Photon 23535 225333096 2008-07-13T03:45:09Z Barak Sh 7058953 Wikification {{otheruses}} {{Infobox Particle | bgcolour = | name = Photon | image = [[Image:Military laser experiment.jpg|275px]] | caption = Photons emitted in a [[Coherence (physics)|coherent]] beam from a [[laser]] | num_types = | composition = [[Elementary particle]] | family = [[Boson]] | group = [[Gauge boson]] | generation = | interaction = [[Electromagnetism|Electromagnetic]] | theorized = [[Albert Einstein]] (1905–17) | discovered = | symbol = <math>\gamma</math> or <math>h\nu</math> | mass = 0<ref name=BJ545>{{cite book |author=B.H. Bransden and C.J. Joachain |title=Quantum Mechanics |edition=2e |id=ISBN 0-582-35691-1 |pages=545}}</ref> | mean_lifetime = Stable<ref>[http://pdg.lbl.gov/2005/tables/gxxx.pdf Official particle table for gauge and Higgs bosons] Retrieved [[October 24]], [[2006]]</ref> | decay_particle = | electric_charge = 0 | color_charge = | spin = 1<ref name=BJ545 /> | num_spin_states = }} In [[physics]], the '''photon''' is the [[elementary particle]] responsible for [[electromagnetism|electromagnetic]] phenomena. It is the [[Force carrier|carrier]] of [[electromagnetic radiation]] of all [[wavelength]]s, including [[gamma ray]]s, [[X-ray]]s, [[ultraviolet|ultraviolet light]], [[visible light]], [[infrared|infrared light]], [[microwave]]s, and [[radio|radio waves]]. The photon differs from many other elementary particles, such as the [[electron]] and the [[quark]], in that it has zero [[rest mass]];<ref name="rel_mass" /> therefore, it travels (in a [[vacuum]]) at the [[speed of light]], ''c''. Like all [[quantum|quanta]], the photon has both wave and particle properties (“[[wave–particle duality]]”). Photons show wave-like phenomena, such as [[refraction]] by a [[Lens (optics)|lens]] and destructive [[interference]] when reflected waves cancel each other out; however, as a particle, it can only interact with [[matter]] by transferring the amount of [[energy]] :<math>E = \frac{hc}{\lambda},</math> where <math>h</math> is [[Planck's constant]], <math>c</math> is the [[speed of light]], and <math>\lambda</math> is its [[wavelength]]. This is different from a classical wave, which may gain or lose arbitrary amounts of energy. For visible light the energy carried by a single photon is around 4×10<sup>–19</sup> [[joule]]s; this energy is just sufficient to excite a single [[molecule]] in a [[photoreceptor cell]] of an [[eye]], thus contributing to [[visual perception|vision]].<ref>Vimal, R. L. P., Pokorny, J., Smith, V. C., & Shevell, S. K. (1989). Foveal cone thresholds. Vision Res, 29(1), 61-78.http://www.geocities.com/vri98/Vimal-foveal-cone-ratio-VR-1989</ref> Apart from having energy, a photon also carries [[momentum]] and has a [[polarization]]. It follows the laws of [[quantum mechanics]], which means that often these properties do not have a well-defined value for a given photon. Rather, they are defined as a probability to measure a certain polarization, position, or momentum. For example, although a photon can excite a single molecule, it is often impossible to predict beforehand ''which'' molecule will be excited. The above description of a photon as a carrier of electromagnetic radiation is commonly used by physicists. However, in theoretical physics, a photon can be considered as a mediator for any type of electromagnetic interactions, including magnetic fields and electrostatic repulsion between like charges. The modern concept of the photon was developed gradually (1905–17) by [[Albert Einstein]]<ref name="Einstein1905">{{cite journal | last = Einstein | first = A | authorlink = Albert Einstein | year = 1905 | title = Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt (trans. A Heuristic Model of the Creation and Transformation of Light) | journal = [[Annalen der Physik]] | volume = 17 | pages = 132–148}} {{de icon}}. An [[s:A Heuristic Model of the Creation and Transformation of Light|English translation]] is available from [[Wikisource]].</ref><ref name="Einstein1909">{{cite journal | last = Einstein | first = A | authorlink = Albert Einstein | year = 1909 | title = Über die Entwicklung unserer Anschauungen über das Wesen und die Konstitution der Strahlung (trans. The Development of Our Views on the Composition and Essence of Radiation) | journal = Physikalische Zeitschrift | volume = 10|pages = 817–825}} {{de icon}}. An [[s:The Development of Our Views on the Composition and Essence of Radiation|English translation]] is available from [[Wikisource]]. </ref><ref name="Einstein1916a">{{cite journal | last = Einstein | first = A | authorlink = Albert Einstein | year = 1916a | title = Strahlungs-emission und -absorption nach der Quantentheorie | journal = Verhandlungen der Deutschen Physikalischen Gesellschaft | volume = 18 | pages = 318}} {{de icon}}</ref><ref name="Einstein1916b">{{cite journal | last = Einstein | first = A | authorlink = Albert Einstein | year = 1916b | title = Zur Quantentheorie der Strahlung | journal = Mitteilungen der Physikalischen Gesellschaft zu Zürich | volume = 16 | pages = 47}} Also ''Physikalische Zeitschrift'', '''18''', 121–128 (1917). {{de icon}}</ref> to explain experimental observations that did not fit the classical [[electromagnetic wave equation|wave model]] of light. In particular, the photon model accounted for the frequency dependence of light's energy, and explained the ability of [[matter]] and [[electromagnetic radiation|radiation]] to be in [[thermal equilibrium]]. Other physicists sought to explain these anomalous observations by ''semiclassical models'', in which light is still described by [[Maxwell's equations]], but the material objects that emit and absorb light are quantized. Although these semiclassical models contributed to the development of [[quantum mechanics]], further experiments proved Einstein's hypothesis that ''light itself'' is [[quantization (physics)|quantized]]; the [[quantum|quanta]] of light are photons. The photon concept has led to momentous advances in experimental and theoretical physics, such as [[laser]]s, [[Bose–Einstein condensation]], [[quantum field theory]], and the [[probability amplitude|probabilistic interpretation]] of quantum mechanics. According to the [[Standard Model]] of [[particle physics]], photons are responsible for producing all [[electric field|electric]] and [[magnetic field]]s, and are themselves the product of requiring that physical laws have a certain [[Symmetry in physics|symmetry]] at every point in [[spacetime]]. The intrinsic properties of photons—such as [[electric charge|charge]], [[invariant mass|mass]] and [[spin (physics)|spin]]—are determined by the properties of this [[gauge theory|gauge symmetry]]. The concept of photons is applied to many areas such as [[photochemistry]], [[two-photon excitation microscopy|high-resolution microscopy]], and [[fluorescence resonance energy transfer|measurements of molecular distances]]. Recently, photons have been studied as elements of [[quantum computer]]s and for sophisticated applications in [[optical communication]] such as [[quantum cryptography]]. ==Nomenclature== The photon was originally called a '''“light quantum”''' (''das Lichtquant'') by [[Albert Einstein]].<ref name="Einstein1905" /> The modern name “photon” derives from the [[Greek language|Greek word]] for light, ''{{polytonic|φῶς}}'', (transliterated ''phôs''), and was coined in 1926 by the physical chemist [[Gilbert N. Lewis]], who published a speculative theory<ref name="Lewis1926">{{cite journal | last=Lewis | first = GN | authorlink = Gilbert N. Lewis | title = The conservation of photons|journal = [[Nature (journal)|Nature]] | year = 1926 | volume = 118 | pages = 874–875 | doi = 10.1038/118874a0}}</ref> in which photons were “uncreatable and indestructible”. Although Lewis' theory was never accepted—being contradicted by many experiments—his new name, ''photon'', was adopted immediately by most physicists. [[Isaac Asimov]] credits [[Arthur Compton]] with defining quanta of light as photons in 1927.<ref>{{cite book | title = The Neutrino, Ghost Particle of the Atom | author = Isaac Asimov | publisher = Doubleday | year = 1966 }}</ref><ref>{{cite book | title = The Universe From Flat Earth To Quasar | author = Isaac Asimov | publisher = Avon Books | year = 1968 | isbn = | url = }}</ref> In physics, a photon is usually denoted by the symbol <math>\gamma</math>, the [[Greek alphabet|Greek letter]] [[gamma]]. This symbol for the photon probably derives from [[gamma ray]]s, which were discovered and named in 1900 by [[Paul Ulrich Villard|Villard]]<ref>{{cite journal | last = Villard | first = P | authorlink = Paul Ulrich Villard | year = 1900 | title = Sur la réflexion et la réfraction des rayons cathodiques et des rayons déviables du radium | journal = Comptes Rendus | volume = 130 | pages = 1010–1012}} {{fr icon}}</ref><ref>{{cite journal | last = Villard | first = P | authorlink = Paul Ulrich Villard | year = 1900 | title = Sur le rayonnement du radium | journal = Comptes Rendus | volume = 130 | pages = 1178–1179}} {{fr icon}}</ref> and shown to be a form of [[electromagnetic radiation]] in 1914 by [[Ernest Rutherford|Rutherford]] and [[Edward Andrade|Andrade]].<ref>{{cite journal | last = Rutherford | first = E | authorlink = Ernest Rutherford | coauthors = [[Edward Andrade|Andrade ENC]] | year = 1914 | title = The Wavelength of the Soft Gamma Rays from Radium B | journal = Philosophical Magazine | volume = 27|pages = 854–868}}</ref> In [[chemistry]] and [[optical engineering]], photons are usually symbolized by <math>h \nu</math>, the energy of a photon, where <math>h</math> is [[Planck's constant]] and the [[Greek alphabet|Greek letter]] <math>\nu</math> ([[Nu (letter)|nu]]) is the photon's [[frequency]]. Much less commonly, the photon can be symbolized by ''hf'', where its frequency is denoted by ''f''. ==Physical properties== [[Image:Electron-positron-scattering.svg|220px|thumb|right|A [[Feynman diagram]] of the exchange of a virtual photon (symbolized by a wavy-line and a gamma, <math>\gamma</math>) between a [[positron]] and an [[electron]].]] {{See also|Special relativity}} The photon is [[invariant mass|massless]],<ref name="rel_mass">The [[invariant mass|mass]] of the photon is believed to be exactly zero, based on experiment and theoretical considerations described in the article. Some sources also refer to the "[[relativistic mass]]" concept, which is just the energy scaled to units of mass. For a photon with wavelength ''&lambda;'' or energy ''E'', this is ''h/&lambda;c'' or ''E''/''c''<sup>2</sup>. This usage for the term "mass" is no longer common in scientific literature.</ref> has no [[electric charge]]<ref name="chargeless">{{cite journal | last = Kobychev | first = V V | coauthors = Popov, S B | year = 2005 | title = Constraints on the photon charge from observations of extragalactic sources | journal = Astronomy Letters | volume = 31 | pages = 147–151|doi = 10.1134/1.1883345 }}</ref> and does not decay spontaneously in empty space. A photon has two possible [[polarization]] states and is described by exactly three continuous parameters: the components of its [[wave vector]], which determine its wavelength <math>\lambda</math> and its direction of propagation. The photon is the [[gauge boson]] for [[electromagnetism]], and therefore all other quantum numbers—such as [[lepton number]], [[baryon number]], or [[strangeness]]—are exactly zero. Photons are emitted in many natural processes, e.g., when a charge is accelerated, during a molecular, atomic or nuclear transition to a lower energy level, or when [[electron-positron annihilation|a particle and its antiparticle are annihilated]]. Photons are absorbed in the [[T-symmetry|time-reversed]] processes which correspond to those mentioned above: for example, in the [[pair production|production of particle–antiparticle pairs]] or in molecular, atomic or nuclear transitions to a higher energy level. In empty space, the photon moves at <math>c</math> (the [[speed of light]]) and its [[energy]] <math>E</math> and [[momentum]] '''p''' are related by <math>E = c p</math>, where <math>p</math> is the magnitude of the momentum. For comparison, the corresponding equation for particles with a [[mass]] <math>m </math> is <math>E^{2} = c^{2} p^{2} + m^{2} c^{4}</math>, as shown in [[special relativity]]. The energy and momentum of a photon depend only on its [[frequency]] <math>\nu</math> or, equivalently, its [[wavelength]] <math>\lambda</math> :<math> E = \hbar\omega = h\nu = \frac{h c}{\lambda} </math> :<math>\mathbf{p} = \hbar\mathbf{k}</math> and consequently the magnitude of the momentum is :<math> p = \hbar k = \frac{h}{\lambda} = \frac{h\nu}{c} </math> where <math>\hbar = h/2\pi \!</math> (known as [[Planck's constant|Dirac's constant or Planck's reduced constant]]); '''k''' is the [[wave vector]] (with the wave number <math>k = 2\pi/\lambda</math> as its magnitude) and <math>\omega = 2\pi\nu</math> is the [[angular frequency]]. Notice that '''k''' points in the direction of the photon's propagation. The photon also carries [[spin (physics)|spin angular momentum]] that does not depend on its frequency.<ref name="spin">This property was proved experimentally by Raman in 1931. {{Citation | last = Raman | first = C.V. | author-link = C. V. Raman | journal = Ind. Jour. Phy. | volume = 6 | pages = 353 | year = Oct. 1931 }} ; {{Citation | last = Raman | first = C.V. | author-link = C. V. Raman | journal = Nature | volume = 128 | pages = 576 and 727 | year = 1931 }} ; {{Citation | last = Raman | first = C.V. | author-link = C. V. Raman | last2 = Bhagavantam | first2 = S. | journal = Nature | volume = 129 | pages = 22–23 | year = 1932 | url = http://www.nature.com/physics/looking-back/raman2/index.html | doi = 10.1038/129022a0 | title = Experimental Proof of the Spin of the Photon. }}. </ref> The magnitude of its spin is <math>\sqrt{2} \hbar </math> and the component measured along its direction of motion, its [[helicity (particle physics)|helicity]], must be <math>\pm\hbar</math>. These two possible helicities correspond to the two possible [[circular polarization]] states of the photon (right-handed and left-handed). To illustrate the significance of these formulae, the [[electron-positron annihilation|annihilation of a particle with its antiparticle]] must result in the creation of at least ''two'' photons for the following reason. In the [[center of mass]] [[frame of reference|frame]], the colliding antiparticles have no net momentum, whereas a single photon always has momentum. Hence, [[momentum|conservation of momentum]] requires that at least two photons are created, with zero net momentum. The energy of the two photons—or, equivalently, their frequency—may be determined from [[conservation law|conservation of four-momentum]]. Seen another way, the photon can be considered as its own antiparticle. The reverse process, [[pair production]], is the dominant mechanism by which high-energy photons such as [[gamma ray]]s lose energy while passing through matter. The classical formulae for the energy and momentum of [[electromagnetic radiation]] can be re-expressed in terms of photon events. For example, the [[radiation pressure|pressure]] of electromagnetic radiation on an object derives from the transfer of photon momentum per unit time and unit area to that object, since pressure is force per unit area and force is the change in [[momentum]] per unit time. ==Historical development== {{main|Light}} [[Image:Young Diffraction.png|thumb|200px|left|[[Thomas Young (scientist)|Thomas Young]]'s [[double-slit experiment]] in 1805 showed that light can act as a [[wave]], helping to defeat early [[elementary particle|particle]] theories of light.]] In most theories up to the eighteenth century, light was pictured as being made up of particles. One of the earliest particle thoeries was described in the ''[[Book of Optics]]'' (1021) by [[Ibn al-Haytham|Alhazen]], who held [[Ray (optics)|light rays]] to be streams of minute particles that "lack all sensible qualities except energy."<ref name=Rashed>{{Citation |last=Rashed |first=Roshdi |year=2007 |title=The Celestial Kinematics of Ibn al-Haytham |journal=Arabic Sciences and Philosophy |volume=17 |pages=7–55 [19] |publisher=[[Cambridge University Press]] |doi=10.1017/S0957423907000355 }}: {{quote|"In his optics ‘‘the smallest parts of light’’, as he calls them, retain only properties that can be treated by geometry and verified by experiment; they lack all sensible qualities except energy."}}</ref> Since particle models cannot easily account for the [[refraction]], [[diffraction]] and [[birefringence]] of light, wave theories of light were proposed by [[René Descartes]] (1637),<ref>{{cite book | last = Descartes | first = R | authorlink = René Descartes | title = Discours de la méthode ([[Discourse on Method]]) | year = 1637}} {{fr icon}}</ref> [[Robert Hooke]] (1665),<ref>{{cite book | last = Hooke | first = R | authorlink = Robert Hooke | year = 1665 | url = http://digital.library.wisc.edu/1711.dl/HistSciTech.HookeMicro '| title = "Micrographia: or some physiological descriptions of minute bodies made by magnifying glasses with observations and inquiries thereupon...''}}</ref> and [[Christian Huygens]] (1678);<ref>{{cite book | last = Huygens | first = C | authorlink = Christian Huygens | year = 1678 | title = Traite de la lumiere (trans. Treatise on Light) }} {{fr icon}}. An [http://www.gutenberg.org/etext/14725 English translation] is available from [[Project Gutenberg]]</ref> however, particle models remained dominant, chiefly due to the influence of [[Isaac Newton]].<ref name="Newton1730">{{cite book | last = Newton | first = I | authorlink = Isaac Newton | year = 1730 | title = Opticks | edition=4th edition | pages=Book II, Part III, Propositions XII–XX; Queries 25–29 | publisher = Dover Publications | id=ISBN 0-486-60205-2}}</ref> In the early nineteenth century, [[Thomas Young (scientist)|Thomas Young]] and [[Augustin-Jean Fresnel|August Fresnel]] clearly demonstrated the [[interference]] and diffraction of light and by 1850 wave models were generally accepted.<ref>{{cite book | last = Buchwald | first = Jed Z. | year = 1989 | title = The Rise of the Wave Theory of Light: Optical Theory and Experiment in the Early Nineteenth Century | publisher = University of Chicago Press | id=ISBN 0-226-07886-8}}</ref> In 1865, [[James Clerk Maxwell]]'s [[Maxwell's equations|prediction]]<ref name="maxwell">{{cite journal | last = Maxwell | first = JC | authorlink = James Clerk Maxwell | year = 1865 | title = [[A Dynamical Theory of the Electromagnetic Field]] | journal = Philosophical Transactions of the Royal Society of London | volume = 155|pages = 459–512 | doi = 10.1098/rstl.1865.0008}} This article followed a presentation by Maxwell on [[8 December]] [[1864]] to the Royal Society.</ref> that light was an electromagnetic wave—which was confirmed experimentally in 1888 by [[Heinrich Hertz]]'s detection of [[radio|radio waves]]<ref name="hertz">{{cite journal | last = Hertz | first = H | authorlink = Heinrich Hertz | year = 1888 | title = Über Strahlen elektrischer Kraft | journal = Sitzungsberichte der Preussischen Akademie der Wissenschaften (Berlin) | volume = 1888 | pages = 1297–1307}} {{de icon}}</ref>—seemed to be the final blow to particle models of light. [[Image:Light-wave.svg|thumb|340px|right|In 1900, [[James Clerk Maxwell|Maxwell's]] [[Maxwell's equations|theoretical model of light]] as oscillating [[electric field|electric]] and [[magnetic field]]s seemed complete. However, several observations could not be explained by any wave model of [[electromagnetic radiation]], leading to the idea that light-energy was packaged into ''quanta'' described by E=hν. Later experiments showed that these light-quanta also carry momentum and, thus, can be considered [[elementary particle|particles]]: the ''photon'' concept was born, leading to a deeper understanding of the electric and magnetic fields themselves.]] The [[electromagnetic wave equation|Maxwell wave theory]], however, does not account for ''all'' properties of light. The Maxwell theory predicts that the energy of a light wave depends only on its [[intensity]], not on its [[frequency]]; nevertheless, several independent types of experiments show that the energy imparted by light to atoms depends only on the light's frequency, not on its intensity. For example, [[photochemistry|some chemical reactions]] are provoked only by light of frequency higher than a certain threshold; light of frequency lower than the threshold, no matter how intense, does not initiate the reaction. Similarly, electrons can be ejected from a metal plate by shining light of sufficiently high frequency on it (the [[photoelectric effect]]); the energy of the ejected electron is related only to the light's frequency, not to its intensity. At the same time, investigations of [[blackbody radiation]] carried out over four decades (1860–1900) by various researchers<ref name="Wien1911">{{cite web | url = http://nobelprize.org/nobel_prizes/physics/laureates/1911/wien-lecture.html | title = Wilhelm Wien Nobel Lecture}} Delivered [[11 December]] [[1911]].</ref> culminated in [[Max Planck]]'s [[Planck's constant|hypothesis]]<ref name="Planck1901">{{cite journal | last = Planck | first = M | authorlink = Max Planck | year = 1901 | title = Über das Gesetz der Energieverteilung im Normalspectrum | journal = [[Annalen der Physik]] | volume = 4 | pages = 553–563 | doi = 10.1002/andp.19013090310}} {{de icon}}</ref><ref name="Planck1918">{{cite web | url = http://nobelprize.org/nobel_prizes/physics/laureates/1918/planck-lecture.html|title = Max Planck's Nobel Lecture}} Delivered [[2 June]] [[1920]].</ref> that the energy of ''any'' system that absorbs or emits electromagnetic radiation of frequency <math>\nu </math> is an integer multiple of an energy quantum <math>E = h\nu </math>. As shown by [[Albert Einstein]],<ref name="Einstein1905" /><ref name="Einstein1909" /> some form of energy quantization ''must'' be assumed to account for the thermal equilibrium observed between matter and [[electromagnetic radiation]]; for this explanation of the [[photoelectric effect]], Einstein received the 1921 [[Nobel Prize]] in physics. Since the Maxwell theory of light allows for all possible energies of electromagnetic radiation, most physicists assumed initially that the energy quantization resulted from some unknown constraint on the matter that absorbs or emits the radiation. In 1905, Einstein was the first to propose that energy quantization was a property of electromagnetic radiation itself.<ref name="Einstein1905" /> Although he accepted the validity of Maxwell's theory, Einstein pointed out that many anomalous experiments could be explained if the ''energy'' of a Maxwellian light wave were localized into point-like quanta that move independently of one another, even if the wave itself is spread continuously over space.<ref name="Einstein1905" /> In 1909<ref name="Einstein1909" /> and 1916,<ref name="Einstein1916b" /> Einstein showed that, if [[Planck's law of black-body radiation]] is accepted, the energy quanta must also carry [[momentum]] <math>p=h/\lambda</math>, making them full-fledged [[elementary particle|particles]]. This photon momentum was observed experimentally<ref name="Compton1923">{{cite journal | last = Compton | first = A | authorlink = Arthur Compton | year = 1923 | title = [http://www.aip.org/history/gap/Compton/01_Compton.html A Quantum Theory of the Scattering of X-rays by Light Elements] | journal = [[Physical Review]] | volume = 21 | pages = 483–502 | doi = 10.1103/PhysRev.21.483}}</ref> by [[Arthur Compton]], for which he received the [[Nobel Prize]] in 1927. The pivotal question was then: how to unify Maxwell's wave theory of light with its experimentally observed particle nature? The answer to this question occupied [[Albert Einstein]] for the rest of his life,<ref name="Pais1982">{{cite book | last = Pais | first = A | authorlink = Abraham Pais | year = 1982 | title = Subtle is the Lord: The Science and the Life of Albert Einstein|publisher = Oxford University Press }}</ref> and was solved in [[quantum electrodynamics]] and its successor, the [[Standard Model]]. ==Early objections== [[Image:Bohr-atom-PAR.svg|thumb|250px|left|Up to 1923, most physicists were reluctant to accept that light itself was quantized. Instead, they tried to explain photon behavior by quantizing only ''matter'', as in the [[Bohr model]] of the [[hydrogen atom]] (shown here). Even though these semiclassical models were only a first approximation, they were accurate for simple systems and they led to [[quantum mechanics]].]] Einstein's 1905 predictions were verified experimentally in several ways in the first two decades of the 20th century, as recounted in [[Robert Millikan]]'s Nobel lecture.<ref name="Millikan1923">{{cite web | url = http://nobelprize.org/nobel_prizes/physics/laureates/1923/millikan-lecture.html|title = Robert A. Millikan's Nobel Lecture}} Delivered [[23 May]] [[1924]].</ref> However, before [[Compton scattering|Compton's experiment]]<ref name="Compton1923" /> showing that photons carried [[momentum]] proportional to their [[frequency]] (1922), most physicists were reluctant to believe that [[electromagnetic radiation]] itself might be particulate. (See, for example, the Nobel lectures of [[Wilhelm Wien|Wien]],<ref name="Wien1911" /> [[Max Planck|Planck]]<ref name="Planck1918" /> and Millikan.<ref name="Millikan1923" />). Many physicists assumed instead that energy quantization resulted from some unknown constraint on the matter that absorbs or emits radiation. This was [[Niels Bohr]]'s point of view, when he developed his atomic model with discrete energy levels that could account qualitatively for the sharp spectral lines and energy quantization observed in the [[Emission (electromagnetic radiation)|emission]] and [[Absorption (electromagnetic radiation)|absorption]] of light by atoms. It was only the Compton scattering of a photon by a ''free'' electron, that convinced most physicists that light itself was quantized. A free electron has no energy levels since it has no internal structure. Even after Compton's experiment, Bohr, [[Hendrik Anthony Kramers|Hendrik Kramers]] and [[John C. Slater|John Slater]] made one last attempt to preserve the Maxwellian continuous electromagnetic field model of light, the so-called [[BKS theory|BKS model]].<ref name="Bohr1924">{{cite journal | last = Bohr | first = N | authorlink = Niels Bohr | coauthors = [[Hendrik Anthony Kramers|Kramers HA]] and [[John C. Slater|Slater JC]] | year = 1924 | title = The Quantum Theory of Radiation|journal = [[Philosophical Magazine]] | volume = 47 | pages = 785–802}} Also ''[[Zeitschrift für Physik]]'', '''24''', 69 (1924).</ref> To account for the then-available data, two drastic hypotheses had to be made: * ''Energy and momentum are conserved only on the average in interactions between matter and radiation, not in elementary processes such as absorption and emission''. This allows one to reconcile the discontinuously changing energy of the atom (jump between energy states) with the continuous release of energy into radiation. * ''Causality is abandoned''. For example, [[spontaneous emission]]s are merely [[stimulated emission|emissions induced]] by a "virtual" electromagnetic field. However, refined Compton experiments showed that energy-momentum is conserved extraordinarily well in elementary processes; and also that the jolting of the electron and the generation of a new photon in [[Compton scattering]] obey causality to within 10 [[picosecond|ps]]. Accordingly, Bohr and his co-workers gave their model “as honorable a funeral as possible“.<ref name="Pais1982" /> Nevertheless, the failures of the BKS model inspired [[Werner Heisenberg]] in his development<ref name="Heisenberg1932">[http://nobelprize.org/nobel_prizes/physics/laureates/1932/heisenberg-lecture.html Heisenberg Nobel lecture], delivered [[11 December]] [[1933]].</ref> of [[matrix mechanics]]. A few physicists persisted<ref name="Mandel1976">{{cite journal | last = Mandel | first = L | authorlink = Leonard Mandel | year = 1976 | title = The case for and against semiclassical radiation theory | journal = Progress in Optics | editor = E. Wolf, ed. | publisher = North-Holland | volume = XIII | pages = 27–69}}</ref> in developing semiclassical models in which [[electromagnetic radiation]] is not quantized, but matter obeys the laws of [[quantum mechanics]]. Although the evidence for photons from chemical and physical experiments was overwhelming by the 1970s, this evidence could not be considered as ''absolutely'' definitive; since it relied on the interaction of light with matter, a sufficiently complicated theory of matter could in principle account for the evidence. Nevertheless, ''all'' semiclassical theories were refuted definitively in the 1970s and 1980s by elegant photon-correlation experiments.<ref name="exp_proof">These experiments produce results that cannot be explained by any classical theory of light, since they involve anticorrelations that result from the [[measurement in quantum mechanics|quantum measurement process]]. In 1974, the first such experiment was carried out by Clauser, who reported a violation of a classical [[Cauchy–Schwarz inequality]]. In 1977, Kimble ''et al.'' demonstrated an analogous anti-bunching effect of photons interacting with a beam splitter; this approach was simplified and sources of error eliminated in the photon-anticorrelation experiment of Grangier ''et al.'' (1986). This work is reviewed and simplified further in Thorn ''et al.'' (2004). (These references are listed below under '''Additional references'''.)</ref> Hence, Einstein's hypothesis that quantization is a property of light itself is considered to be proven. ==Wave–particle duality and uncertainty principles== {{See also|Wave–particle duality|Squeezed coherent state|Uncertainty principle}} Photons, like all quantum objects, exhibit both wave-like and particle-like properties. Their dual wave–particle nature can be difficult to visualize. The photon displays clearly wave-like phenomena such as [[diffraction]] and [[interference]] on the length scale of its wavelength. For example, a single photon passing through a [[double-slit experiment]] lands on the screen with a [[probability distribution]] given by its interference pattern determined by [[Maxwell's equations]].<ref name="Taylor1909">{{cite journal | last = Taylor | first = GI | authorlink = Geoffrey Ingram Taylor | year = 1909 | title = Interference fringes with feeble light | journal = Proceedings of the Cambridge Philosophical Society | volume = 15|pages = 114–115}}</ref> However, experiments confirm that the photon is ''not'' a short pulse of electromagnetic radiation; it does not spread out as it propagates, nor does it divide when it encounters a [[beam splitter]]. Rather, the photon seems like a [[point-like particle]], since it is absorbed or emitted ''as a whole'' by arbitrarily small systems, systems much smaller than its wavelength, such as an atomic nucleus (≈10<sup>–15</sup> m across) or even the point-like [[electron]]. Nevertheless, the photon is ''not'' a point-like particle whose trajectory is shaped probabilistically by the [[electromagnetic field]], as conceived by [[Albert Einstein|Einstein]] and others; that hypothesis was also refuted by the photon-correlation experiments cited above.<ref name="exp_proof" /> According to our present understanding, the electromagnetic field itself is produced by photons, which in turn result from a local [[gauge symmetry]] and the laws of [[quantum field theory]] (see the [[Photon#Second quantization|Second quantization]] and [[Photon#The photon as a gauge boson|Gauge boson]] sections below). [[Image:Heisenberg gamma ray microscope.png|thumb|200px|right|[[Werner Heisenberg|Heisenberg's]] [[thought experiment]] for locating an [[electron]] (shown in blue) with a high-resolution gamma-ray microscope. The incoming [[gamma ray]] (shown in green) is scattered by the electron up into the microscope's [[angular aperture|aperture angle]] θ. The scattered gamma ray is shown in red. [[Optics|Classical optics]] shows that the electron position can be resolved only up to an uncertainty Δx that depends on θ and the [[wavelength]] λ of the incoming light.]] A key element of [[quantum mechanics]] is [[Werner Heisenberg|Heisenberg's]] [[uncertainty principle]], which forbids the simultaneous measurement of the position and momentum of a particle along the same direction. Remarkably, the uncertainty principle for charged, material particles ''requires'' the quantization of light into photons, and even the frequency dependence of the photon's energy and momentum. An elegant illustration is Heisenberg's [[thought experiment]] for locating an electron with an ideal microscope.<ref name="Heisenberg1927">{{cite journal | last = Heisenberg | first = W | authorlink = Werner Heisenberg | year = 1927 | title = Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik | journal = Zeitschrift für Physik | volume = 43|pages = 172–198 | doi = 10.1007/BF01397280}} {{de icon}}</ref> The position of the electron can be determined to within the [[angular resolution|resolving power]] of the microscope, which is given by a formula from classical [[optics]] :<math> \Delta x \sim \frac{\lambda}{\sin \theta} </math> where <math>\theta</math> is the [[angular aperture|aperture angle]] of the microscope. Thus, the position uncertainty <math>\Delta x</math> can be made arbitrarily small by reducing the wavelength. The momentum of the electron is uncertain, since it received a “kick” <math>\Delta p</math> from the light scattering from it into the microscope. If light were ''not'' quantized into photons, the uncertainty <math>\Delta p</math> could be made arbitrarily small by reducing the light's intensity. In that case, since the wavelength and intensity of light can be varied independently, one could simultaneously determine the position and momentum to arbitrarily high accuracy, violating the [[uncertainty principle]]. By contrast, Einstein's formula for photon momentum preserves the uncertainty principle; since the photon is scattered anywhere within the aperture, the uncertainty of momentum transferred equals :<math> \Delta p \sim p_{\mathrm{photon}} \sin\theta = \frac{h}{\lambda} \sin\theta </math> giving the product <math>\Delta x \Delta p \, \sim \, h</math>, which is Heisenberg's uncertainty principle. Thus, the entire world is quantized; both matter and fields must obey a consistent set of quantum laws, if either one is to be quantized. The analogous uncertainty principle for photons forbids the simultaneous measurement of the number <math>n</math> of photons (see [[Fock state]] and the [[Photon#Second quantization|Second quantization]] section below) in an electromagnetic wave and the phase <math>\phi</math> of that wave :<math> \Delta n \Delta \phi > 1 </math> See [[coherent state]] and [[squeezed coherent state]] for more details. Both photons and material particles such as electrons create analogous [[interference|interference patterns]] when passing through a [[double-slit experiment]]. For photons, this corresponds to the interference of a [[electromagnetic wave equation|Maxwell light wave]] whereas, for material particles, this corresponds to the interference of the [[Schrödinger equation|Schrödinger wave equation]]. Although this similarity might suggest that [[Maxwell's equations]] are simply Schrödinger's equation for photons, most physicists do not agree.<ref>{{cite book | last = Kramers | first = HA | authorlink = Hendrik Anthony Kramers | year = 1958 | title = Quantum Mechanics | publisher = North-Holland | location = Amsterdam}}</ref><ref>{{cite book | last = Bohm | first = D | authorlink = David Bohm | year = 1954 | title = Quantum Theory | publisher = Constable | location = London}}</ref> For one thing, they are mathematically different; most obviously, Schrödinger's one equation solves for a [[complex number|complex]] [[field (physics)|field]], whereas Maxwell's four equations solve for [[real number|real]] [[field (physics)|field]]s. More generally, the normal concept of a Schrödinger [[probability amplitude|probability]] [[wave function]] cannot be applied to photons.<ref>{{cite journal | last = Newton | first = TD | coauthors = [[Eugene Wigner|Wigner EP]] | year = 1949 | title = Localized states for elementary particles | journal = Reviews of Modern Physics | volume = 21 | pages = 400–406 | doi = 10.1103/RevModPhys.21.400}}</ref> Being massless, they cannot be localized without being destroyed; technically, photons cannot have a position eigenstate <math>|\mathbf{r} \rangle</math>, and, thus, the normal Heisenberg uncertainty principle <math>\Delta x \Delta p > h/2</math> does not pertain to photons. A few substitute wave functions have been suggested for the photon,<ref>{{cite journal | last = Bialynicki-Birula | first = I | year = 1994 | title = On the wave function of the photon | journal = Acta Physica Polonica A | volume = 86 | pages = 97–116}}</ref><ref>{{cite journal | last = Sipe | first = JE | year = 1995 | title = Photon wave functions | journal = Physical Review A | volume = 52 | pages = 1875–1883 | doi = 10.1103/PhysRevA.52.1875}}</ref><ref>{{cite journal | last = Bialynicki-Birula | first = I | year = 1996 | title = Photon wave function | journal = Progress in Optics | volume = 36 | pages = 245–294}}</ref><ref>{{cite book | last = Scully | first = MO | coauthors = Zubairy MS | year = 1997 | title = Quantum Optics | publisher = Cambridge University Press|location = Cambridge}}</ref> but they have not come into general use. Instead, physicists generally accept the second-quantized theory of photons described below, [[quantum electrodynamics]], in which photons are quantized excitations of electromagnetic modes. ==Bose–Einstein model of a photon gas== {{Main|Bose gas|Bose–Einstein statistics|Spin-statistics theorem}} In 1924, [[Satyendra Nath Bose]] derived [[Planck's law of black-body radiation]] without using any electromagnetism, but rather a modification of coarse-grained counting of [[phase space]].<ref name="Bose1924">{{cite journal | last = Bose | first = SN | authorlink = Satyendra Nath Bose | year = 1924 | title = Plancks Gesetz und Lichtquantenhypothese | journal = [[Zeitschrift für Physik]] | volume = 26 | pages = 178–181 | doi = 10.1007/BF01327326}} {{de icon}}</ref> Einstein showed that this modification is equivalent to assuming that photons are rigorously identical and that it implied a “mysterious non-local interaction”,<ref name="Einstein1924">{{cite journal | last = Einstein | first = A | authorlink = Albert Einstein | year = 1924 | title = Quantentheorie des einatomigen idealen Gases | journal = Sitzungsberichte der Preussischen Akademie der Wissenschaften (Berlin), Physikalisch-mathematische Klasse | volume = 1924 | pages = 261–267}} {{de icon}}</ref><ref name="Einstein1925">{{cite journal | last = Einstein | first = A | authorlink = Albert Einstein | year = 1925 | title = Quantentheorie des einatomigen idealen Gases, Zweite Abhandlung | journal = Sitzungsberichte der Preussischen Akademie der Wissenschaften (Berlin), Physikalisch-mathematische Klasse | volume = 1925 | pages = 3–14}} {{de icon}}</ref> now understood as the requirement for a [[identical particles|symmetric quantum mechanical state]]. This work led to the concept of [[coherent state]]s and the development of the laser. In the same papers, Einstein extended Bose's formalism to material particles ([[boson]]s) and predicted that they would condense into their lowest quantum state at low enough temperatures; this [[Bose–Einstein condensate|Bose–Einstein condensation]] was observed experimentally in 1995.<ref>{{cite journal | last = Anderson | first = MH|coauthors = Ensher JR, Matthews MR, [[Carl Wieman|Wieman CE]], and [[Eric Allin Cornell|Cornell EA]] | title=Observation of Bose–Einstein Condensation in a Dilute Atomic Vapor | journal=Science | year=1995 | volume=269 | pages=198–201 | url=http://links.jstor.org/sici?sici=0036-8075%2819950714%293%3A269%3A5221%3C198%3AOOBCIA%3E2.0.CO%3B2-G | doi=10.1126/science.269.5221.198 | pmid=17789847 }}</ref> Photons must obey [[Bose–Einstein statistics]] if they are to allow the [[superposition principle]] of [[electromagnetic field]]s, the condition that [[Maxwell's equations]] are linear. All particles are divided into [[boson]]s and [[fermion]]s, depending on whether they have integer or half-integer [[spin (physics)|spin]], respectively. The [[spin-statistics theorem]] shows that all bosons obey Bose–Einstein statistics, whereas all fermions obey [[Fermi-Dirac statistics]] or, equivalently, the [[Pauli exclusion principle]], which states that at most one particle can occupy any given state. Thus, if the photon were a fermion, only one photon could move in a particular direction at a time. This is inconsistent with the experimental observation that lasers can produce coherent light of arbitrary intensity, that is, with many photons moving in the same direction. Hence, the photon must be a boson and obey Bose–Einstein statistics. ==Stimulated and spontaneous emission== {{main|Stimulated emission|Laser}} [[Image:Stimulatedemission.png|thumb|400px|right|[[Stimulated emission]] (in which photons “clone” themselves) was predicted by Einstein in his kinetic analysis, and led to the development of the [[laser]]. Einstein's derivation inspired further developments in the quantum treatment of light, which led to the statistical interpretation of quantum mechanics.]] In 1916, Einstein showed that Planck's radiation law implied a relation between the rates at which atoms emit and absorb photons. The condition follows from the assumption that light is emitted and absorbed by atoms independently, and that the thermal equilibrium is preserved by interaction with atoms.<ref name="Einstein1916a" /> Consider a cavity in [[thermal equilibrium]] and filled with [[electromagnetic radiation]] and atoms that can emit and absorb that radiation. Thermal equilibrium requires that the number density <math>\rho(\nu)</math> of photons with frequency <math>\nu</math> is constant in time; hence, the rate of ''emitting'' photons of that frequency must equal the rate of ''absorbing'' them. Einstein hypothesized that the rate <math>R_{ji}</math> for a system to ''absorb'' a photon of frequency <math>\nu</math> and transition from a lower energy <math>E_{j}</math> to a higher energy <math>E_{i}</math> was proportional to the number <math>N_{j}</math> of molecules with energy <math>E_{j}</math> and to the number density <math>\rho(\nu)</math> of ambient photons with that frequency :<math> R_{ji} = N_{j} B_{ji} \rho(\nu) \! </math> where <math>B_{ji}</math> is the [[rate constant]] for absorption. More daringly, Einstein hypothesized that the reverse rate <math>R_{ij}</math> for a system to ''emit'' a photon of frequency <math>\nu</math> and transition from a higher energy <math>E_{i}</math> to a lower energy <math>E_{j}</math> was composed of two terms: :<math> R_{ij} = N_{i} A_{ij} + N_{i} B_{ij} \rho(\nu) \! </math> where <math>A_{ij}</math> is the rate constant for [[spontaneous emission|emitting a photon spontaneously]], and <math>B_{ij}</math> is the rate constant for emitting it in response to ambient photons ([[stimulated emission|induced or stimulated emission]]). This simple kinetic model was a powerful stimulus for research, since it was the first statistical interpretation of single-particle quantum mechanical events. Einstein was able to show that <math>B_{ij} = B_{ji}</math> (i.e., the rate constants for induced emission and absorption are equal) and, perhaps more remarkably, :<math> A_{ij} = \frac{8 \pi h \nu^{3}}{c^{3}} B_{ij}. </math> Einstein could not fully justify his rate equations, because <math>A_{ij}</math> and <math>B_{ij}</math> could only be derivable from what he called a “mechanics and electrodynamics modified to accommodate the quantum hypothesis”. In [[quantum mechanics]], Einstein's rate constant relations are a consequence of the simple form of the matrix elements for the position and momentum of a harmonic oscillator, since the electromagnetic field is a collection of harmonic oscillators. [[Paul Dirac]] derived the <math>B_{ij}</math> rate constants in 1926 using a semiclassical approach,<ref name="Dirac1926">{{cite journal | last = Dirac | first = PAM | authorlink = Paul Dirac | year = 1926 | title = On the Theory of Quantum Mechanics | journal = Proc. Roy. Soc. A | volume = 112 | pages = 661–677 | doi = 10.1098/rspa.1926.0133}}</ref> and, in 1927, succeeded in deriving ''all'' the rate constants from first principles.<ref name="Dirac1927a">{{cite journal | last = Dirac | first = PAM | authorlink = Paul Dirac | year = 1927a | title = The Quantum Theory of the Emission and Absorption of Radiation | journal = Proc. Roy. Soc. A | volume = 114 | pages = 243–265}}</ref><ref name="Dirac1927b">{{cite journal | last = Dirac | first = PAM | authorlink = Paul Dirac | year = 1927b | title = The Quantum Theory of Dispersion | journal = Proc. Roy. Soc. A | volume = 114 | pages = 710–728}}</ref> Dirac's work was the foundation of quantum electrodynamics, i.e., the quantization of the electromagnetic field itself. Dirac's approach is also called ''second quantization'' or [[quantum field theory]];<ref name="Heisenberg1929">{{cite journal | last = Heisenberg | first = W | authorlink = Werner Heisenberg | coauthors = [[Wolfgang Pauli|Pauli W]] | year = 1929 | title = Zur Quantentheorie der Wellenfelder | journal = Zeitschrift für Physik | volume = 56 | pages = 1 | doi = 10.1007/BF01340129}} {{de icon}}</ref><ref name="Heisenberg1930">{{cite journal | last = Heisenberg | first = W | authorlink = Werner Heisenberg|coauthors = [[Wolfgang Pauli|Pauli W]] | year = 1930 | title = Zur Quantentheorie der Wellenfelder | journal = Zeitschrift für Physik | volume = 59 | pages = 139 | doi = 10.1007/BF01341423}} {{de icon}}</ref><ref name="Fermi1932">{{cite journal | last = Fermi | first = E | authorlink = Enrico Fermi | year = 1932 | title = Quantum Theory of Radiation | journal = Reviews of Modern Physics | volume = 4 | pages = 87 | doi = 10.1103/RevModPhys.4.87}}</ref>, earlier quantum mechanical treatments only treat material particles as quantum mechanical, not the electromagnetic field. Einstein was troubled by the fact that his theory seemed incomplete, since it did not determine the ''direction'' of a spontaneously emitted photon. A probabilistic nature of light-particle motion was first considered by [[Isaac Newton|Newton]] in his treatment of [[birefringence]] and, more generally, of the splitting of light beams at interfaces into a transmitted beam and a reflected beam. Newton hypothesized that hidden variables in the light particle determined which path it would follow.<ref name="Newton1730" /> Similarly, Einstein hoped for a more complete theory that would leave nothing to chance, beginning his separation<ref name="Pais1982" /> from quantum mechanics. Ironically, [[Max Born]]'s [[probability amplitude|probabilistic interpretation]] of the [[wave function]]<ref name="Born1926a">{{cite journal | last = Born | first = M | authorlink = Max Born | year = 1926a | title = Zur Quantenmechanik der Stossvorgänge | journal = Zeitschrift für Physik | volume = 37 | pages = 863–867 | doi = 10.1007/BF01397477}} {{de icon}}</ref><ref name="Born1926b">{{cite journal | last = Born | first = M | authorlink = Max Born | year = 1926b | title = Zur Quantenmechanik der Stossvorgänge | journal = Zeitschrift für Physik | volume = 38 | pages = 803}} {{de icon}}</ref> was inspired by Einstein's later work searching for a more complete theory.<ref name="ghost_field">{{cite book | last = Pais | first = A | authorlink = Abraham Pais | year = 1986 | title = Inward Bound: Of Matter and Forces in the Physical World | publisher = Oxford University Press}} Specifically, Born claimed to have been inspired by Einstein's never-published attempts to develop a “ghost-field” theory, in which point-like photons are guided probabilistically by ghost fields that follow Maxwell's equations.</ref> ==Second quantization== {{main|Quantum field theory}} [[Image:Visible EM modes.png|thumb|200px|right|Different ''electromagnetic modes'' (such as those depicted here) can be treated as independent [[quantum harmonic oscillator|simple harmonic oscillators]]. A photon corresponds to a unit of energy E=hν in its electromagnetic mode.]] In 1910, [[Peter Debye]] derived [[Planck's law of black-body radiation]] from a relatively simple assumption.<ref name="Debye1910">{{cite journal | last = Debye | first = P | authorlink = Peter Debye | year = 1910 | title = Der Wahrscheinlichkeitsbegriff in der Theorie der Strahlung | journal = [[Annalen der Physik]] | volume = 33|pages = 1427–34 | doi = 10.1002/andp.19103381617}} {{de icon}}</ref> He correctly decomposed the electromagnetic field in a cavity into its [[Fourier series|Fourier modes]], and assumed that the energy in any mode was an integer multiple of <math>h\nu</math>, where <math>\nu</math> is the frequency of the electromagnetic mode. Planck's law of black-body radiation follows immediately as a geometric sum. However, Debye's approach failed to give the correct formula for the energy fluctuations of blackbody radiation, which were derived by Einstein in 1909.<ref name="Einstein1909" /> In 1925, [[Max Born|Born]], [[Werner Heisenberg|Heisenberg]] and [[Pascual Jordan|Jordan]] reinterpreted Debye's concept in a key way.<ref name="Born1925">{{cite journal | last = Born | first = M | authorlink = Max Born | coauthors = [[Werner Heisenberg|Heisenberg W]] and [[Pascual Jordan|Jordan P]] | year = 1925 | title = Quantenmechanik II | journal = Zeitschrift für Physik | volume = 35|pages = 557–615 | doi = 10.1007/BF01379806}} {{de icon}}</ref> As may be shown classically, the [[Fourier series|Fourier modes]] of the [[electromagnetic four-potential|electromagnetic field]]—a complete set of electromagnetic plane waves indexed by their wave vector '''k''' and polarization state—are equivalent to a set of uncoupled simple harmonic oscillators. Treated quantum mechanically, the energy levels of such oscillators are known to be <math>E = nh\nu</math>, where <math>\nu</math> is the oscillator frequency. The key new step was to identify an electromagnetic mode with energy <math>E = nh\nu</math> as a state with <math>n</math> photons, each of energy <math>h\nu</math>. This approach gives the correct energy fluctuation formula. [[Image:vertex correction.svg|thumb|left|In quantum field theory, the probability of an event is computed by summing the [[probability amplitude]] (a [[complex number]]) for all possible ways in which the event can occur, as in the [[Feynman diagram]] shown here; the probability equals the square of the [[absolute value|modulus]] of the total amplitude.]] [[Paul Dirac|Dirac]] took this one step further.<ref name="Dirac1927a" /><ref name="Dirac1927b" /> He treated the interaction between a charge and an electromagnetic field as a small perturbation that induces transitions in the photon states, changing the numbers of photons in the modes, while conserving energy and momentum overall. Dirac was able to derive Einstein's <math>A_{ij}</math> and <math>B_{ij}</math> coefficients from first principles, and showed that the Bose–Einstein statistics of photons is a natural consequence of quantizing the electromagnetic field correctly (Bose's reasoning went in the opposite direction; he derived [[Planck's law of black body radiation]] by ''assuming'' BE statistics). In Dirac's time, it was not yet known that all bosons, including photons, must obey BE statistics. Dirac's second-order [[perturbation theory (quantum mechanics)|perturbation theory]] can involve [[virtual particle|virtual photons]], transient intermediate states of the electromagnetic field; the static [[Coulomb's law|electric]] and [[magnetism|magnetic]] interactions are mediated by such virtual photons. In such [[quantum field theory|quantum field theories]], the [[probability amplitude]] of observable events is calculated by summing over ''all'' possible intermediate steps, even ones that are unphysical; hence, virtual photons are not constrained to satisfy <math>E = pc</math>, and may have extra [[polarization]] states; depending on the [[gauge fixing|gauge]] used, virtual photons may have three or four polarization states, instead of the two states of real photons. Although these transient virtual photons can never be observed, they contribute measurably to the probabilities of observable events. Indeed, such second-order and higher-order perturbation calculations can give apparently [[infinity|infinite]] contributions to the sum. Such unphysical results are corrected for using the technique of [[renormalization]]. Other virtual particles may contribute to the summation as well; for example, two photons may interact indirectly through virtual [[electron]]-[[positron]] [[pair production|pairs]]. In modern physics notation, the [[quantum state]] of the electromagnetic field is written as a [[Fock state]], a [[tensor product]] of the states for each electromagnetic mode :<math>|n_{k_0}\rangle\otimes|n_{k_1}\rangle\otimes\dots\otimes|n_{k_n}\rangle\dots</math> where <math>|n_{k_i}\rangle</math> represents the state in which <math>\, n_{k_i}</math> photons are in the mode <math>k_i</math>. In this notation, the creation of a new photon in mode <math>k_i</math> (e.g., emitted from an atomic transition) is written as <math>|n_{k_i}\rangle \rightarrow |n_{k_i}+1\rangle</math>. This notation merely expresses the concept of Born, Heisenberg and Jordan described above, and does not add any physics. ==The photon as a gauge boson== {{main|Gauge theory}} The electromagnetic field can be understood as a [[gauge theory]], i.e., as a field that results from requiring that symmetry hold independently at every position in [[spacetime]].<ref name="Ryder">{{cite book | last = Ryder | first = LH | year = 1996 | title = Quantum field theory | edition = 2nd edition | publisher = Cambridge University Press|id=ISBN 0-521-47814-6}}</ref> For the [[electromagnetic field]], this gauge symmetry is the [[Abelian group|Abelian]] [[unitary group|U(1) symmetry]] of a [[complex number]], which reflects the ability to vary the [[complex geometry|phase]] of a complex number without affecting [[real number]]s made from it, such as the [[energy]] or the [[Lagrangian]]. The quanta of an [[gauge theory|Abelian gauge field]] must be massless, uncharged bosons, as long as the symmetry is not broken; hence, the photon is predicted to be massless, and to have zero [[electric charge]] and integer spin. The particular form of the [[electromagnetic interaction]] specifies that the photon must have [[spin (physics)|spin]] ±1; thus, its [[helicity (particle physics)|helicity]] must be <math>\pm \hbar</math>. These two spin components correspond to the classical concepts of [[circular polarization|right-handed and left-handed circularly polarized]] light. However, the transient [[virtual photon]]s of [[quantum electrodynamics]] may also adopt unphysical polarization states.<ref name="Ryder" /> In the prevailing [[Standard Model]] of physics, the photon is one of four [[gauge bosons]] in the [[electroweak interaction]]; the [[W and Z bosons|other three]] are denoted W<sup>+</sup>, W<sup>−</sup> and Z<sup>0</sup> and are responsible for the [[weak interaction]]. Unlike the photon, these gauge bosons have [[invariant mass]], owing to a [[Higgs mechanism|mechanism]] that breaks their [[special unitary group|SU(2) gauge symmetry]]. The unification of the photon with W and Z gauge bosons in the electroweak interaction was accomplished by [[Sheldon Glashow]], [[Abdus Salam]] and [[Steven Weinberg]], for which they were awarded the 1979 [[Nobel Prize]] in physics.<ref name="Glashow">[http://nobelprize.org/nobel_prizes/physics/laureates/1979/glashow-lecture.html Sheldon Glashow Nobel lecture], delivered [[8 December]] [[1979]].</ref><ref name="Salam">[http://nobelprize.org/nobel_prizes/physics/laureates/1979/salam-lecture.html Abdus Salam Nobel lecture], delivered [[8 December]] [[1979]].</ref><ref name="Weinberg">[http://nobelprize.org/nobel_prizes/physics/laureates/1979/weinberg-lecture.html Steven Weinberg Nobel lecture], delivered [[8 December]] [[1979]].</ref> Physicists continue to hypothesize [[grand unification theory|grand unified theories]] that connect these four [[gauge bosons]] with the eight [[gluon]] gauge bosons of [[quantum chromodynamics]]; however, key predictions of these theories, such as [[proton decay]], have not been observed experimentally. ==Photon structure== {{main|Quantum Chromodynamics}} According to [[Quantum Chromodynamics]], a real photon can interact both as a point-like particle, or as a collection of [[quark]]s and [[gluon]]s, i.e., like a [[hadron]]. The structure of the photon is determined not by the traditional valence quark distributions as in a [[proton]], but by fluctuations of the point-like photon into a collection of [[parton (particle physics)|parton]]s.<ref name="sm2001">[http://www.slac.stanford.edu/grp/th/LCBook/qcd.ps.gz QCD and Two-Photon Physics], in Linear Collider Physics Resource Book for Snowmass 2001, Chapter 7, LC-REV-2001-074-US.</ref> ==Contributions to the mass of a system== {{See also|Mass in special relativity|Gravitation}} The energy of a system that emits a photon is ''decreased'' by the energy <math>E</math> of the photon as measured in the rest frame of the emitting system, which may result in a reduction in mass in the amount <math>{E}/{c^2}</math>. Similarly, the mass of a system that absorbs a photon is ''increased'' by a corresponding amount. This concept is applied in a key prediction of QED, the theory of [[quantum electrodynamics]] begun by [[Paul Dirac|Dirac]] (described above). QED is able to predict the [[anomalous magnetic dipole moment|magnetic dipole moment]] of [[lepton]]s to extremely high accuracy; experimental measurements of these magnetic dipole moments have agreed with these predictions perfectly. The predictions, however, require counting the contributions of virtual photons to the mass of the lepton. Another example of such contributions verified experimentally is the QED prediction of the [[Lamb shift]] observed in the [[hyperfine structure]] of bound lepton pairs, such as [[muonium]] and [[positronium]]. Since photons contribute to the [[stress-energy tensor]], they exert a [[gravity|gravitational attraction]] on other objects, according to the theory of [[general relativity]]. Conversely, photons are themselves affected by gravity; their normally straight trajectories may be bent by warped [[spacetime]], as in [[gravitational lensing]], and [[gravitational redshift|their frequencies may be lowered]] by moving to a higher [[potential energy|gravitational potential]], as in the [[Pound-Rebka experiment]]. However, these effects are not specific to photons; exactly the same effects would be predicted for classical [[electromagnetic radiation|electromagnetic waves]]. ==Photons in matter== {{See also|Group velocity|Photochemistry}} Light that travels through transparent matter does so at a lower speed than ''c'', the speed of light in a vacuum. For example, photons suffer so many collisions on the way from the core of the sun that radiant energy can take about a million years to reach the surface;<ref>{{cite book | title = Through the Eyes of Hubble: Birth, Life and Violent Death of Stars | author = Robert Naeye | publisher = CRC Press | year = 1998 | isbn = 0750304847 | url = http://books.google.com/books?id=06_9B7S_q_YC&pg=PA16&dq=million-year+surface+sun+photon&as_brr=3&ei=gYsyR6iELpLgtgOkttmvAQ&sig=70D3g1ajnoVyvnoY1qBAIV0yIf4 }}</ref> however, once in open space, a photon takes only 8.3 minutes to reach Earth. The factor by which the speed is decreased is called the [[refractive index]] of the material. In a classical wave picture, the slowing can be explained by the light inducing electric polarization in the matter, the polarized matter radiating new light, and the new light interfering with the original light wave to form a delayed wave. In a particle picture, the slowing can instead be described as a blending of the photon with quantum excitations of the matter ([[quasi-particle]]s such as [[phonon]]s and [[exciton]]s) to form a [[polariton]]; this polariton has a nonzero [[effective mass]], which means that it cannot travel at ''c''. Light of different frequencies may travel through matter at [[variable speed of light|different speeds]]; this is called [[dispersion (optics)|dispersion]]. The polariton propagation speed <math>v</math> equals its [[group velocity]], which is the [[derivative]] of the energy with respect to momentum. :<math> v = \frac{d\omega}{dk} = \frac{dE}{dp} </math> [[Image:RetinalCisandTrans.png|250px|right|thumb|[[Retinal]] straightens after absorbing a photon γ of the correct wavelength]] where, as above, <math>E</math> and <math>p</math> are the polariton's energy and momentum magnitude, and <math>\omega</math> and <math>k</math> are its angular frequency and wave number, respectively. In some cases, the dispersion can result in [[slow light|extremely slow speeds of light]] in matter. The effects of photon interactions with other quasi-particles may be observed directly in [[Raman scattering]] and [[Brillouin scattering]]. Photons can also be [[absorption (electromagnetic radiation)|absorbed]] by nuclei, atoms or molecules, provoking transitions between their [[energy level]]s. A classic example is the molecular transition of [[retinal]] (C<sub>20</sub>H<sub>28</sub>O, Figure at right), which is responsible for [[Visual perception|vision]], as discovered in 1958 by Nobel laureate biochemist [[George Wald]] and co-workers. As shown here, the absorption provokes a ''cis-trans'' isomerization that, in combination with other such transitions, is transduced into nerve impulses. The absorption of photons can even break chemical bonds, as in the [[photodissociation]] of [[chlorine]]; this is the subject of [[photochemistry]]. ==Technological applications== Photons have many applications in technology. These examples are chosen to illustrate applications of photons ''per se'', rather than general optical devices such as lenses, etc. that could operate under a classical theory of light. The laser is an extremely important application and is discussed above under [[stimulated emission]]. Individual photons can be detected by several methods. The classic [[photomultiplier]] tube exploits the [[photoelectric effect]]: a photon landing on a metal plate ejects an electron, initiating an ever-amplifying avalanche of electrons. [[Charge-coupled device]] chips use a similar effect in [[semiconductor]]s: an incident photon generates a charge on a microscopic capacitor that can be detected. Other detectors such as [[Geiger counter]]s use the ability of photons to ionize gas molecules, causing a detectable change in conductivity. Planck's energy formula <math>E=h\nu</math> is often used by engineers and chemists in design, both to compute the change in energy resulting from a photon absorption and to predict the frequency of the light emitted for a given energy transition. For example, the [[emission spectrum]] of a [[fluorescent lamp|fluorescent light bulb]] can be designed using gas molecules with different electronic energy levels and adjusting the typical energy with which an electron hits the gas molecules within the bulb. Under some conditions, an energy transition can be excited by ''two'' photons that individually would be insufficient. This allows for higher resolution microscopy, because the sample absorbs energy only in the region where two beams of different colors overlap significantly, which can be made much smaller than the excitation volume of a single beam (see [[two-photon excitation microscopy]]). Moreover, these photons cause less damage to the sample, since they are of lower energy. In some cases, two energy transitions can be coupled so that, as one system absorbs a photon, another nearby system "steals" its energy and re-emits a photon of a different frequency. This is the basis of [[fluorescence resonance energy transfer]], which is used to measure molecular distances. ==Recent research== {{See also|Quantum optics}} The fundamental nature of the photon is believed to be understood theoretically; the prevailing [[Standard Model]] predicts that the photon is a gauge boson of spin 1, without mass and without charge, that results from a local [[unitary group|U(1) gauge symmetry]] and mediates the electromagnetic interaction. However, physicists continue to check for discrepancies between experiment and the Standard Model predictions, in the hope of finding clues to physics beyond the Standard Model. In particular, experimental physicists continue to set ever better upper limits on the charge and mass of the photon; a non-zero value for either parameter would be a serious violation of the Standard Model. However, all experimental data hitherto are consistent with the photon having zero charge<ref name="chargeless" /> and mass.<ref name="massless">(a) {{cite journal | last = Goldhaber | first = AS | year = 1971 | title = [http://prola.aps.org/abstract/RMP/v43/i3/p277_1 Terrestrial and Extraterrestrial Limits on The Photon Mass] | journal = [[Reviews of Modern Physics]] | volume = 43 | pages = 277–96. | doi = 10.1103/RevModPhys.43.277}}<br />(b) {{cite journal | last = Fischbach | first = E | coauthors = Kloor H, Langel RA, Lui ATY, and Peredo M | year = 1994 | title = New Geomagnetic Limits on the Photon Mass and on Long-Range Forces Coexisting with Electromagnetism | journal = [[Physical Review Letters]] | volume = 73 | pages = 514–17. | doi = 10.1103/PhysRevLett.73.514}}<br />(c) [http://pdg.lbl.gov/2005/tables/gxxx.pdf Official particle table for gauge and Higgs bosons] S. Eidelman ''et al.'' (Particle Data Group) ''Physics Letters B'' '''592''', 1 (2004)<br />(d) {{cite journal | last = Davis | first = L | coauthors = Goldhaber AS and Nieto MM | year = 1975 | title = Limit on Photon Mass Deduced from Pioneer-10 Observations of Jupiter's Magnetic Field | journal = Physical Review Letters | volume = 35 | pages = 1402–1405. | doi = 10.1103/PhysRevLett.35.1402}}<br />(e) {{cite journal | last = Luo | first = J | coauthors = Shao CG, Liu ZZ, and Hu ZK | year = 1999 | title = Determination of the limit of photon mass and cosmic magnetic vector with rotating torsion balance | journal = Physical Review A | volume = 270 | pages = 288–292.}}<br />(f) {{cite journal | last = Schaeffer | first = BE | year = 1999 | title = Severe limits on variations of the speed of light with frequency | journal = Physical Review Letters | volume = 82 | pages = 4964–4966. | doi = 10.1103/PhysRevLett.82.4964}}<br />(g) {{cite journal | last = Luo | first = J | coauthors = Tu LC, Hu ZK, and Luan EJ | year = 2003 | title = New experimental limit on the photon rest mass with a rotating torsion balance | journal = Physical Review Letters | volume = 90 | pages = Art. No. 081801 | doi = 10.1103/PhysRevLett.90.081801}}<br />(h) {{cite journal | last = Williams | first = ER | coauthors = Faller JE and Hill HA | year = 1971 | title = [http://link.aps.org/abstract/PRL/v26/p721 New Experimental Test of Coulomb's Law: A Laboratory Upper Limit on the Photon Rest Mass] | journal = Physical Review Letters | volume = 26 | pages = 721–724 | doi = 10.1103/PhysRevLett.26.721}}<br />(i) {{cite journal | last = Lakes | first = R | year = 1998 | title = [http://prola.aps.org/abstract/PRL/v80/i9/p1826_1 Experimental Limits on the Photon Mass and Cosmic Magnetic Vector Potential] | journal = Physical Review Letters | volume = 80 | pages = 1826 | doi = 10.1103/PhysRevLett.80.1826}}<br />(j) [http://pdg.lbl.gov/2006/listings/s000.pdf 2006 PDG listing for photon] W.-M. Yao ''et al.'' (Particle Data Group) ''Journal of Physics G'' '''33''', 1 (2006).<br />(k) {{cite journal | last = Adelberger | first = E | coauthors = Dvali, G and Gruzinov, A | title = [http://link.aps.org/abstract/PRL/v98/e010402 Photon Mass Bound Destroyed by Vortices] | journal = Physical Review Letters | volume = 98 | pages = Art. No. 010402 | doi = 10.1103/PhysRevLett.98.010402 | year = 2007}}</ref> The best universally accepted upper limits on the photon charge and mass are 5×10<sup>−52</sup> [[coulomb|C]] (or 3×10<sup>−33</sup> times the [[elementary charge]]) and 1.1×10<sup>−52</sup> [[kilogram|kg]] (6x10<sup>-17</sup> eV), respectively .<ref>[http://pdg.lbl.gov/2005/tables/gxxx.pdf Official particle table for gauge and Higgs bosons] Retrieved [[October 24]], [[2006]]</ref> Much research has been devoted to applications of photons in the field of [[quantum optics]]. Photons seem well-suited to be elements of an ultra-fast [[quantum computer]], and the [[quantum entanglement]] of photons is a focus of research. [[Nonlinear optics|Nonlinear optical processes]] are another active research area, with topics such as two-photon absorption, [[self-phase modulation]] and [[optical parametric oscillator]]s. However, such processes generally do not require the assumption of photons ''per se''; they may often be modeled by treating atoms as nonlinear oscillators. The nonlinear process of [[spontaneous parametric down conversion]] is often used to produce single-photon states. Finally, photons are essential in some aspects of [[optical communication]], especially for [[quantum cryptography]]. ==See also== {{portal|Physics}} {{Col-begin|width=67%}} {{Col-1-of-3}} *[[Light]] *[[Electromagnetic radiation]] *[[Quantum optics]] *[[Photonics]] {{Col-2-of-3}} *[[Photon polarization]] *[[Ballistic photon]] *[[Photography]] *[[Laser]] *[[Advanced Photon Source]] at Argonne National Laboratory {{Col-end}} ==References and footnotes== <div class="references-small" style="-moz-column-count: 2; column-count: 2;"> <references/> </div> ==Additional references== <!-- Ordered by date published; two general histories cited at end --> <div class="references-small"> * {{cite journal | last = Clauser | first = JF. | year = 1974 | title = Experimental distinction between the quantum and classical field-theoretic predictions for the photoelectric effect | journal = Phys. Rev. D | volume = 9 | pages = 853–860 | doi = 10.1103/PhysRevD.9.853}} * {{cite journal | last = Kimble | first = HJ | coauthors = Dagenais M, and Mandel L. | year = 1977 | title = Photon Anti-bunching in Resonance Fluorescence | journal = Phys. Rev. Lett. | volume = 39 | pages = 691–695 | doi = 10.1103/PhysRevLett.39.691}} [http://prola.aps.org/abstract/PRL/v39/i11/p691_1 article web link] * {{cite journal | last = Grangier | first = P | coauthors = Roger G, and Aspect A. | year = 1986 | title = Experimental Evidence for a Photon Anticorrelation Effect on a Beam Splitter: A New Light on Single-Photon Interferences | journal = Europhysics Letters | volume = 1 | pages = 501–504 | doi = 10.1209/0295-5075/1/4/004}} * {{cite journal | last = Thorn | first = JJ | coauthors = Neel MS, Donato VW, Bergreen GS, Davies RE and Beck M. | year = 2004 | title = Observing the quantum behavior of light in an undergraduate laboratory | journal = American Journal of Physics | volume = 72 | pages = 1210–1219 | doi = 10.1119/1.1737397}} http://people.whitman.edu/~beckmk/QM/grangier/grangier.html * {{cite book | last = Pais | first = A. | authorlink = Abraham Pais | year = 1982 | title = Subtle is the Lord: The Science and the Life of Albert Einstein | publisher = Oxford University Press }} An excellent history of the photon's early development. * {{cite web | url = http://nobelprize.org/nobel_prizes/physics/laureates/2005/glauber-lecture.html | title = Roy Glauber's Nobel Lecture, “100 Years of Light Quanta”}} Delivered [[8 December]] [[2005]]. Another history of the photon, summarized by a key physicist who developed the concepts of [[coherent state]]s of photons. * {{cite journal | last = Lamb | first = WE | authorlink = Willis Lamb | year = 1995 | title = Anti-photon | journal = Applied Physics B | volume = 60 | pages = 77–84 | doi = 10.1007/BF01135846}} Feisty, fun and sometimes snarky history of the photon, with a strong argument for allowing only its second-quantized definition, by [[Willis Lamb]], the [[Nobel Prize in Physics|1955 Nobel laureate in Physics]]. * Special supplemental issue of ''Optics and Photonics News'' (vol. 14, October 2003) ** {{cite journal | last = Roychoudhuri | first = C | coauthors = Rajarshi R | title = The nature of light: what is a photon? | journal = Optics and Photonics News | volume = 14 | pages = S1 (Supplement)}} ** {{cite journal | last = Zajonc | first = A | title = Light reconsidered | journal = Optics and Photonics News | volume = 14 | pages = S2–S5 (Supplement)}} ** {{cite journal | last = Loudon | first = R | title = What is a photon? | journal = Optics and Photonics News | volume = 14 | pages = S6–S11 (Supplement)}} ** {{cite journal | last = Finkelstein | first = D | title = What is a photon? | journal = Optics and Photonics News | volume = 14 | pages = S12–S17 (Supplement)}} ** {{cite journal | last = Muthukrishnan | first = A | coauthors = Scully MO, Zubairy MS | title = The concept of the photon—revisited | journal = Optics and Photonics News | volume = 14 | pages = S18–S27 (Supplement)}} ** {{cite journal | last = Mack | first = H | coauthors = Schleich WP | title = A photon viewed from Wigner phase space | journal = Optics and Photonics News | volume = 14 | pages = S28–S35 (Supplement)}} </div> {{particles}} {{QED}} {{featured article}} [[Category:Fundamental physics concepts]] [[Category:Bosons]] [[Category:Electromagnetism]] [[Category:Optics]] [[Category:Quantum electrodynamics]] [[Category:Photons]] {{Link FA|zh}} [[ar:فوتون]] [[az:Foton]] [[bs:Foton]] [[bg:Фотон]] [[ca:Fotó]] [[cs:Foton]] [[da:Foton]] [[de:Photon]] [[et:Footon]] [[el:Φωτόνιο]] [[es:Fotón]] [[eo:Fotono]] [[fa:فوتون]] [[fr:Photon]] [[ga:Fótón]] [[gl:Fotón]] [[ko:광자]] [[hr:Foton]] [[id:Foton]] [[it:Fotone]] [[he:פוטון]] [[ku:Foton]] [[la:Photon]] [[lv:Fotons]] [[lt:Fotonas]] [[hu:Foton]] [[ml:ഫോട്ടോണ്‍]] [[ms:Foton]] [[nl:Foton]] [[ja:光子]] [[no:Foton]] [[nn:Foton]] [[nds:Photon]] [[pl:Foton]] [[pt:Fóton]] [[ro:Foton]] [[ru:Фотон]] [[sq:Fotoni]] [[scn:Fotoni]] [[simple:Photon]] [[sk:Fotón]] [[sl:Foton]] [[sr:Фотон]] [[sh:Foton]] [[fi:Fotoni]] [[sv:Foton]] [[vi:Photon]] [[tr:Foton]] [[uk:Фотон]] [[ur:نوریہ]] [[zh:光子]]