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{{About|the elementary particle of light}}
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{{merge from|Photon molecule|date=September 2013}}
{{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]]
|statistics=[[Bose–Einstein statistics|Bosonic]]
|group=[[Gauge boson]]
|generation=
|interaction=[[Electromagnetism|Electromagnetic]]
|theorized=[[Albert Einstein]]
|discovered=
|symbol=&gamma;, [[Planck constant|''h'']][[frequency|&nu;]], or [[Reduced Planck constant|ħ]][[angular frequency|&omega;]]
|mass=0<br>{{nowrap|<{{val|1|e=-18|ul=eV/c<sup>2</sup>}}}}<ref name="Particle_table_2009">
{{cite journal
|author=Amsler, C. ''et al.'' ([[Particle Data Group]])
|year=2008 +2009 partial update
|url=http://pdg.lbl.gov/2009/tables/rpp2009-sum-gauge-higgs-bosons.pdf
|title=Review of Particle Physics: Gauge and Higgs bosons
|journal=[[Physics Letters B]]
|volume=667|page=1|doi=10.1016/j.physletb.2008.07.018
|bibcode=2008PhLB..667....1P
}}</ref>
|mean_lifetime=Stable<ref name="Particle_table_2009"/>
|decay_particle=
|electric_charge=0<br>{{nowrap|<{{val|1|e=-35|ul=e}}}}<ref name="Particle_table_2009"/>
|color_charge=
|spin=1
|num_spin_states=
|parity=−1<ref name="Particle_table_2009"/>
|g_parity=
|c_parity=−1<ref name="Particle_table_2009"/>
|r_parity=
|condensed_symmetries=''[[Isospin|I]]''(''[[Total angular momentum|J]]''<sup>''[[Parity (physics)|P]][[C parity|C]]''</sup>)=0,1(1<sup>−−</sup>)<ref name="Particle_table_2009"/>
}}
A '''photon''' is an [[elementary particle]], the [[quantum]] of [[light]] and all other forms of [[electromagnetic radiation]], and the [[force carrier]] for the [[electromagnetic force]], even when [[Static forces and virtual-particle exchange|static]] via [[virtual photons]]. The effects of this [[force]] are easily observable at both the [[microscopic scale|microscopic]] and [[macroscopic scale|macroscopic]] level, because the photon has zero [[rest mass]]; this allows long distance [[fundamental interaction|interaction]]s. Like all elementary particles, photons are currently best explained by [[quantum mechanics]] and exhibit [[wave–particle duality]], exhibiting properties of both [[wave]]s and [[wikt:particle|particles]]. For example, a single photon may be [[refraction|refracted]] by a [[Lens (optics)|lens]] or exhibit [[Interference (wave propagation)|wave interference]] with itself, but also act as a particle giving a definite result when its [[position (vector)|position]] is measured.


The modern photon concept was developed gradually by [[Albert Einstein]] 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]]. It also accounted for anomalous observations, including the properties of [[black body radiation]], that other physicists, most notably [[Max Planck]], had sought to explain using ''semiclassical models'', in which light is still described by [[Maxwell's equations]], but the material objects that emit and absorb light, do so in amounts of energy that are ''quantized'' (i.e., they change energy only by certain particular discrete amounts and cannot change energy in any arbitrary way). Although these semiclassical models contributed to the development of quantum mechanics, many further experiments<ref>{{cite journal|author=Kimble, H.J.; Dagenais, M.; Mandel, L.|title=Photon Anti-bunching in Resonance Fluorescence|journal=[[Physical Review Letters]]|volume=39|issue=11|pages=691–695|year=1977|doi=10.1103/PhysRevLett.39.691|bibcode=1977PhRvL..39..691K}}</ref><ref>{{cite journal|author=Grangier, P.; Roger, G.; Aspect, A.|title=Experimental Evidence for a Photon Anticorrelation Effect on a Beam Splitter: A New Light on Single-Photon Interferences|journal=[[EPL (journal)|Europhysics Letters]]|volume=1|issue=4|pages=173–179|year=1986|doi=10.1209/0295-5075/1/4/004|bibcode=1986EL......1..173G}}</ref> starting with [[Compton scattering]] of single photons by electrons, first observed in 1923, validated Einstein's hypothesis that ''light itself'' is [[quantization (physics)|quantized]]. In 1926 the chemist [[Gilbert N. Lewis]] coined the name ''photon'' for these particles, and after 1927, when [[Arthur H. Compton]] won the Nobel Prize for his scattering studies, most scientists accepted the validity that [[quantum|quanta]] of light have an independent existence, and Lewis' term ''photon'' for light quanta was accepted.
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In the [[Standard Model]] of [[particle physics]], photons are described as a necessary consequence of physical laws having 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 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. It has been applied to [[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]]. In the [[fine arts]], photons are what makes [[photography]] possible, and in [[biology]] photons enable humans and other organisms to have [[visual perception|vision]].
 
==Nomenclature==
{{Standard model|QED}}
In 1900, Max Planck was working on black-body radiation and suggested that the energy in electromagnetic waves could only be released in "packets" of energy. In his 1901 article <ref name="Planck1901">{{cite journal|last=Planck|first=M.|authorlink=Max Planck|year=1901|title=On the Law of Distribution of Energy in the Normal Spectrum|url=http://dbhs.wvusd.k12.ca.us/webdocs/Chem-History/Planck-1901/Planck-1901.html|archiveurl=http://web.archive.org/web/20080418002757/http://dbhs.wvusd.k12.ca.us/webdocs/Chem-History/Planck-1901/Planck-1901.html|archivedate=2008-04-18|journal=[[Annalen der Physik]]|volume=4|pages=553–563|doi=10.1002/andp.19013090310|issue=3|bibcode=1901AnP...309..553P}}</ref> in [[Annalen der Physik]] he called these packets "energy elements". The word [[Quantum|''quanta'']] (singular ''quantum'') was used even before 1900 to mean particles or amounts of different [[Quantity|quantities]], including [[electron|electricity]]. Later, in 1905, [[Albert Einstein]] went further by suggesting that electromagnetic waves could only exist in these discrete wave-packets.<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|url=http://www.physik.uni-augsburg.de/annalen/history/einstein-papers/1905_17_132-148.pdf|journal=[[Annalen der Physik]]|volume=17|pages=132–148|doi=10.1002/andp.19053220607|bibcode=1905AnP...322..132E|issue=6}} {{de icon}}. An [[s:A Heuristic Model of the Creation and Transformation of Light|English translation]] is available from [[Wikisource]].</ref> He called such a [[wave]]-packet ''the light quantum'' (German: ''das Lichtquant''). The name ''photon'' derives from the [[Greek language|Greek word]] for light, ''{{lang|grc|φῶς}}'' (transliterated ''phôs''), and was coined<ref group="Note">Although the 1967 [http://nobelprize.org/nobel_prizes/physics/laureates/1918/planck-lecture.html Elsevier translation] of Planck's Nobel Lecture interprets Planck's ''Lichtquant'' as "photon", the more literal 1922 translation by Hans Thacher Clarke and Ludwik Silberstein [http://books.google.com/books?id=4UC4AAAAIAAJ ''The origin and development of the quantum theory''], The Clarendon Press, 1922 (here [http://www.readanybook.com/ebook/the-origin-and-development-of-the-quantum-theory-48217#downloadable]) uses "light-quantum". No evidence is known that Planck himself used the term "photon" by 1926 (see also [http://www.nobeliefs.com/photon.htm this note]).</ref> in 1926 by the physical chemist [[Gilbert N. Lewis|Gilbert Lewis]], who published a speculative theory in which photons were "uncreatable and indestructible".<ref name="Lewis1926">{{cite journal|last=Lewis|first=G.N.|authorlink=Gilbert N. Lewis|title=The conservation of photons|journal=[[Nature (journal)|Nature]]|year=1926|url=http://www.nobeliefs.com/photon.htm|volume=118|pages=874–875|doi=10.1038/118874a0|bibcode=1926Natur.118..874L|issue=2981}}</ref> Although Lewis' theory was never accepted as it was contradicted by many experiments, his new name, ''photon'', was adopted immediately by most physicists. [[Isaac Asimov]] credits [[Arthur Compton]] with defining quanta of energy as photons in 1923.<ref>{{cite book|title=The Neutrino, Ghost Particle of the Atom|first=I.|last=Asimov|authorlink=Isaac Asimov|location=Garden City (NY)|publisher=[[Doubleday (publisher)|Doubleday]]|year=1966|isbn=0-380-00483-6|lccn=6603}}</ref><ref>{{cite book|title=The Universe From Flat Earth To Quasar|first=I.|last=Asimov|authorlink=Isaac Asimov|location=New York (NY)|publisher=[[Walker (publisher)|Walker]]|year=1966|isbn=0-8027-0316-X|lccn=6605}}</ref>
 
In physics, a photon is usually denoted by the symbol ''γ'' (the [[Greek alphabet|Greek letter]] [[gamma]]). This symbol for the photon probably derives from [[gamma ray]]s, which were discovered in 1900 by [[Paul Ulrich Villard|Paul 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 des Séances de l'Académie des Sciences]]|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 des Séances de l'Académie des Sciences]]|volume=130|pages=1178–1179}} {{fr icon}}</ref> named by [[Ernest Rutherford]] in 1903, and shown to be a form of [[electromagnetic radiation]] in 1914 by Rutherford and [[Edward Andrade]].<ref>{{cite journal|last=Rutherford|first=E.|authorlink=Ernest Rutherford|coauthors=[[Edward Andrade|Andrade, E.N.C.]]|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 ''hν'', the energy of a photon, where ''h'' is [[Planck's constant]] and the [[Greek alphabet|Greek letter]] ''ν'' ([[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==
{{See also|Special relativity}}
<!--
[[Image:Electron-positron-scattering.svg|220px|thumb|right|A [[Feynman diagram]] of the exchange of a virtual photon (symbolized by an oscillating line labelled γ (gamma) between a [[positron]] and an [[electron]].]]
-->[[Image:Light cone colour.svg|thumb|right|The cone shows possible values of wave 4-vector of a photon. Green and indigo represent left and right<!-- I do not know a "correct" assignment --> polarization]]
A photon is [[massless particle|massless]],<ref group="Note">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 ''λ'' or energy ''E'', this is ''h/λc'' or ''E''/''c''<sup>2</sup>. This usage for the term "mass" is no longer common in scientific literature. Further info: What is the mass of a photon? http://math.ucr.edu/home/baez/physics/ParticleAndNuclear/photon_mass.html</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|arxiv=hep-ph/0411398|bibcode=2005AstL...31..147K|issue=3 }}</ref> and is [[stable particle|stable]]. A photon has two possible [[photon polarization|polarization]] states. In the [[position and momentum space|momentum representation]], which is preferred in quantum field theory, a photon is described by its [[wave vector]], which determines its wavelength ''λ'' and its direction of propagation. A photon's wave vector may not be zero and can be represented either as a [[Euclidean vector|spatial 3-vector]] or as a (relativistic) [[four-vector]]; in the latter case it belongs to the [[light cone]] (pictured). Different signs of the four-vector denote different [[circular polarization]]s, but in the 3-vector representation one should account for the polarization state separately; it actually is a [[spin quantum number]]. In both cases the space of possible wave vectors is three-dimensional.
 
The photon is the [[gauge boson]] for [[electromagnetism]],<ref>Role as gauge boson and polarization section 5.1 in{{Cite book|last=Aitchison|first=I.J.R.|last2=Hey|first2=A.J.G.|title=Gauge Theories in Particle Physics|publisher=[[IOP Publishing]]|year=1993|isbn=0-85274-328-9}}</ref> and therefore all other quantum numbers of the photon (such as [[lepton number]], [[baryon number]], and [[flavor (particle physics)#Flavour quantum numbers|flavour quantum numbers]]) are zero.<ref>See p.31 in{{Cite journal|doi=10.1016/j.physletb.2008.07.018|last=Amsler|first=C.|coauthors=et al.|title=Review of Particle Physics|journal=[[Physics Letters]] B|volume=667|pages=1–1340|year=2008|bibcode=2008PhLB..667....1P}}</ref>
 
Photons are emitted in many natural processes. For example, when a charge is [[acceleration|accelerated]] it emits [[synchrotron radiation]]. During a [[molecule|molecular]], [[atom]]ic or [[atomic nucleus|nuclear]] transition to a lower [[energy level]], photons of various energy will be emitted, from [[radio wave]]s to [[gamma ray]]s. A photon can also be emitted when a particle and its corresponding [[antiparticle]] are [[annihilation|annihilated]] (for example, [[electron–positron annihilation]]).
 
In empty space, the photon moves at ''c'' (the [[speed of light]]) and its [[energy]] and [[momentum]] are related by {{nowrap|''E'' {{=}} ''pc''}}, where ''p'' is the [[magnitude (mathematics)|magnitude]] of the momentum vector '''p'''. This derives from the following relativistic relation, with {{nowrap|''m'' {{=}} 0}}:<ref>See section 1.6 in {{Cite book|last=Alonso|first=M.|last2=Finn|first2=E.J.|title=Fundamental University Physics Volume III: Quantum and Statistical Physics|publisher=[[Addison-Wesley]]|year=1968|isbn=0-201-00262-0}}</ref>
:<math>E^{2}=p^{2} c^{2} + m^{2} c^{4}.</math>
 
The energy and momentum of a photon depend only on its [[frequency]] (''ν'') or inversely, its [[wavelength]] (''λ''):
 
:<math>E=\hbar\omega=h\nu=\frac{hc}{\lambda}</math>
 
:<math>\boldsymbol{p}=\hbar\boldsymbol{k},</math>
 
where '''''k''''' is the [[wave vector]] (where the wave number {{nowrap|''k'' {{=}} {{!}}'''''k'''''{{!}} {{=}} 2π/''λ''}}), {{nowrap|''ω'' {{=}} 2π''ν''}} is the [[angular frequency]], and {{nowrap|ħ {{=}} ''h''/2π}} is the [[Planck constant|reduced Planck constant]].<ref>Davison E. Soper, [http://physics.uoregon.edu/~soper/Light/photons.html Electromagnetic radiation is made of photons], Institute of Theoretical Science, University of Oregon</ref>
 
Since '''''p''''' points in the direction of the photon's propagation, the magnitude of the momentum is
 
:<math>p=\hbar k=\frac{h\nu}{c}=\frac{h}{\lambda}.</math>
 
The photon also carries [[spin (physics)|spin angular momentum]] that does not depend on its frequency.<ref name="spin">This property was experimentally verified by Raman and Bhagavantam in 1931: {{Cite journal|last=Raman|first=C.V.|last2=Bhagavantam|first2=S.|title=Experimental proof of the spin of the photon|url=http://dspace.rri.res.in/bitstream/2289/2123/1/1931%20IJP%20V6%20p353.pdf|format=PDF|authorlink=C. V. Raman|journal=[[Indian Journal of Physics]]|volume=6|page=353|year=1931}}</ref> The magnitude of its spin is <math>\scriptstyle{\sqrt{2} \hbar}</math> and the component measured along its direction of motion, its [[helicity (particle physics)|helicity]], must be ±ħ. These two possible helicities, called right-handed and left-handed,<!-- Which is which again?--> correspond to the two possible [[circular polarization]] states of the photon.<ref>E.g., section 1.3.3.2 in {{Cite book|last=Burgess|first=C.|url=http://books.google.com/books?id=PLYECqs2geEC&pg=PA27|last2=Moore|first2=G.|title=The Standard Model. A Primer|publisher=[[Cambridge University Press]]|year=2007|isbn=0-521-86036-9}}</ref>
 
To illustrate the significance of these formulae, the annihilation of a particle with its antiparticle in free space 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 (since it is determined, as we have seen, only by the photon's frequency or wavelength—which cannot be zero). Hence, [[momentum|conservation of momentum]] (or equivalently, [[translational invariance]]) requires that at least two photons are created, with zero net momentum. (However, it is possible if the system interacts with another particle or field for annihilation to produce one photon, as when a positron annihilates with a bound atomic electron, it is possible for only one photon to be emitted, as the nuclear Coulomb field breaks translational symmetry.) 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.<ref>E.g., section 9.3 in {{Cite book|last=Alonso|first=M.|last2=Finn|first2=E.J.|title=Fundamental University Physics Volume III: Quantum and Statistical Physics|publisher=[[Addison-Wesley]]|year=1968}}</ref> That process is the reverse of "annihilation to one photon" allowed in the electric field of an atomic nucleus.
 
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.<ref>E.g., Appendix XXXII in {{Cite book|last=Born|first=M.|authorlink=Max Born|title=Atomic Physics|publisher=[[Blackie & Son]]|year=1962|isbn=0-486-65984-4}}</ref>
 
===Experimental checks on photon mass===
The photon is currently understood to be strictly massless, but this is an experimental question. If the photon is not a strictly massless particle, it would not move at the exact speed of light in vacuum, ''c''. Its speed would be lower and depend on its frequency. Relativity would be unaffected by this; the so-called speed of light, ''c'', would then not be the actual speed at which light moves, but a constant of nature which is the maximum speed that any object could theoretically attain in space-time.<ref>{{cite journal|author=Mermin, David|title=Relativity without light|doi=10.1119/1.13917|journal=American Journal of Physics|date=February 1984|volume=52|issue=2|pages=119–124|bibcode=1984AmJPh..52..119M }}</ref> Thus, it would still be the speed of space-time ripples ([[gravitational waves]] and [[graviton]]s), but it would not be the speed of photons.
 
A massive photon would have other effects as well. [[Coulomb's law]] would be modified and the electromagnetic field would have an extra physical degree of freedom. These effects yield more sensitive experimental probes of the photon mass than the frequency dependence of the speed of light. If Coulomb's law is not exactly valid, then that would cause the presence of an [[electric field]] inside a hollow conductor when it is subjected to an external electric field. This thus allows one to [[Tests of electromagnetism|test]] Coulomb's law to very high precision.<ref>{{cite journal|last1=Plimpton|first1=S.|last2=Lawton|first2=W.|title=A Very Accurate Test of Coulomb's Law of Force Between Charges|journal=Physical Review|volume=50|page=1066|year=1936|doi=10.1103/PhysRev.50.1066|bibcode=1936PhRv...50.1066P|issue=11 }}</ref> A null result of such an experiment has set a limit of ''m'' ≲ 10<sup>−14</sup>&nbsp;eV/c<sup>2</sup>.<ref>{{cite journal|last1=Williams|first1=E.|last2=Faller|first2=J.|last3=Hill|first3=H.|title=New Experimental Test of Coulomb's Law: A Laboratory Upper Limit on the Photon Rest Mass|journal=Physical Review Letters|volume=26|page=721|year=1971|doi=10.1103/PhysRevLett.26.721|bibcode=1971PhRvL..26..721W|issue=12}}</ref>
 
Sharper upper limits have been obtained in experiments designed to detect effects caused by the galactic [[magnetic potential|vector potential]]. Although the galactic vector potential is very large because the galactic [[magnetic field]] exists on very long length scales, the magnetic field is only observable if the photon is massless. In case of a massive photon, the mass term <math>\scriptstyle\frac{1}{2} m^2 A_{\mu}A^{\mu}</math> would affect the galactic plasma. The fact that no such effects are seen implies an upper bound on the photon mass of ''m'' < {{val|3|e=-27|u=eV/c<sup>2</sup>}}.<ref>{{cite journal|last1=Chibisov|first1=G V|title=Astrophysical upper limits on the photon rest mass|journal=Soviet Physics Uspekhi|volume=19|page=624|year=1976|doi=10.1070/PU1976v019n07ABEH005277|bibcode=1976SvPhU..19..624C|issue=7 }}</ref> The galactic vector potential can also be probed directly by measuring the torque exerted on a magnetized ring.<ref>{{cite journal|last1=Lakes|first1=Roderic|title=Experimental Limits on the Photon Mass and Cosmic Magnetic Vector Potential|journal=Physical Review Letters|volume=80|page=1826|year=1998|doi=10.1103/PhysRevLett.80.1826|bibcode=1998PhRvL..80.1826L|issue=9}}</ref> Such methods were used to obtain the sharper upper limit of 10<sup>−18</sup>eV/c<sup>2</sup> (the equivalent of {{val|1.07|e=-27|u=atomic mass units}}) given by the [[Particle Data Group]].<ref name=amsler>{{cite journal|last1=Amsler|first1=C|last2=Doser|first2=M|last3=Antonelli|first3=M|last4=Asner|first4=D|last5=Babu|first5=K|last6=Baer|first6=H|last7=Band|first7=H|last8=Barnett|first8=R|last9=Bergren|first9=E|title=Review of Particle Physics⁎|journal=Physics Letters B|volume=667|page=1|year=2008|doi=10.1016/j.physletb.2008.07.018|bibcode=2008PhLB..667....1P}} [http://pdg.lbl.gov/2009/tables/contents_tables.html Summary Table]</ref>
 
These sharp limits from the non-observation of the effects caused by the galactic vector potential have been shown to be model dependent.<ref>{{cite journal|last1=Adelberger|first1=Eric|last2=Dvali|first2=Gia|last3=Gruzinov|first3=Andrei|title=Photon-Mass Bound Destroyed by Vortices|journal=Physical Review Letters|volume=98|issue=1|page=010402|year=2007|pmid=17358459|doi=10.1103/PhysRevLett.98.010402|bibcode=2007PhRvL..98a0402A|arxiv=hep-ph/0306245 }} [http://arxiv.org/abs/hep-ph/0306245 preprint]</ref> If the photon mass is generated via the [[Higgs mechanism]] then the upper limit of ''m''≲10<sup>−14</sup>&nbsp;eV/c<sup>2</sup> from the test of Coulomb's law is valid.
 
Photons inside [[superconductors]] do develop a nonzero [[effective mass (solid-state physics)|effective rest mass]]; as a result, electromagnetic forces become short-range inside superconductors.<ref>{{cite book
|last=Wilczek |first=Frank
|title=The Lightness of Being: Mass, Ether, and the Unification of Forces
|year=2010
|publisher=Basic Books
|page=212
|isbn=978-0-465-01895-6
|url=http://books.google.nl/books?id=22Z36Qoz664C&pg=PA212}}</ref>
 
{{See also|Supernova/Acceleration Probe}}
 
==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. Since [[Subatomic particle|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]])|publisher=[[Imprimerie de Ian Maire]]|year=1637|isbn=0-268-00870-1}} {{fr icon}}</ref> [[Robert Hooke]] (1665),<ref>{{cite book|last=Hooke|first=R.|authorlink=Robert Hooke|year=1667|location=London (UK)|publisher=[[Royal Society of London]]|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&nbsp;... |isbn=0-486-49564-7}}</ref> and [[Christiaan Huygens]] (1678);<ref>{{cite book|last=Huygens|first=C.|authorlink=Christiaan Huygens|year=1678|title=Traité de la lumière}} {{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|origyear=1730|year=1952|title=Opticks|edition=4th|pages=Book II, Part III, Propositions XII–XX; Queries 25–29|nopp=true|location=Dover (NY)|publisher=[[Dover Publications]]|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 (wave propagation)|interference]] and diffraction of light and by 1850 wave models were generally accepted.<ref>{{cite book|last=Buchwald|first=J.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]]|isbn=0-226-07886-8|oclc=18069573}}</ref> In 1865, [[James Clerk Maxwell]]'s [[Maxwell's equations|prediction]]<ref name="maxwell">{{cite journal|last=Maxwell|first=J.C.|authorlink=James Clerk Maxwell|year=1865|title=[[A Dynamical Theory of the Electromagnetic Field]]|journal=[[Philosophical Transactions of the Royal Society]]|volume=155|pages=459–512|doi=10.1098/rstl.1865.0008|bibcode=1865RSPT..155..459C}} 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 (physics)|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.<ref>Frequency-dependence of luminiscence p. 276f., photoelectric effect section 1.4 in {{Cite book|last=Alonso|first=M.|last2=Finn|first2=E.J.|title=Fundamental University Physics Volume III: Quantum and Statistical Physics|publisher=[[Addison-Wesley]]|isbn=0-201-00262-0|year=1968}}</ref><ref group="Note">The phrase "no matter how intense" refers to intensities below approximately 10<sup>13</sup> W/cm<sup>2</sup> at which point [[perturbation theory]] begins to break down. In contrast, in the intense regime, which for visible light is above approximately 10<sup>14</sup> W/cm<sup>2</sup>, the classical wave description correctly predicts the energy acquired by electrons, called [[ponderomotive energy]]. (See also: Boreham ''et al.'' (1996). "[http://adsabs.harvard.edu/abs/1996AIPC..369.1234B Photon density and the correspondence principle of electromagnetic interaction]".) By comparison, sunlight is only about 0.1 W/cm<sup>2</sup>.</ref>
 
At the same time, investigations of [[blackbody radiation]] carried out over four decades (1860–1900) by various researchers<ref name="Wien1911">{{cite web|last=Wien|first=W.|authorlink=Wilhelm Wien|year=1911|url=http://nobelprize.org/nobel_prizes/physics/laureates/1911/wien-lecture.html|title=Wilhelm Wien Nobel Lecture}}</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|bibcode=1901AnP...309..553P
|issue=3 }} {{de icon}} [http://web.archive.org/web/20080418002757/http://dbhs.wvusd.k12.ca.us/webdocs/Chem-History/Planck-1901/Planck-1901.html English translation]</ref><ref name="Planck1918">
{{cite web|last=Planck|first=M.|authorlink=Max Planck|year=1920|url=http://nobelprize.org/nobel_prizes/physics/laureates/1918/planck-lecture.html|title=Max Planck's Nobel Lecture}}</ref> that the energy of ''any'' system that absorbs or emits electromagnetic radiation of frequency ''ν'' is an integer multiple of an energy quantum ''E=hν''. 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.<ref>Presentation speech by [[Svante Arrhenius]] for the 1921 Nobel Prize in Physics, December 10, 1922. [http://nobelprize.org/nobel_prizes/physics/laureates/1921/press.html Online text] from [nobelprize.org], The Nobel Foundation 2008. Access date 2008-12-05.</ref>
 
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">{{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|url=http://www.ekkehard-friebe.de/EINSTEIN-1909-P.pdf|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> and 1916,<ref name="Einstein1916b">{{cite journal|last=Einstein|first=A.|authorlink=Albert Einstein|year=1916|title=Zur Quantentheorie der Strahlung|journal=[[Mitteilungen der Physikalischen Gesellschaft zu Zürich]]|volume=16|page=47}} Also ''Physikalische Zeitschrift'', '''18''', 121–128 (1917). {{de icon}}</ref> Einstein showed that, if [[Planck's law of black-body radiation]] is accepted, the energy quanta must also carry [[momentum]] ''p=h/λ'', 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=A Quantum Theory of the Scattering of X-rays by Light Elements|url=http://www.aip.org/history/gap/Compton/01_Compton.html|journal=[[Physical Review]]|volume=21|pages=483–502|doi=10.1103/PhysRev.21.483|bibcode=1923PhRv...21..483C|issue=5}}</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|url=http://www.questia.com/PM.qst?a=o&d=74596612|publisher=[[Oxford University Press]]|isbn=0-19-853907-X}}</ref> and was solved in [[quantum electrodynamics]] and its successor, the [[Standard Model]] (see [[Photon#Second quantization|Second quantization]] and [[Photon#The photon as a gauge boson|The photon as a gauge boson]], below).
 
==Early objections==
[[Image:Bohr-atom-PAR.svg|thumb|250px|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|last=Millikan|first=R.A|authorlink=Robert Millikan|year=1924|url=http://nobelprize.org/nobel_prizes/physics/laureates/1923/millikan-lecture.html|title=Robert A. Millikan's Nobel Lecture}}</ref> However, before [[Compton scattering|Compton's experiment]]<ref name="Compton1923" /> showing that photons carried [[momentum]] proportional to their [[wave number]] (or 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" />). Instead, there was a widespread belief that energy quantization resulted from some unknown constraint on the matter that absorbs or emits radiation. Attitudes changed over time. In part, the change can be traced to experiments such as [[Compton scattering]], where it was much more difficult not to ascribe quantization to light itself to explain the observed results.<ref>{{Cite journal|last=Hendry|first=J.|year=1980|title=The development of attitudes to the wave-particle duality of light and quantum theory, 1900–1920|journal=[[Annals of Science]]|volume=37|issue=1|pages=59–79|doi=10.1080/00033798000200121}}</ref>
 
Even after Compton's experiment, [[Niels 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, H.A.]]; [[John C. Slater|Slater, J.C.]]|year=1924|title=The Quantum Theory of Radiation|journal=[[Philosophical Magazine]]|volume=47|pages=785–802}} Also ''[[European Physical Journal|Zeitschrift für Physik]]'', '''24''', 69 (1924).</ref> To account for the data then available, 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 of [[matrix mechanics]].<ref name="Heisenberg1932">{{cite web|last=Heisenberg|first=W.|authorlink=Werner Heisenberg|url=http://nobelprize.org/nobel_prizes/physics/laureates/1932/heisenberg-lecture.html|title=Heisenberg Nobel lecture|year=1933}}</ref>
 
A few physicists persisted<ref name="Mandel1976">
{{cite journal|last=Mandel|first=L.|authorlink=Leonard Mandel|year=1976|doi=10.1016/S0079-6638(08)70018-0|title=The case for and against semiclassical radiation theory|journal=[[Progress in Optics]]|editor=E. Wolf|publisher=[[North-Holland]]|volume=13|pages=27–69|series=Progress in Optics|isbn=978-0-444-10806-7}}</ref> in developing semiclassical models in which [[electromagnetic radiation]] is not quantized, but matter appears to obey 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 photon-correlation experiments.<ref group="Note">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 (wave propagation)|interference]] on the length scale of its wavelength. For example, a single photon passing through a [[double-slit experiment]] lands on the screen exhibiting interference phenomena but only if no measure was made on the actual slit being run across. To account for the particle interpretation that phenomenon is called [[probability distribution]] but behaves according to [[Maxwell's equations]].<ref name="Taylor1909">{{cite conference|last=Taylor|first=G.I.|authorlink=Geoffrey Ingram Taylor|year=1909|title=Interference fringes with feeble light|booktitle=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]].<ref name="Saleh">{{cite book|author=Saleh, B. E. A. and Teich, M. C.|title=Fundamentals of Photonics|publisher=Wiley|year=2007|isbn=0-471-35832-0}}</ref> Rather, the photon seems to be 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. 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:Gamma-ray-microscope.svg|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|url=http://osulibrary.oregonstate.edu/specialcollections/coll/pauling/bond/papers/corr155.1.html|title=Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik|journal=[[European Physical Journal|Zeitschrift für Physik]]|volume=43|pages=172–198|doi=10.1007/BF01397280|bibcode=1927ZPhy...43..172H|issue=3–4}} {{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 λ. 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_{\text{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.<ref>E.g., p. 10f. in {{Cite book|last=Schiff|first=L.I.|title=Quantum Mechanics|edition=3rd|publisher=[[McGraw-Hill]]|year=1968|isbn=0-07-055287-8|asin=B001B3MINM}}</ref>
 
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 (wave propagation)|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=H.A.|authorlink=Hendrik Anthony Kramers|year=1958|title=Quantum Mechanics|publisher=[[North-Holland]]|location=Amsterdam|isbn=0-486-49533-7|asin=B0006AUW5C}}</ref><ref>{{cite book|last=Bohm|first=D.|authorlink=David Bohm|origyear=1954|year=1989|url=http://books.google.com/books?id=9DWim3RhymsC&pg=PA592|title=Quantum Theory|publisher=[[Dover Publications]]|isbn=0-486-65969-0}}</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]] fields. More generally, the normal concept of a Schrödinger [[probability amplitude|probability]] [[wave function]] cannot be applied to photons.<ref>{{cite journal|last=Newton|first=T.D.|coauthors=[[Eugene Wigner|Wigner, E.P.]]|year=1949|title=Localized states for elementary particles|journal=[[Reviews of Modern Physics]]|volume=21|pages=400–406|doi=10.1103/RevModPhys.21.400|bibcode=1949RvMP...21..400N|issue=3}}</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|url=http://www.cft.edu.pl/~birula/publ/APPPwf.pdf|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=J.E.|year=1995|title=Photon wave functions|journal=[[Physical Review]] A|volume=52|pages=1875–1883|doi=10.1103/PhysRevA.52.1875|bibcode=1995PhRvA..52.1875S|issue=3}}</ref><ref>{{cite journal|last=Bialynicki-Birula|first=I.|year=1996|title=Photon wave function|journal=[[Progress in Optics]]|volume=36|pages=245–294|doi=10.1016/S0079-6638(08)70316-0|series=Progress in Optics|isbn=978-0-444-82530-8}}</ref><ref>{{cite book|last=Scully|first=M.O.|coauthors=Zubairy, M.S.|year=1997|title=Quantum Optics|publisher=[[Cambridge University Press]]|location=Cambridge (UK)|isbn=0-521-43595-1|url=http://books.google.com/books?id=20ISsQCKKmQC&printsec=frontcover}}</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|Gas in a box}}
 
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=S.N.|authorlink=Satyendra Nath Bose|year=1924|title=Plancks Gesetz und Lichtquantenhypothese|journal=[[European Physical Journal|Zeitschrift für Physik]]|volume=26|pages=178–181|doi=10.1007/BF01327326|bibcode=1924ZPhy...26..178B}} {{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|doi=10.1002/3527608958.ch28|title=Quantentheorie des einatomigen idealen Gases, Zweite Abhandlung|journal=[[Sitzungsberichte der Preussischen Akademie der Wissenschaften]] (Berlin), Physikalisch-mathematische Klasse|volume=1925|pages=3–14|isbn=978-3-527-60895-9}} {{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=M.H.|coauthors=Ensher, J.R.; Matthews, M.R.; [[Carl Wieman|Wieman, C.E.]]; [[Eric Allin Cornell|Cornell, E.A.]]|title=Observation of Bose–Einstein Condensation in a Dilute Atomic Vapor|journal=[[Science (journal)|Science]]|year=1995|volume=269|pages=198–201|doi=10.1126/science.269.5221.198|pmid=17789847|issue=5221|jstor=2888436|bibcode=1995Sci...269..198A}}</ref> It was later used by [[Lene Hau]] to slow, and then completely stop, light in 1999<ref>[http://news.harvard.edu/gazette/1999/02.18/light.html]</ref> and 2001.<ref>[http://www.photonics.com/Article.aspx?AID=28520]</ref>
 
The modern view on this is that photons are, by virtue of their integer spin, [[boson]]s (as opposed to [[fermion]]s with half-integer spin). By the [[spin-statistics theorem]], all bosons obey Bose–Einstein statistics (whereas all fermions obey [[Fermi-Dirac statistics]]).<ref>{{Cite book|last=Streater|first=R.F.|last2=Wightman|first2=A.S.|title=PCT, Spin and Statistics, and All That|publisher=[[Addison-Wesley]]|year=1989|isbn=0-201-09410-X}}</ref>
 
==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 could be derived from a semi-classical, statistical treatment of photons and atoms, which implies 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. 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 energy density <math>\rho(\nu)</math> of photons with frequency <math>\nu</math> (which is proportional to their [[number density]]) is, on average, constant in time; hence, the rate at which photons of any particular frequency are ''emitted'' must equal the rate of ''absorbing'' them.<ref name="Einstein1916a">{{cite journal|last=Einstein|first=A.|authorlink=Albert Einstein|year=1916|title=Strahlungs-emission und -absorption nach der Quantentheorie|journal=[[Verhandlungen der Deutschen Physikalischen Gesellschaft]]|volume=18|pages=318–323|bibcode=1916DPhyG..18..318E}} {{de icon}}</ref>
 
Einstein began by postulating simple proportionality relations for the different reaction rates involved. In his model, 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> is proportional to the number <math>N_{j}</math> of atoms with energy <math>E_{j}</math> and to the energy 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. For the reverse process, there are two possibilities: spontaneous emission of a photon, and a return to the lower-energy state that is initiated by the interaction with a passing photon. Following Einstein's approach, the corresponding rate <math>R_{ij}</math> for the emission of photons of frequency <math>\nu</math> and transition from a higher energy <math>E_{i}</math> to a lower energy <math>E_{j}</math> is
 
:<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]]). In thermodynamic equilibrium, the number of atoms in state i and that of atoms in state j must, on average, be constant; hence, the rates <math>R_{ji}</math> and <math>R_{ij}</math> must be equal. Also, by arguments analogous to the derivation of [[Boltzmann statistics]], the ratio of <math>N_{i}</math> and <math>N_{j}</math> is <math>g_i/g_j\exp{(E_j-E_i)/kT)},</math> where <math>g_{i,j}</math> are the [[degenerate energy level|degeneracy]] of the state i and that of j, respectively, <math>E_{i,j}</math> their energies, k the [[Boltzmann constant]] and T the system's [[temperature]]. From this, it is readily derived that
<math>g_iB_{ij}=g_jB_{ji}</math> and
:<math>
A_{ij}=\frac{8 \pi h \nu^{3}}{c^{3}} B_{ij}.
</math>
The A and Bs are collectively known as the ''Einstein coefficients''.<ref>Section 1.4 in {{cite book|last=Wilson|first=J.|last2=Hawkes|first2=F.J.B.|title=Lasers: Principles and Applications|publisher=[[Prentice Hall]]|location=New York|year=1987|isbn=0-13-523705-X}}</ref>
 
Einstein could not fully justify his rate equations, but claimed that it should be possible to calculate the coefficients <math>A_{ij}</math>, <math>B_{ji}</math> and <math>B_{ij}</math> once physicists had obtained "mechanics and electrodynamics modified to accommodate the quantum hypothesis".<ref>P. 322 in {{cite journal|last=Einstein|first=A.|authorlink=Albert Einstein|year=1916|title=Strahlungs-emission und -absorption nach der Quantentheorie|journal=[[Verhandlungen der Deutschen Physikalischen Gesellschaft]]|volume=18|pages=318–323|bibcode=1916DPhyG..18..318E}} {{de icon}}: {{quote|Die Konstanten <math>A^n_m</math> and <math>B^n_m</math> w&uuml;rden sich direkt berechnen lassen, wenn wir im Besitz einer im Sinne der Quantenhypothese modifizierten Elektrodynamik und Mechanik w&auml;ren."}}</ref> In fact, in 1926, [[Paul Dirac]] derived the <math>B_{ij}</math> rate constants in using a semiclassical approach,<ref name="Dirac1926">{{cite journal|last=Dirac|first=P.A.M.|authorlink=Paul Dirac|year=1926|title=On the Theory of Quantum Mechanics|journal=[[Proceedings of the Royal Society]] A|volume=112|pages=661–677|doi=10.1098/rspa.1926.0133|bibcode=1926RSPSA.112..661D|issue=762}}</ref> and, in 1927, succeeded in deriving ''all'' the rate constants from first principles within the framework of quantum theory.<ref name="Dirac1927a">
{{cite journal|last=Dirac|first=P.A.M.|authorlink=Paul Dirac|doi=10.1098/rspa.1927.0039|year=1927|url=http://dieumsnh.qfb.umich.mx/archivoshistoricosmq/ModernaHist/Dirac1927.pdf|title=The Quantum Theory of the Emission and Absorption of Radiation|journal=[[Proceedings of the Royal Society]] A|volume=114|pages=243–265|bibcode=1927RSPSA.114..243D|issue=767}}</ref><ref name="Dirac1927b">
{{cite conference|last=Dirac|first=P.A.M.|authorlink=Paul Dirac|year=1927b|title=The Quantum Theory of Dispersion|journal=[[Proceedings of the Royal Society]] 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=[[European Physical Journal|Zeitschrift für Physik]]|volume=56|page=1|doi=10.1007/BF01340129|bibcode=1929ZPhy...56....1H}} {{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=[[European Physical Journal|Zeitschrift für Physik]]|volume=59|page=139|doi=10.1007/BF01341423|bibcode=1930ZPhy...59..168H|issue=3–4}} {{de icon}}</ref><ref name="Fermi1932">{{cite journal|last=Fermi|first=E.|authorlink=Enrico Fermi|year=1932|title=Quantum Theory of Radiation|url=http://www.physics.princeton.edu/~mcdonald/examples/QM/fermi_rmp_4_87_32.pdf|journal=[[Reviews of Modern Physics]]|volume=4|page=87|doi=10.1103/RevModPhys.4.87|bibcode=1932RvMP....4...87F}}</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=1926|title=Zur Quantenmechanik der Stossvorgänge|journal=[[European Physical Journal|Zeitschrift für Physik]]|volume=37|pages=863–867|url=http://www.physics.princeton.edu/~mcdonald/examples/QM/born_zp_37_863_26.pdf|doi=10.1007/BF01397477|bibcode=1926ZPhy...37..863B|issue=12}} {{de icon}}</ref><ref name="Born1926b">{{cite journal|last=Born|first=M.|authorlink=Max Born|year=1926|title=Quantenmechanik der Stossvorgänge|journal=[[European Physical Journal|Zeitschrift für Physik]]|volume=38|page=803|doi=10.1007/BF01397184|bibcode=1926ZPhy...38..803B|issue=11–12}} {{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|url=http://books.google.com/books?id=mREnwpAqz-YC&pg=PA260|page=260|title=Inward Bound: Of Matter and Forces in the Physical World|publisher=[[Oxford University Press]]|isbn=0-19-851997-4}} 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=1427–1434|doi=10.1002/andp.19103381617|bibcode=1910AnP...338.1427D|issue=16}} {{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.]]; [[Pascual Jordan|Jordan, P.]]|year=1925|title=Quantenmechanik II|journal=[[European Physical Journal|Zeitschrift für Physik]]|volume=35|pages=557–615|doi=10.1007/BF01379806|bibcode=1926ZPhy...35..557B|issue=8–9}} {{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 oscillator]]s. 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'' B–E statistics). In Dirac's time, it was not yet known that all bosons, including photons, must obey Bose–Einstein 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 (waves)|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]].<ref>Photon-photon-scattering section 7-3-1, renormalization chapter 8-2 in {{Cite book|last=Itzykson|first=C.|last2=Zuber|first2=J.-B.|title=Quantum Field Theory|publisher=[[McGraw-Hill]]|year=1980|isbn=0-07-032071-3}}</ref> In fact, such photon-photon scattering, as well as electron-photon scattering, is meant to be one of the modes of operations of the planned particle accelerator, the [[International Linear Collider]].<ref>{{Cite journal|last=Weiglein|first=G.|title=Electroweak Physics at the ILC|journal=[[Journal of Physics: Conference Series]]|volume=110|page=042033|year=2008|doi=10.1088/1742-6596/110/4/042033|bibcode=2008JPhCS.110d2033W|issue=4}}</ref>
 
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 hadronic properties of the photon==
Measurements of the interaction between energetic photons and hadrons show that the interaction is much more intense than expected by the interaction of merely photons with the hadron's electric charge. Furthermore, the interaction of energetic photons with protons is similar to the interaction of photons with neutrons<ref>Bauer, T. H., Spital, R. D., Yennie, D. R. and Pipkin, F. M, The hadronic properties of the photon in high-energy interactions, Rev. Mod. Phys. 50, 261–436 (1978), pages&nbsp;292–293</ref> in spite of the fact that the electric charge structures of protons and neutrons are substantially different.
 
A theory called [[Vector Meson Dominance]] (VMD) was developed to explain this effect. According to VMD, the photon is a superposition of the pure electromagnetic photon (which interacts only with electric charges) and vector meson.<ref>Theory of strong interactions, J. J. Sakurai, Ann. Phys., 11 (1960)</ref>
 
However, if experimentally probed at very short distances, the intrinsic structure of the photon is recognized as a flux of quark and gluon components, quasi-free according to asymptotic freedom in [[Quantum chromodynamics|QCD]] and described by the [[Photon Structure Function|photon structure function]].<ref>T.F. Walsh and P.M. Zerwas, Two-photon processes in the parton model, Physics Letters B44 (1973) 195</ref><ref>E.Witten, Anomalous cross-section for photon - photon scattering  in gauge theories, Nuclear Physics B120 (1977) 189</ref> A comprehensive comparison of data with theoretical predictions is presented in a recent review.<ref>R. Nisius, The photon structure from deep inelastic electron photon scattering, Physics Report 332 (2000) 165</ref>
 
==The photon as a gauge boson==
{{Main|Gauge theory}}
 
The electromagnetic field can be understood as a [[gauge field]], i.e., as a field that results from requiring that a gauge symmetry holds independently at every position in [[spacetime]].<ref name="Ryder">{{cite book|last=Ryder|first=L.H.|year=1996|url=http://books.google.com/books?id=nnuW_kVJ500C&printsec=frontcover|title=Quantum field theory|edition=2nd|publisher=[[Cambridge University Press]]|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 [[observable]]s or [[real number|real valued functions]] 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|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 boson]]s with the eight [[gluon]] gauge bosons of [[quantum chromodynamics]]; however, key predictions of these theories, such as [[proton decay]], have not been observed experimentally.<ref>E.g., chapter 14 in {{cite book|last=Hughes|first=I. S.|title=Elementary particles|edition=2nd|publisher=[[Cambridge University Press]]|year=1985|isbn=0-521-26092-2}}</ref>
 
==Contributions to the mass of a system==
{{See also|Mass in special relativity|General relativity}}
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. As an application, the energy balance of nuclear reactions involving photons is commonly written in terms of the masses of the nuclei involved, and terms of the form <math>{E}/{c^2}</math> for the gamma photons (and for other relevant energies, such as the recoil energy of nuclei).<ref>E.g., section 10.1 in {{Cite book|last=Dunlap|first=R.A.|title=An Introduction to the Physics of Nuclei and Particles|publisher=[[Cengage Learning#Brands/imprints|Brooks/Cole]]|year=2004|isbn=0-534-39294-6}}</ref>
 
This concept is applied in key predictions of [[quantum electrodynamics]] (QED, see above). In that theory, the mass of electrons (or, more generally, leptons) is modified by including the mass contributions of virtual photons, in a technique known as [[renormalization]]. Such "radiative corrections" contribute to a number of predictions of QED, such as the [[anomalous magnetic dipole moment|magnetic dipole moment]] of [[lepton]]s, the [[Lamb shift]], and the [[hyperfine structure]] of bound lepton pairs, such as [[muonium]] and [[positronium]].<ref>Radiative correction to electron mass section 7-1-2, anomalous magnetic moments section 7-2-1, Lamb shift section 7-3-2 and hyperfine splitting in positronium section 10-3 in {{Cite book|last=Itzykson|first=C.|last2=Zuber|first2=J.-B.|title=Quantum Field Theory|publisher=[[McGraw-Hill]]|year=1980|isbn=0-07-032071-3}}</ref>
 
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]].<ref>E. g. sections 9.1 (gravitational contribution of photons) and 10.5 (influence of gravity on light) in {{Cite book|last=Stephani|first=H.|url=http://books.google.com/books?id=V04_vLQvstcC&pg=PA86|last2=Stewart|first2=J.|pages=86 ff, 108 ff.|title=General Relativity: An Introduction to the Theory of Gravitational Field|isbn=0-521-37941-5|publisher=[[Cambridge University Press]]|year=1990}}</ref>
 
==Photons in matter==
{{See also|Group velocity|Photochemistry}}
 
Any 'explanation' of how photons travel through matter has to explain why different arrangements of matter are transparent or opaque at different wavelengths (light through carbon as diamond or not, as graphite) and why individual photons behave in the same way as large groups. Explanations that invoke 'absorption' and 're-emission' have to provide an explanation for the directionality of the photons (diffraction, reflection) and further explain how entangled photon pairs can travel through matter without their quantum state collapsing.
 
The simplest explanation is that light that travels through transparent matter does so at a lower speed than ''c'', the speed of light in a vacuum. In addition, light can also undergo [[scattering]] and [[Absorption (electromagnetic radiation)|absorption]]. There are circumstances in which heat transfer through a material is mostly radiative, involving emission and absorption of photons within it. An example would be in the [[Solar core|core]] of the Sun. 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|first=R.|last=Naeye|publisher=[[CRC Press]]|year=1998|isbn=0-7503-0484-7|url=http://books.google.com/?id=06_9B7S_q_YC&pg=PA16&dq=million-year+surface+sun+photon|oclc=40180195}}</ref> However, this phenomenon is distinct from scattered radiation passing diffusely through matter, as it involves local equilibrium between the radiation and the temperature. Thus, the time is how long it takes the ''energy'' to be transferred, not the ''photons'' themselves. Once in open space, a photon from the Sun takes only 8.3 minutes to reach Earth. The factor by which the speed of light is decreased in a material 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 excitation of the matter ([[quasi-particle]]s such as [[phonon]]s and [[exciton]]s) to form a [[polariton]]; this polariton has a nonzero [[effective mass (solid-state physics)|effective mass]], which means that it cannot travel at ''c''.
 
Alternatively, photons may be viewed as ''always'' traveling at ''c'', even in matter, but they have their phase shifted (delayed or advanced) upon interaction with atomic scatters: this modifies their wavelength and momentum, but not speed.<ref>Ch 4 in {{Cite book|last=Hecht|first=Eugene|title=Optics|publisher=[[Addison Wesley]]|year=2001|isbn=978-0-8053-8566-3}}</ref> A light wave made up of these photons does travel slower than the speed of light. In this view the photons are "bare", and are scattered and phase shifted, while in the view of the preceding paragraph the photons are "dressed" by their interaction with matter, and move without scattering or phase shifting, but at a lower speed.
 
Light of different frequencies may travel through matter at [[variable speed of light|different speeds]]; this is called [[dispersion (optics)|dispersion]]. In some cases, it 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]].<ref>Polaritons section 10.10.1, Raman and Brillouin scattering section 10.11.3 in {{Cite book|last=Patterson|first=J.D.|pages=569 ff, 580 ff|last2=Bailey|first2=B.C.|url=http://books.google.com/books?id=uRQg87Mb6DoC&pg=569|title=Solid-State Physics: Introduction to the Theory|publisher=[[Springer Science+Business Media|Springer]]|year=2007|isbn=3-540-24115-9}}</ref>
 
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, which is responsible for [[Visual perception|vision]], as discovered in 1958 by Nobel laureate [[biochemistry|biochemist]] [[George Wald]] and co-workers. 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]].<ref>E.g., section 11-5 C in {{Cite book|last=Pine|first=S.H.|last2=Hendrickson|first2=J.B.|last3=Cram|first3=D.J.|last4=Hammond|first4=G.S.|title=Organic Chemistry|edition=4th|publisher=[[McGraw-Hill]]|year=1980|isbn=0-07-050115-7}}</ref><ref>Nobel lecture given by G. Wald on December 12, 1967, online at nobelprize.org: [http://nobelprize.org/nobel_prizes/medicine/laureates/1967/wald-lecture.html The Molecular Basis of Visual Excitation].</ref> Analogously, [[gamma ray]]s can in some circumstances dissociate atomic nuclei in a process called [[photodisintegration]].
 
==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 [[Electrical conductivity|conductivity]].<ref>Photomultiplier section 1.1.10, CCDs section 1.1.8, Geiger counters section 1.3.2.1 in {{cite book|first=C.R.|last=Kitchin|title=Astrophysical Techniques|publisher=[[CRC Press]]|location=Boca Raton (FL)|year=2008|isbn=1-4200-8243-4}}</ref>
 
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.<ref group="Note">An example is US Patent Nr. [http://patft.uspto.gov/netacgi/nph-Parser?Sect1=PTO1&Sect2=HITOFF&d=PALL&p=1&u=%2Fnetahtml%2FPTO%2Fsrchnum.htm&r=1&f=G&l=50&s1=5212709.PN.&OS=PN/5212709&RS=PN/5212709 5212709].</ref>
 
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.<ref>{{cite journal|author=Denk, W.; Svoboda, K.|title=Photon upmanship: Why multiphoton imaging is more than a gimmick|journal=[[Neuron (journal)|Neuron]]|volume=18|issue=3|pages=351–357|year=1997|pmid=9115730|doi=10.1016/S0896-6273(00)81237-4}}</ref>
 
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]], a technique that is used in [[molecular biology]] to study the interaction of suitable [[protein]]s.<ref>{{Cite book|first=J.R.|last=Lakowicz|url=http://books.google.com/books?id=-PSybuLNxcAC&pg=PA529|title=Principles of Fluorescence Spectroscopy|pages=529 ff|publisher=[[Springer Science+Business Media|Springer]]|year=2006|isbn=0-387-31278-1}}</ref>
 
Several different kinds of [[hardware random number generator]] involve the detection of single photons. In one example, for each bit in the random sequence that is to be produced, a photon is sent to a [[beam-splitter]]. In such a situation, there are two possible outcomes of equal probability. The actual outcome is used to determine whether the next bit in the sequence is "0" or "1".<ref>{{Cite journal|first=T.|last=Jennewein|first2=U.|last2=Achleitner|first3=G.|last3=Weihs|first4=H.|last4=Weinfurter|first5=A.|last5=Zeilinger|title=A fast and compact quantum random number generator|doi=10.1063/1.1150518|journal=[[Review of Scientific Instruments]]|volume=71|pages=1675–1680|year=2000|arxiv=quant-ph/9912118|bibcode=2000RScI...71.1675J|issue=4 }}</ref><ref>{{Cite journal|first=A.|last=Stefanov|first2=N.|last2=Gisin|first3=O.|last3=Guinnard|first4=L.|last4=Guinnard|first5=H.|last5=Zbiden|title=Optical quantum random number generator|journal=[[Journal of Modern Optics]]|volume=47|pages=595–598|year=2000|doi=10.1080/095003400147908|issue=4}}</ref>
 
==Recent research==
{{See also|Quantum optics}}
 
Much research has been devoted to applications of photons in the field of [[quantum optics]]. Photons seem well-suited to be elements of an extremely 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]], [[modulational instability]] 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]].<ref group="Note">Introductory-level material on the various sub-fields of quantum optics can be found in {{Cite book
|last=Fox|first=M.
|title=Quantum Optics: An Introduction
|publisher=[[Oxford University Press]]
|year=2006|url=http://books.google.com/books?id=Q-4dIthPuL4C&printsec=frontcover
|isbn=0-19-856673-5
}}</ref>
 
==See also==
{{Portal|Physics}}
{{cmn|3|
*[[Advanced Photon Source]] at Argonne National Laboratory
*[[Ballistic photon]]
*[[Doppler shift]]
*[[Electromagnetic radiation]]
*[[HEXITEC]]
*[[Laser]]
*[[Light]]
*[[Luminiferous aether]]
*[[Medipix]]
*[[Phonons]]
*[[Photon counting]]
*[[Photon polarization]]
*[[Photography]]
*[[Photonics]]
*[[Quantum optics]]
*[[Static forces and virtual-particle exchange]]
*[[Two-photon physics]]
*[[EPR paradox]]
*[[Dirac equation]]
}}
 
==Notes==
<references group="Note"/>
 
==References==
{{Reflist|2}}
 
==Additional references==
{{Commons category}}
<!-- Ordered by date published; two general histories cited at end -->
<div class="references-small">
By date of publication:
*{{Cite journal|last=Clauser|first=J.F.|year=1974|title=Experimental distinction between the quantum and classical field-theoretic predictions for the photoelectric effect|journal=[[Physical Review D]]|volume=9|pages=853–860|doi=10.1103/PhysRevD.9.853|bibcode=1974PhRvD...9..853C|issue=4}}
*{{Cite journal|last=Kimble|first=H.J.|last2=Dagenais|first2=M.|last3=Mandel|first3=L.|year=1977|title=Photon Anti-bunching in Resonance Fluorescence|journal=[[Physical Review Letters]]|volume=39|pages=691–695|doi=10.1103/PhysRevLett.39.691|bibcode=1977PhRvL..39..691K|issue=11}}
*{{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]]}}
*{{cite book |last=Feynman |first=Richard |authorlink=Richard Feynman |year=1985 |isbn=978-0-691-12575-6 |title=[[QED: The Strange Theory of Light and Matter]] |publisher=Princeton University Press}}
*{{Cite journal|last=Grangier|first=P.|last2=Roger|first2=G.|last3=Aspect|first3=A.|year=1986|title=Experimental Evidence for a Photon Anticorrelation Effect on a Beam Splitter: A New Light on Single-Photon Interferences|journal=[[EPL (journal)|Europhysics Letters]]|volume=1|pages=173–179|doi=10.1209/0295-5075/1/4/004|bibcode=1986EL......1..173G|issue=4}}
*{{Cite journal|last=Lamb|first=W.E.|authorlink=Willis Lamb|year=1995|title=Anti-photon|journal=[[Applied Physics B]]|volume=60|pages=77–84|doi=10.1007/BF01135846|bibcode=1995ApPhB..60...77L|issue=2–3}}
*Special supplemental issue of ''Optics and Photonics News'' (vol. 14, October 2003) [http://www.sheffield.ac.uk/polopoly_fs/1.14183!/file/photon.pdf article web link]
**{{cite journal|last=Roychoudhuri|first=C.|last2=Rajarshi|first2=R.|title=The nature of light: what is a photon?|journal=[[Optics and Photonics News]]|volume=14|pages=S1 (Supplement)|year=2003}}
**{{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.|last2=Scully|first2=M.O.|last3=Zubairy|first3=M.S.|title=The concept of the photon—revisited|journal=[[Optics and Photonics News]]|volume=14|pages=S18–S27 (Supplement)}}
**{{cite journal|last=Mack|first=H.|last2=Schleich|authorlink2=Wolfgang P. Schleich|first2=W.P.|title=A photon viewed from Wigner phase space|journal=[[Optics and Photonics News]]|volume=14|pages=S28–S35 (Supplement)}}
*{{cite web|last=Glauber|first=R.|title=One Hundred Years of Light Quanta|work=2005 Physics Nobel Prize Lecture|url=http://nobelprize.org/nobel_prizes/physics/laureates/2005/glauber-lecture.pdf|year=2005}}
Education with single photons:
*{{Cite journal|last=Thorn|first=J.J.|last2=Neel|first2=M.S.|last3=Donato|first3=V.W.|last4=Bergreen|first4=G.S.|last5=Davies|first5=R.E.|last6=Beck|year=2004|title=Observing the quantum behavior of light in an undergraduate laboratory|url=http://people.whitman.edu/~beckmk/QM/grangier/Thorn_ajp.pdf|journal=[[American Journal of Physics]]|volume=72|pages=1210–1219|doi=10.1119/1.1737397|first6=M.|bibcode=2004AmJPh..72.1210T|issue=9}}
*{{Cite journal|last=Bronner|first=P.|last2=et al.|year=2009|title=Interactive screen experiments with single photons|url=http://www.QuantumLab.de|journal=[[European Journal of Physics]]|volume=30|pages=345–353|doi=10.1088/0143-0807/30/2/014|first2=Andreas|last3=Silberhorn|first3=Christine|last4=Meyn|first4=Jan-Peter|bibcode=2009EJPh...30..345B|issue=2}}
 
</div>
 
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