermal_field_theoryhttps://en.wikipedia.org/wiki/General_relativityhttps://en.wikipedia.org/wiki/Particle_physicshttps://en.wikipedia.org/wiki/Particle_physicshttps://en.wikipedia.org/wiki/Standard_model

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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 densityin terms of the field components; the

dynamics can be obtained by using theaction principle.

It is possible to construct simple fields without any a priori

knowledge of physics using only mathematics fromseveral

variable calculus, potential theory and partial differential

equations. 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

overvector fields(as are these three quantities, and those

for vector PDEs in general). More generally problems

incontinuum mechanicsmay involve for example,

directionalelasticity(from which comes the term tensor,

derived from theLatinword for stretch),complex fluidflows

oranisotropic diffusion, which are framed as matrix-tensor

PDEs, and then require matrices or tensor fields,

hencematrixortensor calculus. It should be noted that the

scalars (and hence the vectors, matrices and tensors) can

be real or complex as both arefieldsin the abstract-

algebraic/ring-theoreticsense.

In a general setting, classical fields are described by

sections offiber bundlesand their dynamics is formulated in

the terms ofjet manifolds(covariant classical field theory).[14]

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Inmodern physics, the most often studied fields are those

that model the fourfundamental forceswhich one day may

lead to theUnified Field Theory.

Symmetries of fields [edit]

A convenient way of classifying a field (classical or quantum)

is by thesymmetriesit possesses. Physical symmetries are

usually of two types:

Spacetime symmetries [edit]

Main article:Spacetime symmetries

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

theforceat each point in amagnetic field) which are

specified by attaching a vector to each point of space.

The components of this vector transform between

themselves as usual under rotations in space.

tensor fields, (such as thestress tensorof a crystal)

specified by a tensor at each point of space. The

components of the tensor transform between

themselves as usual under rotations in space.

spinor fields(such as theDirac spinor) arise inquantum

field theoryto describe particles withspin.

Internal symmetries [edit]

Main article:Gauge symmetry

Fields may have internal symmetries in addition to spacetime

symmetries. For example, in many situations one needs

fields which are a list of space-time scalars: (1, 2, ... N).

For example, in weather prediction these may be

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temperature, pressure, humidity, etc. Inparticle physics,

thecolorsymmetry of the interaction ofquarksis an example

of an internal symmetry of thestrong interaction, as is

theisospinorflavoursymmetry.

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 [edit]

Main article:Statistical field theory

Statistical field theory attempts to extend the field-

theoreticparadigmtoward 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.

Continuous random fields [edit]

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 functionsare not continuous. When

dealing carefully with classical fields at finite temperature,

the mathematical methods of continuous random fields are

used, becausethermally fluctuatingclassical fields

arenowhere differentiable.Random fieldsare 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 spaceof functions as its

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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 almost

everywhere, but such that when we take aweighted

averageof all theinfinitiesover 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 mapfrom a space of functions into

thereal numbers.

Mathematics of fields [edit]

The continuum view (hence the term "field") can be

approached by letting the system have an infinite number

ofdegrees of freedom. The dimension of avector ordinary

differential equationis simply thedimensionof the

vectordependent variable, or thevector function. In this

sensepartial differential equationsso can be thought of as

(coupled)ODEsof infinite dimension (a mathematical

interpretation of the degrees of freedom argument).[15]In

addition, vector fields calledslope fieldsare important tools

in analyzing results in ODEs (see alsophase plane).

The exact nature of the object (and its arguments) in the

differential equation

(e.g.realscalar,complexmatrix,Euclidean vectororfour

vectoretc.) determines the kind of analysis (in our examples

- calculus of a real single variable, a complex matrix and

over real vector fields) needed.

Other than partial differential equations, other parts of

(classical)real analysisandcomplex analysiswere either

inspired by or have techniques applied (or both) in field

theory. Examples of such areas arespectral

theoryandharmonic analysis(vibrations and waves) or the

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