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| [[File:Field-illustrations-add.png|thumb|The magnitude and direction of a two-dimensional electric field surrounding two equally charged (repelling) particles. Brightness represents magnitude and hue represents direction.]]
| | Name: Arnoldo Wootton<br>My age: 38 years old<br>Country: Italy<br>City: Riva Di Solto <br>ZIP: 24060<br>Address: Via Lombardi 100<br><br>My blog post: [http://topseosoft.com/category/freebies-goodies/free-linkslist-freebies-goodies/ Free linkslist] |
| [[File:Field-illustrations-sub.png|thumb|Oppositely charged (attracting) particles.]]
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| A '''field''' is a [[physical quantity]] that has a value for each [[Point (geometry)|point]] in space and time.<ref name=Gribbin>
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| {{cite book |author=John Gribbin|title=Q is for Quantum: Particle Physics from A to Z|publisher=Weidenfeld & Nicolson|location=London|year=1998|isbn=0-297-81752-3|page=138}}</ref> For example, in a weather forecast, the wind [[velocity]] is described by assigning a vector to each point in space. Each vector represents the speed and direction of the movement of air at that point.
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| A field can be classified as a [[scalar field]], a [[vector field]], a [[spinor field]] or a [[tensor field]] according to whether the value of the field at each point is a [[scalar (physics)|scalar]], a [[Euclidean vector|vector]], a [[spinor]] or a [[tensor]], respectively. For example, the [[Newtonian gravity|Newtonian]] [[gravitational 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 a ''classical field'' or a ''quantum field'', depending on whether it is characterized by numbers or [[operator (physics)|quantum operators]] respectively.
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| 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, in [[Newton's law of universal gravitation|Newton's theory 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 on [[cosmos|cosmic]] scales.
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| Defining the field as "numbers in space" shouldn't detract from the idea that it has [[physical property|physical]] [[reality]]. “It occupies space. It contains energy. Its presence eliminates a true vacuum.”<ref name=Wheeler>
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| {{cite book |author=John Archibald Wheeler|title=Geons, Black Holes, and Quantum Foam: A Life in Physics.|publisher=Norton|location=London|year=1998|page=163}}
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| </ref> The field creates a "condition in space"<ref name=Feynman>{{cite book |author=Richard P. Feynman|title=Feynman's Lectures on Physics, Volume 1.|publisher=Caltech| year=1963 |pages=2–4}}</ref> 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 a [[Reaction (physics)|reaction]] force, picking up [[momentum]], but the second charge feels nothing until the influence, traveling at the [[speed of light]], reaches it and gives it the momentum. Where is the momentum before the second charge moves? By the law of [[conservation of momentum]] it must be somewhere. Physicists have found it of "great utility for the analysis of forces"<ref name=Feynman>{{cite book |author=Richard P. Feynman|title=Feynman's Lectures on Physics, Volume 1.|publisher=Caltech| year=1963 |pages=10–9}}</ref> to think of it as being in the field.
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| This utility leads to physicists believing that [[electromagnetic field]]s actually exist, making the field concept a supporting [[paradigm]] of the entire edifice of modern physics. That said, [[John Archibald Wheeler|John Wheeler]] and [[Richard Feynman]] seriously considered Newton's pre-field concept of [[Action at a distance (physics)|action at a distance]] (although they set it aside because of the ongoing utility of the field concept for research in [[general relativity]] and [[quantum electrodynamics]]).
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| "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".<ref name="Feynman"/>
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| ==History==
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| To [[Isaac Newton]] his [[law of universal gravitation]] simply expressed the gravitational [[force]] that acted between any pair of massive objects. When looking at the motion of many bodies all interacting with each other, such as the planets in the [[Solar System]], dealing with the force between each pair of bodies separately rapidly becomes computationally inconvenient. 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 which would be felt by 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.<ref name=Weinberg1977>{{cite journal
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| |title=The Search for Unity: Notes for a History of Quantum Field Theory
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| |first=Steven |last=Weinberg
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| |journal=Daedalus |volume=106 |number=4 |year=1977 |pages=17–35
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| |jstor=20024506
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| }}</ref>
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| The development of the independent concept of a field truly began in the nineteenth century with the development of the theory of [[electromagnetism]]. In the early stages, [[André-Marie Ampère]] and [[Charles-Augustin de Coulomb]] could manage with Newton-style laws that expressed the forces between pairs of [[electric charge]]s or [[electric current]]s. However, it became much more natural to take the field approach and express these laws in terms of [[electric field|electric]] and [[magnetic field]]s; in 1849 [[Michael Faraday]] became the first to coin the term "field".<ref name=Weinberg1977/>
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| The independent nature of the field became more apparent with [[James Clerk Maxwell]]'s discovery that [[electromagnetic wave|waves in these fields]] 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.<ref name=Weinberg1977/>
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| Maxwell, at first, did not adopt the modern concept of a field as fundamental entity that could independently exist. Instead, he supposed that the [[electromagnetic field]] expressed the deformation of some underlying medium—the [[luminiferous 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 the [[theory of special relativity]] by [[Albert Einstein]] in 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.<ref name=Weinberg1977/>
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| In the late 1920s, the new rules of [[quantum mechanics]] were first applied to the electromagnetic fields. In 1927, [[Paul Dirac]] used [[quantum field]]s to successfully explain how the decay of an [[atom]] to lower [[quantum state]] lead to the [[spontaneous emission]] of a [[photon]], the quantum of the electromagnetic field. This was soon followed by the realization (following the work of [[Pascual Jordan]], [[Eugene Wigner]], [[Werner Heisenberg]], and [[Wolfgang Pauli]]) that all particles, including [[electron]]s and [[proton]]s, could be understood as the quanta of some quantum field, elevating fields to the status of the most fundamental objects in nature.<ref name=Weinberg1977/>
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| ==Classical fields==
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| {{Main|Classical field theory}}
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| There are several examples of [[Classical field theory|classical fields]]. Classical field theories remain useful wherever quantum properties do not arise, and can be active areas of research. [[Elasticity (physics)|Elasticity]] of materials, [[fluid dynamics]] and [[Maxwell's equations]] are cases in point.
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| Some of the simplest physical fields are vector force fields. Historically, the first time that fields were taken seriously was with [[Michael Faraday|Faraday's]] [[lines of force]] when describing the [[electric field]]. The [[gravitational field]] was then similarly described.
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| ===Newtonian gravitation===
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| [[File:Newtonian gravity field (physics).svg|thumb|upright|In [[classical gravitation]], mass is the source of an attractive [[gravitational field]] '''g'''.]]
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| A classical field theory describing gravity is [[gravity|Newtonian gravitation]], which describes the gravitational force as a mutual interaction between two [[mass]]es.
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| Any massive body ''M'' has a [[gravitational field]] '''g''' which describes its influence on other massive bodies. The gravitational field of ''M'' at a point '''r''' in space is found by determining the force '''F''' that ''M'' exerts on a small [[test mass]] ''m'' located at '''r''', and then dividing by ''m'':<ref name="kleppner85">{{cite book|last1=Kleppner|first1=David|last2=Kolenkow|first2=Robert|title=An Introduction to Mechanics|page=85}}</ref>
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| :<math> \mathbf{g}(\mathbf{r}) = \frac{\mathbf{F}(\mathbf{r})}{m}.</math> | |
| Stipulating that ''m'' is much smaller than ''M'' ensures that the presence of ''m'' has a negligible influence the behavior of ''M''.
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| According to [[Newton's law of gravitation]], '''F'''('''r''') is given by<ref name="kleppner85" />
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| :<math>\mathbf{F}(\mathbf{r}) = -\frac{G M m}{r^2}\hat{\mathbf{r}},</math>
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| where <math>\hat{\mathbf{r}}</math> is a [[unit vector]] lying along the line joining ''M'' and ''m'' and pointing from ''m'' to ''M''. Therefore, the gravitational field of '''M''' is<ref name="kleppner85" />
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| :<math>\mathbf{g}(\mathbf{r}) = \frac{\mathbf{F}(\mathbf{r})}{m} = -\frac{G M}{r^2}\hat{\mathbf{r}}.</math>
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| The experimental observation that inertial mass and gravitational mass are equal to [[equivalence principle#Tests of the weak equivalence principle|unprecedented levels of accuracy]] leads to the identification of the gravitational field strength as identical to the acceleration experienced by a particle. This is the starting point of the [[equivalence principle]], which leads to [[general relativity]].
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| Because the gravitational force '''F''' is [[conservative field|conservative]], the gravitational field '''g''' can be rewritten in terms of the [[gradient]] of a scalar function, the [[gravitational potential]] Φ('''r'''):
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| :<math>\mathbf{g}(\mathbf{r}) = -\nabla \Phi(\mathbf{r}).</math>
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| ===Electromagnetism===
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| {{Main|Electromagnetism}}
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| [[Michael Faraday]] first realized the importance of a field as a physical object, during his investigations into [[magnetism]]. He realized that [[Electric field|electric]] and [[magnetic field|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.
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| These ideas eventually led to the creation, by [[James Clerk Maxwell]], of the first unified field theory in physics with the introduction of equations for the [[electromagnetic field]]. The modern version of these equations is called [[Maxwell's equations]].
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| ====Electrostatics====
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| {{Main|Electrostatics}}
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| A [[test charge|charged test particle]] with charge ''q'' experiences a force '''F''' based solely on its charge. We can similarly describe the [[electric field]] '''E''' so that {{nowrap|'''F''' {{=}} ''q'''''E'''}}. Using this and [[Coulomb's law]] tells us that the electric field due to a single charged particle as
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| :<math>\mathbf{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{\mathbf{r}}.</math>
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| The electric field is [[conservative field|conservative]], and hence can be described by a scalar potential, ''V''('''r'''):
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| :<math> \mathbf{E}(\mathbf{r}) = -\nabla V(\mathbf{r}).</math>
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| ====Magnetostatics====
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| {{Main|Magnetostatics}}
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| A steady current ''I'' flowing along a path ''ℓ'' will exert a force on nearby charged particles that is quantitatively different from the electric field force described above. The force exerted by ''I'' on a nearby charge ''q'' with velocity '''v''' is
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| :<math>\mathbf{F}(\mathbf{r}) = q\mathbf{v} \times \mathbf{B}(\mathbf{r}),</math>
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| where '''B'''('''r''') is the [[magnetic field]], which is determined from ''I'' by the [[Biot-Savart law]]:
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| :<math>\mathbf{B}(\mathbf{r}) = \frac{\mu_0 I}{4\pi} \int \frac{d\boldsymbol{\ell} \times d\hat{\mathbf{r}}}{r^2}.</math>
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| 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 [[magnetic vector potential|vector potential]], '''A'''('''r'''):
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| :<math> \mathbf{B}(\mathbf{r}) = \boldsymbol{\nabla} \times \mathbf{A}(\mathbf{r}) </math>
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| {{multiple image
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| | align = center
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| | direction = horizontal
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| | footer = The [[electric field|'''E''' fields]] and [[magnetic field|'''B''' fields]] due to [[electric charge]]s (black/white) and [[magnet|magnetic poles]] (red/blue).<ref name="Mc Graw Hill">{{cite book |title=McGraw Hill Encyclopaedia of Physics |first1=C.B. |last1= Parker|edition=2nd|publisher=Mc Graw Hill|year=1994|isbn=0-07-051400-3}}</ref><ref name="M. Mansfield, C. O’Sullivan 2011">{{cite book |author= M. Mansfield, C. O’Sullivan|title= Understanding Physics|edition= 4th |year= 2011|publisher= John Wiley & Sons|isbn=978-0-47-0746370}}</ref>
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| | image1 = em monopoles.svg
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| | caption1 = '''E''' fields due to stationary electric charges and '''B''' fields due to stationary [[magnetic monopole|magnetic charges]]. In motion ([[velocity]] '''v'''), an ''electric'' charge induces a '''B''' field while a ''magnetic'' charge induces an '''E''' field. [[Conventional current]] is used.
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| | width1 = 350
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| | image2 = em dipoles.svg
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| | caption2 = '''Top:''' '''E''' field due to an [[electric dipole moment]] '''d'''. '''Bottom left:''' '''B''' field due to a ''mathematical'' [[magnetic dipole]] '''m''' formed by two magnetic monopoles. '''Bottom right:''' '''B''' field due to a pure [[magnetic dipole moment]] '''m''' found in ordinary matter (''not'' from monopoles).
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| | width2 = 300
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| }}
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| ====Electrodynamics====
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| {{Main|Electrodynamics}}
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| 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]], a set of differential equations which directly relate '''E''' and '''B''' to ρ and '''J'''.<ref name="griffiths326">{{cite book|last=Griffiths|first=David|title=Introduction to Electrodynamics|edition=3rd|page=326}}</ref>
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| Alternatively, one can describe the system in terms of its scalar and vector potentials ''V'' and '''A'''. A set of integral equations known as ''[[retarded potential]]s'' allow one to calculate ''V'' and '''A''' from ρ and '''J''',<ref group="note">This is contingent on the correct choice of [[gauge fixing|gauge]]. ''V'' and '''A''' are not completely determined by ρ and '''J'''; rather, they are only determined up to some scalar function ''f''('''r''', ''t'') known as the gauge. The retarded potential formalism requires one to choose the [[Lorenz gauge]].</ref> and from there the electric and magnetic fields are determined via the relations<ref name="wangsness469">{{cite book|last=Wangsness|first=Roald|title=Electromagnetic Fields|edition=2nd|page=469}}</ref>
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| :<math> \mathbf{E} = -\boldsymbol{\nabla} V - \frac{\partial \mathbf{A}}{\partial t}</math>
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| :<math> \mathbf{B} = \boldsymbol{\nabla} \times \mathbf{A}.</math>
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| At the end of the 19th century, the [[electromagnetic 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.
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| ===Gravitation in general relativity===
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| [[File:Relativistic gravity field (physics).svg|thumb|right|350px|In [[general relativity]], mass-energy warps space time ([[Einstein tensor]] '''G'''),<ref>{{cite book |title=Gravitation|author=J.A. Wheeler, C. Misner, K.S. Thorne|publisher=W.H. Freeman & Co|year=1973|isbn=0-7167-0344-0}}</ref> and rotating asymmetric mass-energy distributions with [[angular momentum]] '''J''' generate [[Gravitoelectromagnetism|GEM fields]] '''H'''<ref>{{cite book |title=Gravitation and Inertia|author=I. Ciufolini and J.A. Wheeler|publisher=Princeton Physics Series|year=1995|isbn=0-691-03323-4}}</ref>]]
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| Einstein's theory of gravity, called [[general relativity]], is another example of a field theory. Here the principal field is the [[metric tensor (general relativity)|metric tensor]], a symmetric 2nd-rank tensor field in [[spacetime]]. This replaces [[Newton's law of universal gravitation]].
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| ===Waves as fields===
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| [[Waves]] can be constructed as physical fields, due to their [[speed of light|finite propagation speed]] and [[causality|causal nature]] when a simplified [[physical model]] of an [[Physical system#The concept of closed systems in physics|isolated closed system]] is set {{clarify|date=March 2013}}. They are also subject to the [[inverse-square law]].
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| For electromagnetic waves, there are [[optical field]]s, and terms such as [[Near and far field|near- and far-field]] limits for diffraction. In practice, though the field theories of optics are superseded by the electromagnetic field theory of Maxwell.
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| ==Quantum fields==<!-- This section is linked from [[Kip Thorne]] -->
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| {{main|Quantum field theory}}
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| It is now believed that [[quantum 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 corresponding [[quantum field theory]]. For example, [[Quantization (physics)|quantizing]] [[classical electrodynamics]] gives [[quantum electrodynamics]]. Quantum electrodynamics is arguably the most successful scientific theory; [[experiment]]al [[data]] confirm its predictions to a higher [[Accuracy and precision|precision]] (to more [[significant digit]]s) than any other theory.<ref>{{Cite book|last1=Peskin |first1=Michael E. |last2=Schroeder |first2=Daniel V. |title=An Introduction to Quantum Fields |page=198 |year=1995 |publisher= Westview Press |isbn=0-201-50397-2|ref=harv|postscript=<!--None-->}}. Also see [[precision tests of QED]].</ref> The two other fundamental quantum field theories are [[quantum chromodynamics]] and the [[electroweak theory]].
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| [[File:Qcd fields field (physics).svg|400px|center|thumb|Fields due to [[color charge]]s, like in [[quark]]s ('''G''' is the [[gluon field strength tensor]]). These are "colorless" combinations. '''Top:''' Color charge has "ternary neutral states" as well as binary neutrality (analogous to [[electric charge]]). '''Bottom:''' The quark/antiquark combinations.<ref name="Mc Graw Hill"/><ref name="M. Mansfield, C. O’Sullivan 2011"/>]]
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| In quantum chromodynamics, the color field lines are coupled at short distances by [[gluon]]s, which are polarized by the field and line up with it. This effect increases within a short distance (around 1 [[femtometre|fm]] from the vicinity of the quarks) making the color force increase within a short distance, [[Color confinement|confining the quarks]] within [[hadron]]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.<ref>{{cite book|title=Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles|edition=2nd|author=R. Resnick, R. Eisberg|publisher=John Wiley & Sons|year=1985|page=684|isbn=978-0-471-87373-0}}</ref>
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| These three quantum field theories can all be derived as special cases of the so-called [[standard model]] of [[particle physics]]. [[General relativity]], the Einsteinian field theory of gravity, has yet to be successfully quantized. However an extension, [[thermal field theory]], deals with quantum field theory at ''finite temperatures'', something seldom considered in quantum field theory.
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| In [[BRST formalism|BRST theory]] one deals with odd fields, e.g. [[Faddeev–Popov ghost]]s. There are different descriptions of odd classical fields both on [[graded manifold]]s and [[supermanifold]]s.
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| As above with classical fields, it is possible to approach their quantum counterparts from a purely mathematical view using similar techniques as before. The equations governing the quantum fields are in fact PDEs (specifically, [[relativistic wave equations]] (RWEs)). Thus one can speak of [[Yang-Mills field|Yang-Mills]], [[Dirac field|Dirac]], [[Klein-Gordon field|Klein-Gordon]] and [[Schroedinger field]]s as being solutions to their respective equations. A possible problem is that these RWEs can deal with complicated [[mathematical objects]] with exotic algebraic properties (e.g. [[spinors]] are not [[tensors]], so may need calculus over [[spinor field]]s), but these in theory can still be subjected to analytical methods given appropriate [[Generalization (mathematics)|mathematical generalization]].
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| ==Field theory==
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| A field theory is a [[physical theory]] that describes how one or more physical fields interact with matter.
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| 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 mechanics|Hamiltonian]] of the field, and treating it as the [[classical mechanics]] (or [[quantum mechanics]]) of a system with an infinite number of [[degrees of freedom (physics and chemistry)|degrees of freedom]]. The resulting field theories are referred to as classical or quantum field theories.
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| The dynamics of a classical field are usually specified by the [[Lagrangian|Lagrangian density]] in terms of the field components; the dynamics can be obtained by using the [[Action (physics)|action principle]].
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| It is possible to construct simple fields without any a priori knowledge of physics using only mathematics from [[multivariable calculus|several variable calculus]], potential theory and [[partial differential equation]]s (PDEs). For example, scalar PDEs might consider quantities such as amplitude, density and pressure fields for the wave equation and [[fluid dynamics]]; temperature/concentration fields for the [[heat equation|heat]]/[[diffusion equation]]s. 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 over [[vector fields]] (as are these three quantities, and those for vector PDEs in general). More generally problems in [[continuum mechanics]] may involve for example, directional [[elasticity tensor|elasticity]] (from which comes the term ''tensor'', derived from the [[Latin]] word for stretch), [[complex fluid]] flows or [[anisotropic diffusion]], which are framed as matrix-tensor PDEs, and then require matrices or tensor fields, hence [[matrix calculus|matrix]] or [[tensor calculus]]. It should be noted that the scalars (and hence the vectors, matrices and tensors) can be real or complex as both are [[field (algebra)|fields]] in the abstract-algebraic/[[ring theory|ring-theoretic]] sense.
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| In a general setting, classical fields are described by sections of [[fiber bundle]]s and their dynamics is formulated in the terms of [[jet bundle|jet manifolds]] ([[covariant classical field theory]]).<ref>Giachetta, G., Mangiarotti, L., [[Gennadi Sardanashvily|Sardanashvily, G.]] (2009) ''Advanced Classical Field Theory''. Singapore: World Scientific, ISBN 978-981-283-895-7 ([http://xxx.lanl.gov/abs/0811.0331 arXiv: 0811.0331v2])</ref>
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| In [[modern physics]], the most often studied fields are those that model the four [[fundamental forces]] which one day may lead to the [[Unified Field Theory]].
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| ===Symmetries of fields=== <!-- the article [[Standard Model (mathematical formulation)]] links here -->
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| A convenient way of classifying a field (classical or quantum) is by the [[Symmetry in physics|symmetries]] it possesses. Physical symmetries are usually of two types:
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| ====Spacetime symmetries====
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| {{main|Spacetime symmetries}}
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| Fields are often classified by their behaviour under transformations of [[spacetime]]. The terms used in this classification are:
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| * [[scalar field]]s (such as [[temperature]]) whose values are given by a single variable at each point of space. This value does not change under transformations of space.
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| * [[vector field]]s (such as the magnitude and direction of the [[force (physics)|force]] at each point in a [[magnetic 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.
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| * [[tensor field]]s, (such as the [[Stress (physics)|stress tensor]] of 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.
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| * [[spinor field]]s (such as the [[Dirac spinor]]) arise in [[quantum field theory]] to describe particles with [[spin (physics)|spin]].
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| ====Internal symmetries====
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| {{main|Gauge symmetry}}
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| 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: (φ<sub>1</sub>, φ<sub>2</sub>, ... φ<sub>''N''</sub>). For example, in weather prediction these may be temperature, pressure, humidity, etc. In [[particle physics]], the [[color charge|color]] symmetry of the interaction of [[quark]]s is an example of an internal symmetry of the [[strong interaction]], as is the [[isospin]] or [[flavour (particle physics)|flavour]] symmetry.
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| 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.
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| ===Statistical field theory===
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| {{main|Statistical field theory}}
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| Statistical field theory attempts to extend the field-theoretic [[paradigm]] toward many-body systems and [[statistical mechanics]]. As above, it can be approached by the usual infinite number of degrees of freedom argument.
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| 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]].
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| ===Continuous random fields===
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| Classical fields as above, such as the [[electromagnetic 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, because [[Thermal fluctuations|thermally fluctuating]] classical fields are [[nowhere differentiable]]. [[Random field]]s are indexed sets of [[random variable]]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 [[Distribution (mathematics)|tempered distribution]].
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| We can think about a continuous random field, in a (very) rough way, as an ordinary function that is <math>\pm\infty</math> almost everywhere, but such that when we take a [[weighted average]] of all the [[infinity|infinities]] over any finite region, we get a finite result. The infinities are not well-defined; but the finite values can be associated with the functions used as the weight functions to get the finite values, and that can be well-defined. We can define a continuous random field well enough as a [[linear map]] from a space of functions into the [[real number]]s.
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| ===Mathematics of fields===
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| The continuum view (hence the term "field") can be approached by letting the system have an infinite number of [[Degrees of freedom (physics and chemistry)|degrees of freedom]]. The dimension of a [[System of ordinary differential equation#System of ODEs|vector ordinary differential equation]] is simply the [[dimension]] of the vector [[dependent variable]], or the [[vector function]]. In this sense [[partial differential equation]]s so can be thought of as (coupled) [[Ordinary differential equation|ODEs]] of infinite dimension (a mathematical interpretation of the degrees of freedom argument).<ref>Nonlinear Dispersive Equations: Local And Global Analysis, Terence Tao.</ref> In addition, vector fields called [[slope field]]s are important tools in analyzing results in ODEs (see also [[phase plane]]).
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| The exact nature of the object (and its arguments) in the differential equation (e.g. [[real number|real]] scalar, [[complex number|complex]] [[matrix (mathematics)|matrix]], [[Euclidean vector]] or [[four vector]] etc.) determines the kind of analysis (in our examples – calculus of a real single variable, a complex matrix and over real vector fields) needed.
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| Other than partial differential equations, other parts of (classical) [[real analysis]] and [[complex analysis]] were either inspired by or have techniques applied (or both) in field theory. Examples of such areas are [[spectral theory]] and [[harmonic analysis]] (vibrations and waves) or the self-descriptive [[potential theory]], all now mathematical subjects in their own right. However perhaps the most prominent examples are [[variational calculus]] (given its connections to the [[Lagrangian mechanics|Lagrangian]] and [[Hamiltonian mechanics|Hamiltonian]] formalisms) and [[multivariable calculus]] with its generalizations [[differential geometry]] – including [[tensor calculus]], and [[gauge theory]] – and its close relative [[differential topology]].
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| ==See also==
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| * [[Covariant Hamiltonian field theory]]
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| * [[Scalar field theory]]
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| ==Notes==
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| {{reflist|group=note}}
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| ==References==
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| <references />
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| ==Further reading==
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| * [[Lev Landau|Landau, Lev D.]] and [[Evgeny Lifshitz|Lifshitz, Evgeny M.]] (1971). ''Classical Theory of Fields'' (3rd ed.). London: Pergamon. ISBN 0-08-016019-0. Vol. 2 of the [[Course of Theoretical Physics]]. <!-- Probably worth noting that this is an advanced undergraduate / graduate level textbook, not something which is aimed at the casual reader! -->
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| ==External links==
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| * [http://www-dick.chemie.uni-regensburg.de/group/stephan_baeurle/index.html Particle and Polymer Field Theories]
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| {{DEFAULTSORT:Field (Physics)}}
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| [[Category:Theoretical physics]]
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| [[Category:Concepts in physics]]
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