View source for Photon

{{short description|Elementary particle or quantum of light}}
{{About|the elementary particle or quantum of light}}
{{Infobox particle
 |bgcolour = 
 |name = Photon
 |num_types =
 |composition = [[Elementary particle]]
 |statistics = [[Bosonic]]
 |group = [[Gauge boson]]
 |generation = 
 |interaction = [[Electromagnetism|Electromagnetic]], [[weak interaction|weak]] (and [[gravity]])
 |theorized = [[Albert Einstein]] (1905) <br/> The name "photon" is generally attributed to [[Gilbert N. Lewis]] (1926)
 |discovered = 
 |symbol={{big| γ}}
 |mass={{nowrap|0 (theoretical value)}}<br/>
   {{nowrap|&lt; {{val|1|e=-18|ul=eV/c2}} (experimental limit)}}<ref name="Particle_table_2009">
    {{cite journal
     |last1=Amsler |first1=C.
     |display-authors=etal
     |collaboration=[[Particle Data Group]]
     |year=2008
     |title=Review of Particle Physics: Gauge and Higgs bosons
     |journal=[[Physics Letters B]]
     |volume=667 |issue=1 |page=1
     |bibcode=2008PhLB..667....1A
     |doi=10.1016/j.physletb.2008.07.018
     |hdl=1854/LU-685594  |s2cid=227119789
 |hdl-access=free
     |url=http://pdg.lbl.gov/2009/tables/rpp2009-sum-gauge-higgs-bosons.pdf
    }}
</ref>
 |mean_lifetime = Stable<ref name="Particle_table_2009"/>
 |decay_particle = 
 |electric_charge = 0
<br/>{{nowrap|&lt; {{val|1|e=-35|ul=e}}}}<ref name="Particle_table_2009"/>
 |color_charge = 0
 |spin = 1&nbsp;[[reduced Planck constant|''ħ'']]
 |num_spin_states = +1&nbsp;''ħ'',&nbsp; −1&nbsp;''ħ''
 |parity = −1<ref name="Particle_table_2009"/>
 |g_parity = 
 |c_parity = −1<ref name="Particle_table_2009"/>
 |r_parity = 
 |condensed_symmetries=''[[Weak 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''' ({{etymology|grc|''{{wikt-lang|grc|φῶς}}'', ''{{wikt-lang|grc|φωτός}}'' ({{grc-transl|φῶς, φωτός}})|light}}) is an [[elementary particle]] that is a [[quantum]] of the [[electromagnetic field]], including [[electromagnetic radiation]] such as [[light]] and [[radio wave]]s, and the [[force carrier]] for the [[electromagnetic force]]. Photons are [[Massless particle|massless]], so they always move at the [[speed of light|speed of light in vacuum]] ({{val|299792458|u=m/s}} (or about {{convert|299792458|m/s|mi/s|disp=out|sigfig=6}})) when they are in vacuum. The photon belongs to the class of [[boson]] particles.

As with other elementary particles, photons are best explained by [[quantum mechanics]] and exhibit [[wave–particle duality]], their behavior featuring properties of both [[wave]]s and [[particle]]s.<ref>{{cite book  |last1=Joos |first1=George  |date=1951  |title=Theoretical Physics  |page=679  |publisher=Blackie and Son Limited  |location=London and Glasgow }}</ref> The modern photon concept originated during the first two decades of the 20th century with the work of [[Albert Einstein]], who built upon the research of [[Max Planck]]. While Planck was trying to explain how [[matter]] and electromagnetic radiation could be in [[thermal equilibrium]] with one another, he proposed that the energy stored within a [[material]] object should be regarded as composed of an [[integer]] number of discrete, equal-sized parts. To explain the [[photoelectric effect]], Einstein introduced the idea that light itself is made of discrete units of energy. In 1926, [[Gilbert N. Lewis]] popularized the term ''photon'' for these energy units.<ref name="www.aps.org">{{Cite web|url=https://www.aps.org/publications/apsnews/201212/physicshistory.cfm|title=December 18, 1926: Gilbert Lewis coins "photon" in letter to Nature|website=www.aps.org|language=en|access-date=2019-03-09}}</ref><ref>{{Cite web|url=https://www.atomicheritage.org/profile/gilbert-n-lewis|title=Gilbert N. Lewis|website=Atomic Heritage Foundation|language=en|access-date=2019-03-09}}</ref><ref name="kragh">{{Cite arXiv |last=Kragh |first=Helge |date=2014 |title=Photon: New light on an old name |eprint=1401.0293 |class=physics.hist-ph }}</ref> Subsequently, many other experiments validated Einstein's approach.<ref name="compton-lecture">{{cite book|last1=Compton|first1=Arthur H.|title=From Nobel Lectures, Physics 1922–1941|publisher=Elsevier Publishing Company|year=1965|location=Amsterdam|chapter=X-rays as a branch of optics|orig-year=12 Dec 1927|chapter-url=https://www.nobelprize.org/uploads/2018/06/compton-lecture.pdf}}</ref><ref>{{Cite journal|last1=Kimble|first1=H.J.|last2=Dagenais|first2=M.|last3=Mandel|first3=L.|year=1977|title=Photon Anti-bunching in Resonance Fluorescence|url=https://authors.library.caltech.edu/6051/1/KIMprl77.pdf|journal=[[Physical Review Letters]]|volume=39|issue=11|pages=691–695|bibcode=1977PhRvL..39..691K|doi=10.1103/PhysRevLett.39.691}}</ref><ref>{{cite journal|last1=Grangier|first1=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|issue=4|pages=173–179|bibcode=1986EL......1..173G|citeseerx=10.1.1.178.4356|doi=10.1209/0295-5075/1/4/004|s2cid=250837011 }}</ref>

In the [[Standard Model]] of [[particle physics]], photons and other elementary particles are described as a necessary consequence of physical laws having a certain [[Symmetry (physics)|symmetry]] at every point in [[spacetime]]. The intrinsic properties of particles, such as [[electric charge|charge]], [[invariant mass|mass]], and [[Spin (physics)|spin]], are determined by [[gauge symmetry]]. The photon concept has led to momentous advances in experimental and theoretical physics, including [[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]]. Moreover, photons have been studied as elements of [[quantum computer]]s, and for applications in [[optical imaging]] and [[optical communication]] such as [[quantum cryptography]].

==Nomenclature==
[[File:Photoelectric_effect_in_a_solid_-_diagram.svg|alt=|thumb|[[Photoelectric effect]]: the emission of electrons from a metal plate caused by light quanta – photons]]
[[File:1926 Gilbert N. Lewis letter which brought the word &quot;photon&quot; into common usage.jpg|thumb|upright=3|1926 [[Gilbert N. Lewis]] letter which brought the word &quot;photon&quot; into common usage]]
The word [[Quantum|''quanta'']] (singular ''quantum,'' Latin for ''[[wikt:quantum|how much]]'') was used before 1900 to mean particles or amounts of different [[Quantity|quantities]], including [[electron|electricity]]. In 1900, the German physicist [[Max Planck]] was studying [[black-body radiation]], and he suggested that the experimental observations, specifically at [[ultraviolet catastrophe|shorter wavelengths]], would be explained if the energy stored within a molecule was a "discrete quantity composed of an integral number of finite equal parts", which he called "energy elements".<ref>{{cite journal|last=Kragh |first=Helge |author-link=Helge Kragh |title=Max Planck: the reluctant revolutionary |journal=[[Physics World]] |date=2000-12-01 |volume=13 |number=12 |pages=31–36 |doi=10.1088/2058-7058/13/12/34}}</ref> In 1905, [[Albert Einstein]] published a paper in which he proposed that many light-related phenomena—including black-body radiation and the [[photoelectric effect]]—would be better explained by modelling electromagnetic waves as consisting of spatially localized, discrete wave-packets.<ref name="Einstein1905">{{cite journal |last=Einstein |first=Albert |author-link=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]] |language=de |volume=17 |issue=6 |pages=132–148 |bibcode=1905AnP...322..132E |doi=10.1002/andp.19053220607 |doi-access=free}} An [[wikisource:Translation:On a Heuristic Point of View about the Creation and Conversion of Light|English translation]] is available from [[Wikisource]].</ref> He called such a wave-packet ''a light quantum'' (German: ''ein Lichtquant'').{{efn|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 {{cite book |first=Max |last=Planck |title=The Origin and Development of the Quantum Theory |year=1922 |publisher=Clarendon Press |section=via Google Books |section-url={{google books |plainurl=y |id=4UC4AAAAIAAJ}} |url=https://archive.org/details/origindevelopmen00planrich |via=Internet Archive (archive.org, 2007-03-01)}} uses "light-quantum". No evidence is known that Planck himself had used the term "photon" as of 1926 ([http://www.nobeliefs.com/photon.htm see also]).}}

The name ''photon'' derives from the [[Greek language|Greek word]] for light, ''{{lang|grc|φῶς}}'' (transliterated ''phôs''). [[Arthur Compton]] used ''photon'' in 1928, referring to [[Gilbert N. Lewis]], who coined the term in a letter to ''[[Nature (journal)|Nature]]'' on 18&nbsp;December 1926.<ref name="www.aps.org" /><ref>{{cite journal |last=Lewis |first=Gilbert N. |author-link=Gilbert N. Lewis |date=18 December 1926 |title=The conservation of photons |journal=Nature |volume=118 |issue=2981 |pages=874–875 |bibcode=1926Natur.118..874L |doi=10.1038/118874a0 |eissn=1476-4687 |s2cid=4110026}} ''see also''<br />
{{cite conference |date=24–29 October 1927 |others=l'Institut International de Physique Solvay (host institution) |title=Discordances entre l'expérience et la théorie électromagnétique du rayonnement |conference=Cinquième Conseil de Physique |publisher=Gauthier-Villars et Cie |publication-place=Paris, France |publication-date=1928 |pages=55–85 |place=Bruxelles, Belgium |book-title=Electrons et photons: Rapports et discussions du cinquième Conseil de Physique tenu à Bruxelles du 24 au 29 octobre 1927 sous les auspices de l'Institut International de Physique Solvay |lang=fr}}</ref> The same name was used earlier but was never widely adopted before Lewis: in 1916 by the American physicist and psychologist [[Leonard T. Troland]], in 1921 by the Irish physicist [[John Joly]], in 1924 by the French physiologist René Wurmser (1890–1993), and in 1926 by the French physicist Frithiof Wolfers (1891–1971).<ref name="kragh"/> The name was suggested initially as a unit related to the illumination of the eye and the resulting sensation of light and was used later in a physiological context. Although Wolfers's and Lewis's theories were contradicted by many experiments and never accepted, the new name was adopted by most physicists very soon after Compton used it.<ref name="kragh"/>{{efn|[[Isaac Asimov|Asimov]]<ref name=Asimov-1971-1983/> credits [[Arthur Compton]] with defining quanta of energy as photons in 1923.<ref name=Asimov-1971-1983>{{cite book |first=Isaac |last=Asimov |title=The Neutrino: Ghost particle of the atom |url={{google books |plainurl=y |id=-SkMngEACAAJ}} |date=1983 |publisher=Avon Books |isbn=978-0-380-00483-6 |author-link=Isaac Asimov |location=Garden City, NY}} and<br/>{{cite book |first=Isaac |last=Asimov |author-link=Isaac Asimov |title=The Universe: From flat Earth to quasar |url={{google books |plainurl=y |id=Eo5xpO83YpoC}} |date=1971 |location=New York |publisher=[[Walker (publisher)|Walker]] |isbn=978-0-8027-0316-3 |lccn=66022515}}</ref>}}

In physics, a photon is usually denoted by the symbol [[gamma|{{math|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=Paul Ulrich |author-link=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]] |language=fr |volume=130 |pages=1010–1012}}</ref><ref>{{cite journal |last=Villard |first=Paul Ulrich |author-link=Paul Ulrich Villard |year=1900 |title=Sur le rayonnement du radium |journal=[[Comptes Rendus des Séances de l'Académie des Sciences]] |language=fr |volume=130 |pages=1178–1179}}</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 |last1=Rutherford |first1=Ernest |author-link=Ernest Rutherford |last2=Andrade |first2=Edward N.C. |author-link2=Edward Andrade |year=1914 |title=The wavelength of the soft gamma rays from Radium B |url=https://zenodo.org/record/2278669 |journal=[[Philosophical Magazine]] |volume=27 |issue=161 |pages=854–868 |doi=10.1080/14786440508635156}}</ref> In [[chemistry]] and [[optical engineering]], photons are usually symbolized by {{mvar|hν}}, which is the [[photon energy]], where {{mvar|h}} is the [[Planck constant]] and the [[Greek alphabet|Greek letter]] {{mvar|ν}} ([[Nu (letter)|nu]]) is the photon's [[frequency]].<ref name="Liddle2015">{{cite book |author=Liddle |first=Andrew |url=https://books.google.com/books?id=6n64CAAAQBAJ&pg=PA16 |title=An Introduction to Modern Cosmology |date=2015 |publisher=John Wiley & Sons |isbn=978-1-118-69025-3 |page=16 |language=en}}</ref>

==Physical properties==
The photon has no [[electric charge]],<ref>{{cite book |last1=Frisch |first1=David H. |title=Elementary Particles |last2=Thorndike |first2=Alan M. |publisher=[[David Van Nostrand]] |year=1964 |location=Princeton, New Jersey |page=22 |language=en-us |author1-link=David H. Frisch}}</ref><ref name="chargeless">{{cite journal |last1=Kobychev |first1=V. V. |last2=Popov |first2=S. B. |year=2005 |title=Constraints on the photon charge from observations of extragalactic sources |journal=[[Astronomy Letters]] |volume=31 |issue=3 |pages=147–151 |arxiv=hep-ph/0411398 |bibcode=2005AstL...31..147K |doi=10.1134/1.1883345 |s2cid=119409823}}</ref> is generally considered to have zero [[rest mass]]<ref>{{cite web |first=John |last=Baez |author-link=John Baez |title=What is the mass of a photon? |publisher=[[University of California, Riverside|U.C. Riverside]] |type=pers. academic site |url=http://math.ucr.edu/home/baez/physics/ParticleAndNuclear/photon_mass.html}}</ref> and is a [[stable particle]]. The experimental upper limit on the photon mass<ref>{{Cite journal |last=Tu |first=Liang-Cheng |last2=Luo |first2=Jun |last3=Gillies |first3=George T |date=2005-01-01 |title=The mass of the photon |url=https://iopscience.iop.org/article/10.1088/0034-4885/68/1/R02 |journal=Reports on Progress in Physics |volume=68 |issue=1 |pages=77–130 |doi=10.1088/0034-4885/68/1/R02 |issn=0034-4885}}</ref><ref>{{Cite journal |last=Goldhaber |first=Alfred Scharff |last2=Nieto |first2=Michael Martin |date=2010-03-23 |title=Photon and graviton mass limits |url=https://link.aps.org/doi/10.1103/RevModPhys.82.939 |journal=Reviews of Modern Physics |language=en |volume=82 |issue=1 |pages=939–979 |doi=10.1103/RevModPhys.82.939 |issn=0034-6861|arxiv=0809.1003 }}</ref> is very small, on the order of 10<sup>−50</sup> kg; its lifetime would be more than 10<sup>18</sup> years.<ref>{{Cite journal |last=Heeck |first=Julian |date=2013-07-11 |title=How Stable is the Photon? |url=https://link.aps.org/doi/10.1103/PhysRevLett.111.021801 |journal=Physical Review Letters |language=en |volume=111 |issue=2 |doi=10.1103/PhysRevLett.111.021801 |issn=0031-9007|arxiv=1304.2821 }}</ref> For comparison the [[age of the universe]] is about {{convert|13.8e9|m|m|disp=number}} years. <!--convert does not support time units, using it only for number in scientific notation --> 

In a vacuum, a photon has two possible [[photon polarization|polarization]] states.<ref name="Schwartz2014">{{cite book |first=Matthew D. |last=Schwartz |title=Quantum Field Theory and the Standard Model |url=https://books.google.com/books?id=HbdEAgAAQBAJ&pg=PA66 |year=2014 |publisher=Cambridge University Press |isbn=978-1-107-03473-0 |pages=66}}</ref> The photon is the [[gauge boson]] for [[electromagnetism]],<ref>"Role as gauge boson and polarization" §5.1 in {{cite book |last1=Aitchison |first1=I.J.R. |last2=Hey |first2=A.J.G. |title=Gauge Theories in Particle Physics|publisher=[[IOP Publishing]] |year=1993 |url={{google books |plainurl=y |id=ZJ-ZY8NW9TIC}} |isbn=978-0-85274-328-7}}</ref>{{rp|29–30}} 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>{{cite journal |doi=10.1016/j.physletb.2008.07.018 |pmid=10020536 |last=Amsler |first=C. |display-authors=etal |title=Review of Particle Physics |journal=[[Physics Letters B]] |volume=667 |issue=1–5 |page=31 |year=2008 |bibcode=2008PhLB..667....1A |url=http://scipp.ucsc.edu/%7Ehaber/pubs/Review_of_Particle_Physics_2014.pdf |hdl=1854/LU-685594 |s2cid=227119789 |hdl-access=free}}</ref> Also, the photon obeys [[Bose–Einstein statistics]], and not [[Fermi–Dirac statistics]]. That is, they do ''not'' obey the [[Pauli exclusion principle]]<ref name=Halliday>{{cite book |last1=Halliday |first1=David |last2=Resnick |first2=Robert |last3=Walker |first3=Jerl |title=Fundamental of Physics |publisher=John Wiley and Sons, Inc. |edition=7th |isbn=978-0-471-23231-5 |year=2005 |url=https://archive.org/details/isbn_0471216437}}</ref>{{rp|1221}} and more than one can occupy the same bound quantum state.

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, ranging from [[radio wave]]s to [[gamma ray]]s. Photons can also be emitted when a particle and its corresponding [[antiparticle]] are [[annihilation|annihilated]] (for example, [[electron–positron annihilation]]).<ref name=Halliday/>{{rp|572,1114,1172}}

===Relativistic energy and momentum===
{{See also|Photon energy|Special relativity}}
[[File:Light cone colour.svg|thumb|right|The cone shows possible values of wave 4-vector of a photon. The "time" axis gives the angular frequency ([[radians per second|rad⋅s<sup>−1</sup>]]) and the "space" axis represents the angular wavenumber (rad⋅m<sup>−1</sup>). Green and indigo represent left and right<!-- I do not know a "correct" assignment --> polarization.]]
In empty space, the photon moves at {{mvar|c}} (the [[speed of light]]) and its [[energy]] and [[momentum]] are related by {{math|1=''E'' = ''pc''}}, where {{mvar|p}} is the [[magnitude (mathematics)|magnitude]] of the momentum vector {{math|'''''p'''''}}. This derives from the following relativistic relation, with {{math|1=''m'' = 0}}:<ref>See {{harvnb|Alonso|Finn|1968|loc=Section 1.6}}.</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]] (<math>\nu</math>) or inversely, its [[wavelength]] ({{mvar|λ}}):
:<math>E = \hbar \, \omega = h \nu = \frac{\, h\,c \,}{\lambda}</math>
:<math>\boldsymbol{p} = \hbar \boldsymbol{k} ~,</math>
where '''{{mvar|k}}''' is the [[wave vector]], where
* {{math| ''k'' ≡ {{abs|'''''k'''''}} {{=}} {{sfrac| 2''π'' |''λ''}} }} &emsp; is the [[wave number]], and
* {{math| ''ω'' ≡ 2 ''πν''}} &emsp; is the [[angular frequency]], and
* {{math| ''ħ'' ≡ {{sfrac|''h''| 2''π'' }} }} &emsp; is the [[reduced Planck constant]].<ref>{{cite web |first=Davison E. |last=Soper |title=Electromagnetic radiation is made of photons |department=Institute of Theoretical Science |publisher=[[University of Oregon]] |url=http://physics.uoregon.edu/~soper/Light/photons.html}}</ref>

Since <math>\boldsymbol{p}</math> points in the direction of the photon's propagation, the magnitude of its momentum is
:<math>p \equiv \left| \boldsymbol{p} \right| = \hbar k = \frac{\, h \nu \,}{c} = \frac{\, h \,}{\lambda} ~.</math>

===Polarization and spin angular momentum===
{{main|Photon polarization|Spin angular momentum of light}}
The photon also carries [[Spin angular momentum of light|spin angular momentum]], which is related to [[photon polarization]]. (Beams of light also exhibit properties described as [[orbital angular momentum of light]]).

The angular momentum of the photon has two possible values, either {{mvar|+ħ}} or {{mvar|−ħ}}. These two possible values correspond to the two possible pure states of [[circular polarization]]. Collections of photons in a light beam may have mixtures of these two values; a linearly polarized light beam will act as if it were composed of equal numbers of the two possible angular momenta.<ref name="Hecht">{{Cite book |last=Hecht |first=Eugene |title=Optics |date=1998 |publisher=Addison-Wesley |isbn=978-0-201-83887-9 |edition=3rd |location=Reading, Massachusetts; Harlow |language=en-us}}</ref>{{rp|325}}

The spin angular momentum of light does not depend on its frequency, and was experimentally verified by [[C. V. Raman]] and S. Bhagavantam in 1931.<ref name="spin">{{Cite journal |last1=Raman |first1=C. V. |author1-link=C. V. Raman |last2=Bhagavantam |first2=S. |year=1931 |title=Experimental proof of the spin of the photon |url=http://dspace.rri.res.in/bitstream/2289/2123/1/1931%20IJP%20V6%20p353.pdf |url-status=dead |journal=Indian Journal of Physics |volume=6 |issue=3244 |page=353 |bibcode=1932Natur.129...22R |doi=10.1038/129022a0 |s2cid=4064852 |archive-url=https://web.archive.org/web/20160603235132/http://dspace.rri.res.in/bitstream/2289/2123/1/1931%20IJP%20V6%20p353.pdf |archive-date=2016-06-03 |access-date=2008-12-28 |hdl-access=free |hdl=10821/664}}</ref>

=== Antiparticle annihilation ===
{{Main | Annihilation | Electron-positron annihilation}}
The collision of a particle with its antiparticle can create photons. In free space at least ''two'' photons must be created since, in the [[center of momentum frame]], the colliding antiparticles have no net momentum, whereas a single photon always has momentum (determined 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.{{efn|
However, it is possible if the system interacts with a third particle or field for the annihilation to produce one photon, since the third particle or field can absorb momentum equal and opposite to the single photon, providing dynamic balance. An example is when a positron annihilates with a bound atomic electron; in that case, it is possible for only one photon to be emitted, as the nuclear Coulomb field breaks translational symmetry.}}<ref name=Griffiths2008>{{cite book |last=Griffiths |first=David J. |year=2008 |title=Introduction to Elementary Particles |edition=2nd revised |publisher=WILEY-VCH |isbn=978-3-527-40601-2}}</ref>{{rp|64–65}} The energy of the two photons, or, equivalently, their frequency, may be determined from [[Conservation law (physics)|conservation of four-momentum]].

{{anchor |antiphoton}}Seen another way, the photon can be considered as [[Truly neutral particle|its own antiparticle]] (thus an "antiphoton" is simply a normal photon with opposite momentum, equal polarization, and 180° out of phase). 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>{{harvnb|Alonso|Finn|1968|loc=Section 9.3}}.</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>{{cite book |last1=Born |first1=Max |url={{google books |plainurl=y |id=NmM-KujxMtoC}} |title=Atomic Physics |last2=Blin-Stoyle |first2=Roger John |last3=Radcliffe |first3=J. M. |date=1989 |publisher=Courier Corporation |isbn=978-0-486-65984-8 |language=en |section=Appendix&nbsp;XXXII}}</ref>

===Experimental checks on photon mass===
Current commonly accepted physical theories imply or assume the photon to be strictly massless. If photons were not purely massless, their speeds would vary with frequency, with lower-energy (redder) photons moving slightly slower than higher-energy photons. 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 [[upper bound]] on speed that any object could theoretically attain in spacetime.<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 spacetime ripples ([[gravitational waves]] and [[graviton]]s), but it would not be the speed of photons.

If a photon did have non-zero mass, there would be 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 allow the presence of an [[electric field]] to exist within a hollow conductor when it is subjected to an external electric field. This provides a means for precision [[Tests of electromagnetism|tests of Coulomb's law]].<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 {{nowrap|''m'' ≲ {{val|e=-14|u=eV/c2}}}}.<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 on the mass of light have been obtained in experiments designed to detect effects caused by the galactic [[magnetic vector potential|vector potential]]. Although the galactic vector potential is large because the galactic [[magnetic field]] exists on great length scales, only the magnetic field would be observable if the photon is massless. In the case that the photon has mass, the mass term {{sfrac|1|2}}''m''{{sup|2}}''A''{{sub|''μ''}}''A''{{sup|''μ''}} would affect the galactic plasma. The fact that no such effects are seen implies an upper bound on the photon mass of {{nowrap|''m'' &lt; {{val|3|e=-27|u=eV/c2}}}}.<ref>{{cite journal |last1=Chibisov |first1=G. V. |year=1976 |title=Astrophysical upper limits on the photon rest mass |journal=Soviet Physics Uspekhi |volume=19 |issue=7 |page=624 |bibcode=1976SvPhU..19..624C |doi=10.1070/PU1976v019n07ABEH005277}}</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 {{val|1.07|e=-27|u=eV/c2}} (the equivalent of {{val|e=-36|u=[[dalton (unit)|dalton]]s}}) 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|display-authors=8|first9=E|title=Review of Particle Physics⁎|journal=[[Physics Letters B]]|volume=667|issue=1–5|page=1|year=2008|doi=10.1016/j.physletb.2008.07.018|bibcode=2008PhLB..667....1A|url=http://scipp.ucsc.edu/%7Ehaber/pubs/Review_of_Particle_Physics_2014.pdf|hdl=1854/LU-685594|s2cid=227119789|hdl-access=free}} [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 |s2cid=31249827}}</ref> If the photon mass is generated via the [[Higgs mechanism]] then the upper limit of {{nowrap|''m'' ≲ {{val|e=-14|u=eV/c2}}}} from the test of Coulomb's law is valid.
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NOTE BEFORE DELETION:Don't confuse absorption and remission with a single photon. This is untrue, an overstretch, and may confuse:
Photons inside [[superconductors]] 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
|journal=Physics Today
|volume=62
|issue=4
|year=2010
|publisher=Basic Books
|page=212
|isbn=978-0-465-01895-6
|url={{google books |plainurl=y |id=22Z36Qoz664C|page=212}}
|bibcode=2009PhT....62d..61W
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==Historical development==
{{Main|Light}}
[[File:Young Diffraction.png|thumb|[[Thomas Young (scientist)|Thomas Young]]'s [[double-slit experiment]] in 1801 showed that light can act as a [[wave]], helping to invalidate early [[elementary particle|particle]] theories of light.<ref name=Halliday/>{{rp|964}}]]
In most theories up to the eighteenth century, light was pictured as being made 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=René |url={{google books |plainurl=y |id=difXAAAAMAAJ}} |title=Discours de la méthode (Discourse on Method) |publisher=Imprimerie de Ian Maire |year=1637 |isbn=978-0-268-00870-3 |language=fr |author-link=René Descartes}}</ref> [[Robert Hooke]] (1665),<ref>{{cite book |last=Hooke |first=Robert |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;... |publisher=[[Royal Society of London]] |year=1667 |isbn=978-0-486-49564-4 |location=London, UK |language=en-uk |author-link=Robert Hooke}}</ref> and [[Christiaan Huygens]] (1678);<ref>{{cite book |last=Huygens |first=Christiaan |title=Traité de la lumière |title-link=Traité de la lumière |year=1678 |language=fr |author-link=Christiaan Huygens}}. An [[gutenberg: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=Isaac |url={{google books |plainurl=y |id=bSiTKcLf07UC}} |title=Opticks |publisher=Dover Publications |year=1952 |isbn=978-0-486-60205-9 |edition=4th |location=Dover, New York |at=Book II, Part III, Propositions XII–XX; Queries 25–29 |language=en |author-link=Isaac Newton |orig-year=1730}}</ref> In the early 19th 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. |url={{google books |plainurl=y |id=EbDw1lV_MKsC}} |title=The Rise of the Wave Theory of Light: Optical theory and experiment in the early nineteenth century |journal=Physics Today |publisher=University of Chicago Press |year=1989 |isbn=978-0-226-07886-1 |volume=43 |pages=78–80 |language=en-us |bibcode=1990PhT....43d..78B |doi=10.1063/1.2810533 |oclc=18069573 |issue=4}}</ref> [[James Clerk Maxwell]]'s 1865 [[Maxwell's equations|prediction]]<ref name="maxwell">{{cite journal |last=Maxwell |first=James Clerk |author-link=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 |bibcode=1865RSPT..155..459C |doi=10.1098/rstl.1865.0008 |s2cid=186207827 |title-link=A dynamical theory of the electromagnetic field}} This article followed a presentation by Maxwell on 8&nbsp;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=Heinrich |author-link=Heinrich Hertz |year=1888 |title=Über Strahlen elektrischer Kraft |journal=Sitzungsberichte der Preussischen Akademie der Wissenschaften |language=de |volume=1888 |pages=1297–1307 |place=Berlin, Deutchland}}</ref> – seemed to be the final blow to particle models of light.

[[File:Light-wave.svg|thumb|upright=1.25|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 {{nobr| {{mvar|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" pp. 276ff., §1.4 "photoelectric effect" in {{harvnb|Alonso|Finn|1968}}.</ref>{{efn|
The phrase "no matter how intense" refers to intensities ''below'' approximately {{10^|13}}&nbsp;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^|14}}&nbsp;W/cm<sup>2</sup>, the classical wave description correctly predicts the energy acquired by electrons, called [[ponderomotive energy]].<ref>See also: {{cite journal |bibcode = 1996AIPC..369.1234B |title = Photon density and the correspondence principle of electromagnetic interaction |journal = AIP Conference Proceedings |volume = 369 |pages = 1234–1243 |last1 = Boreham|first1 = Bruce W. |last2 = Hora|first2 = Heinrich |last3 = Bolton |first3 = Paul R. |year = 1996 |doi = 10.1063/1.50410}}</ref>
By comparison, sunlight is only about 0.1&nbsp;W/cm<sup>2</sup>.
}}

At the same time, investigations of [[black-body radiation]] carried out over four decades (1860–1900) by various researchers<ref name="Wien1911">{{cite web |last=Wien |first=W. |author-link=Wilhelm Wien |year=1911 |url=http://nobelprize.org/nobel_prizes/physics/laureates/1911/wien-lecture.html |title=Wilhelm Wien Nobel Lecture |website=nobelprize.org}}</ref> culminated in [[Max Planck]]'s [[Planck constant|hypothesis]]<ref name="Planck1901">
{{cite journal |last=Planck |first=Max |author-link=Max Planck |year=1901 |title=Über das Gesetz der Energieverteilung im Normalspectrum |journal=[[Annalen der Physik]] |language=de |volume=4 |issue=3 |pages=553–563 |bibcode=1901AnP...309..553P |doi=10.1002/andp.19013090310 |doi-access=free}} [https://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=Max |author-link=Max Planck |year=1920 |title=Max Planck's Nobel Lecture |url=http://nobelprize.org/nobel_prizes/physics/laureates/1918/planck-lecture.html |publisher=nobelprize.org}}</ref> that the energy of ''any'' system that absorbs or emits electromagnetic radiation of frequency {{mvar|ν}} is an integer multiple of an energy quantum {{nobr| {{mvar|E}} {{=}} {{mvar|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=Albert |author-link=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]] |language=de |volume=10 |pages=817–825}} An [[wikisource:Translation: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=Albert |author-link=Albert Einstein |year=1916 |title=Zur Quantentheorie der Strahlung |journal=Mitteilungen der Physikalischen Gesellschaft zu Zürich |language=de |volume=16 |page=47}} Also ''Physikalische Zeitschrift'' (in German), '''18''', 121–128 (1917).</ref> Einstein showed that, if [[Planck's law]] regarding black-body radiation is accepted, the energy quanta must also carry [[momentum]] {{nobr|{{mvar| p {{=}} {{sfrac| h | λ }} }},}} making them full-fledged particles. This photon momentum was observed experimentally by [[Arthur Compton]],<ref name="Compton1923">{{cite journal |last=Compton |first=Arthur |author-link=Arthur Compton |year=1923 |title=A quantum theory of the scattering of X-rays by light elements |url=https://history.aip.org/history/exhibits/gap/Compton/Compton.html#compton1 |journal=[[Physical Review]] |language=en |volume=21 |issue=5 |pages=483–502 |bibcode=1923PhRv...21..483C |doi=10.1103/PhysRev.21.483 |doi-access=free}}</ref> for which he received the Nobel Prize in 1927. The pivotal question then, was 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. |author-link=Abraham Pais |year=1982 |title=Subtle is the Lord: The science and the life of Albert Einstein |url=https://archive.org/details/subtleislordscie00pais |publisher=Oxford University Press |isbn=978-0-19-853907-0}}</ref> and was solved in [[quantum electrodynamics]] and its successor, the [[Standard Model]]. (See ''{{section link||Quantum field theory}}'' and ''{{section link||As a gauge boson}}'', below.)

[[File:Bohr-atom-PAR.svg|thumb|Up to 1923, most physicists were reluctant to accept that light itself was quantized. Instead, they tried to explain photon behaviour 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=Robert A. |author-link=Robert Millikan |year=1924 |title=Robert A. Millikan's Nobel Lecture |url=http://nobelprize.org/nobel_prizes/physics/laureates/1923/millikan-lecture.html}}</ref> However, before Compton's experiment<ref name="Compton1923"/> showed that photons carried momentum proportional to their [[wave number]] (1922),{{Full citation needed|date=November 2020}} 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"/> 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 absorbed or emitted radiation. Attitudes changed over time. In part, the change can be traced to experiments such as those revealing [[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]].<ref name="Bohr1924">{{cite journal |last1=Bohr |first1=Niels |author-link=Niels Bohr |last2=Kramers |first2=Hendrik Anthony |author2-link=Hendrik Anthony Kramers |last3=Slater |first3=John C. |author3-link=John C. Slater |year=1924 |title=The Quantum Theory of Radiation |journal=[[Philosophical Magazine]] |volume=47 |issue=281 |pages=785–802 |doi=10.1080/14786442408565262}} Also ''[[European Physical Journal|Zeitschrift für Physik]]'' (in German), '''24''', p. 69 (1924).</ref> An important feature of the BKS theory is how it treated the [[conservation of energy]] and the [[conservation of momentum]]. In the BKS theory, energy and momentum are only conserved on the average across many interactions between matter and radiation. However, refined Compton experiments showed that the conservation laws hold for individual interactions.<ref>{{Cite journal|last=Howard |first=Don |date=December 2004|title=Who Invented the "Copenhagen Interpretation"? A Study in Mythology |journal=[[Philosophy of Science (journal)|Philosophy of Science]] |language=en |volume=71 |issue=5 |pages=669–682 |doi=10.1086/425941 |issn=0031-8248 |jstor=10.1086/425941 |s2cid=9454552}}</ref> 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=Werner |author-link=Werner Heisenberg |year=1933 |title=Heisenberg Nobel lecture |url=http://nobelprize.org/nobel_prizes/physics/laureates/1932/heisenberg-lecture.html}}</ref>

A few physicists persisted<ref name="Mandel1976">
{{cite book |last=Mandel |first=Leonard |title=The case for and against semiclassical radiation theory |journal=Progess in Optics |publisher=North-Holland |year=1976 |isbn=978-0-444-10806-7 |editor=Wolf |editor-first=E. |series=[[Progress in Optics]] |volume=13 |pages=27–69 |language=en |bibcode=1976PrOpt..13...27M |doi=10.1016/S0079-6638(08)70018-0 |author-link=Leonard Mandel}}</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 from chemical and physical experiments for the existence of photons was overwhelming by the 1970s, this evidence could not be considered as ''absolutely'' definitive; since it relied on the interaction of light with matter, and a sufficiently complete 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.{{efn|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, Roger, & Aspect (1986);<ref>{{cite journal |last1=Grangier |first1=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 |issue=4 |pages=173–179 |doi=10.1209/0295-5075/1/4/004 |bibcode=1986EL......1..173G |citeseerx=10.1.1.178.4356|s2cid=250837011 }}</ref> This work is reviewed and simplified further in Thorn, Neel, ''et al.'' (2004).<ref>{{cite journal |last1=Thorn |first1=J.J. |last2=Neel |first2=M.S. |last3=Donato |first3=V.W. |last4=Bergreen |first4=G.S. |last5=Davies |first5=R.E. |last6=Beck |first6=M. |year=2004 |title=Observing the quantum behavior of light in an undergraduate laboratory |journal=[[American Journal of Physics]] |volume=72 |issue=9 |pages=1210–1219 |doi=10.1119/1.1737397 |bibcode=2004AmJPh..72.1210T |url=http://people.whitman.edu/~beckmk/QM/grangier/Thorn_ajp.pdf}}</ref>
}} Hence, Einstein's hypothesis that quantization is a property of light itself is considered to be proven.

==Wave–particle duality and uncertainty principles==
[[File:Mach-Zehnder photons animation.gif|thumb|upright=1.25|Photons in a [[Mach–Zehnder interferometer]] exhibit wave-like interference and particle-like detection at [[Photon-counting detector|single-photon detectors]].]]
Photons obey the laws of quantum mechanics, and so their behavior has both wave-like and particle-like aspects. When a photon is detected by a measuring instrument, it is registered as a single, particulate unit. However, the ''probability'' of detecting a photon is calculated by equations that describe waves. This combination of aspects is known as [[wave–particle duality]]. For example, the [[probability distribution]] for the location at which a photon might be detected displays clearly wave-like phenomena such as [[diffraction]] and [[Interference (wave propagation)|interference]]. A single photon passing through a [[double-slit experiment|double slit]] has its energy received at a point on the screen with a probability distribution given by its interference pattern determined by [[Maxwell's equations|Maxwell's wave equations]].<ref name="Taylor1909">{{cite conference |last=Taylor |first=Geoffrey Ingram |author-link=Geoffrey Ingram Taylor |year=1909 |title=Interference fringes with feeble light |conference=Cambridge Philosophical Society |volume=15 |pages=114–115 |work=Proceedings of the Cambridge Philosophical Society}}</ref> However, experiments confirm that the photon is ''not'' a short pulse of electromagnetic radiation; a photon's Maxwell waves will diffract, but photon energy does not spread out as it propagates, nor does this energy divide when it encounters a [[beam splitter]].<ref name="Saleh">{{cite book |last1=Saleh |first1=B. E. A. |title=Fundamentals of Photonics |last2=Teich |first2=M. C. |publisher=Wiley |year=2007 |isbn=978-0-471-35832-9 |language=en |name-list-style=amp}}</ref> Rather, the received photon acts like a [[point-like particle]] since it is absorbed or emitted ''as a whole'' by arbitrarily small systems, including systems much smaller than its wavelength, such as an atomic nucleus (≈10<sup>−15</sup> m across) or even the point-like [[electron]].

While many introductory texts treat photons using the mathematical techniques of non-relativistic quantum mechanics, this is in some ways an awkward oversimplification, as photons are by nature intrinsically relativistic. Because photons have zero [[rest mass]], no [[wave function]] defined for a photon can have all the properties familiar from wave functions in non-relativistic quantum mechanics.{{efn|The issue was first formulated by Theodore Duddell Newton and [[Eugene Wigner]].<ref>{{cite journal|last1=Newton|first1=T.D.|last2=Wigner |first2=E.P. |author-link2=Eugene Wigner |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|url=https://cds.cern.ch/record/1062641/files/RevModPhys.21.400.pdf|doi-access=free}}</ref><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|issue=1–2|pages=97–116|doi=10.12693/APhysPolA.86.97|bibcode=1994AcPPA..86...97B|doi-access=free}}</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|pmid=9912446|bibcode=1995PhRvA..52.1875S|issue=3}}</ref> The challenges arise from the fundamental nature of the [[Lorentz group]], which describes the symmetries of [[spacetime]] in special relativity. Unlike the generators of [[Galilean transformation]]s, the generators of [[Lorentz boost]]s do not commute, and so simultaneously assigning low uncertainties to all coordinates of a relativistic particle's position becomes problematic.<ref>{{cite book|last=Bialynicki-Birula|first=I.|year=1996|title=Photon wave function|journal=Progess 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|bibcode=1996PrOpt..36..245B|s2cid=17695022 }}</ref>}} In order to avoid these difficulties, physicists employ the second-quantized theory of photons described below, [[quantum electrodynamics]], in which photons are quantized excitations of electromagnetic modes.<ref name="scully1997">{{cite book |last1=Scully |first1=M. O. |url={{google books |plainurl=y |id=20ISsQCKKmQC}} |title=Quantum Optics |last2=Zubairy |first2=M. S. |publisher=Cambridge University Press |year=1997 |isbn=978-0-521-43595-6 |location=Cambridge, England |language=en-uk}}</ref>

Another difficulty is finding the proper analogue for the [[uncertainty principle]], an idea frequently attributed to Heisenberg, who introduced the concept in analyzing a [[thought experiment]] involving [[Heisenberg's microscope|an electron and a high-energy photon]]. However, Heisenberg did not give precise mathematical definitions of what the "uncertainty" in these measurements meant. The precise mathematical statement of the position–momentum uncertainty principle is due to [[Earle Hesse Kennard|Kennard]], [[Wolfgang Pauli|Pauli]], and [[Hermann Weyl|Weyl]].<ref>{{Cite journal|last1=Busch|first1=Paul|author-link1=Paul Busch (physicist) |last2=Lahti|first2=Pekka|last3=Werner|first3=Reinhard F.|date=2013-10-17|title=Proof of Heisenberg's Error-Disturbance Relation|journal=Physical Review Letters|language=en|volume=111|issue=16|pages=160405|doi=10.1103/PhysRevLett.111.160405|pmid=24182239|arxiv=1306.1565|bibcode=2013PhRvL.111p0405B|s2cid=24507489|issn=0031-9007|url=https://www.repo.uni-hannover.de/bitstream/123456789/8834/1/Proof%20of%20Heisenberg%e2%80%99s%20Error-Disturbance%20Relation.pdf}}</ref><ref>{{Cite journal|last=Appleby|first=David Marcus|date=2016-05-06|title=Quantum Errors and Disturbances: Response to Busch, Lahti and Werner|journal=Entropy|language=en|volume=18|issue=5|pages=174|doi=10.3390/e18050174|arxiv=1602.09002|bibcode=2016Entrp..18..174A|doi-access=free}}</ref> The uncertainty principle applies to situations where an experimenter has a choice of measuring either one of two "canonically conjugate" quantities, like the position and the momentum of a particle. According to the uncertainty principle, no matter how the particle is prepared, it is not possible to make a precise prediction for both of the two alternative measurements: if the outcome of the position measurement is made more certain, the outcome of the momentum measurement becomes less so, and vice versa.<ref name="L&L">{{cite book |last1=Landau |first1=Lev D. |url=https://archive.org/details/QuantumMechanics_104 |title=Quantum Mechanics: Non-Relativistic Theory |last2=Lifschitz |first2=Evgeny M. |publisher=[[Pergamon Press]] |year=1977 |isbn=978-0-08-020940-1 |edition=3rd |volume=3 |language=en |oclc=2284121 |author-link1=Lev Landau |author-link2=Evgeny Lifshitz}}</ref> A [[coherent state]] minimizes the overall uncertainty as far as quantum mechanics allows.<ref name="scully1997"/> [[Quantum optics]] makes use of coherent states for modes of the electromagnetic field. There is a tradeoff, reminiscent of the position–momentum uncertainty relation, between measurements of an electromagnetic wave's amplitude and its phase.<ref name="scully1997"/> This is sometimes informally expressed in terms of the uncertainty in the number of photons present in the electromagnetic wave, <math>\Delta N</math>, and the uncertainty in the phase of the wave, <math>\Delta \phi</math>. However, this cannot be an uncertainty relation of the Kennard–Pauli–Weyl type, since unlike position and momentum, the phase <math>\phi</math> cannot be represented by a [[Hermitian operator]].<ref>{{Cite journal |last1=Busch |first1=P. |last2=Grabowski |first2=M. |last3=Lahti |first3=P. J. |date=January 1995 |title=Who Is Afraid of POV Measures? Unified Approach to Quantum Phase Observables |journal=[[Annals of Physics]] |language=en |volume=237 |issue=1 |pages=1–11 |bibcode=1995AnPhy.237....1B |doi=10.1006/aphy.1995.1001}}</ref>

==Bose–Einstein model of a photon gas==
{{Main|Bose gas|Bose–Einstein statistics|Spin-statistics theorem|Gas in a box|Photon gas}}

In 1924, [[Satyendra Nath Bose]] derived [[Planck's law of black-body radiation]] without using any electromagnetism, but rather by using a modification of coarse-grained counting of [[phase space]].<ref name="Bose1924">{{cite journal |last=Bose |first=Satyendra Nath |author-link=Satyendra Nath Bose |year=1924 |title=Plancks Gesetz und Lichtquantenhypothese |journal=[[European Physical Journal|Zeitschrift für Physik]] |language=de |volume=26 |issue=1 |pages=178–181 |bibcode=1924ZPhy...26..178B |doi=10.1007/BF01327326 |s2cid=186235974}}</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=Albert |author-link=Albert Einstein |year=1924 |title=Quantentheorie des einatomigen idealen Gases |journal=Sitzungsberichte der Preussischen Akademie der Wissenschaften (Berlin), Physikalisch-mathematische Klasse |language=de |volume=1924 |pages=261–267}}</ref><ref name="Einstein1925">{{cite book |last=Einstein |first=Albert |title=Quantentheorie des einatomigen idealen Gases, Zweite Abhandlung |journal=Sitzungsberichte der Preussischen Akademie der Wissenschaften (Berlin), Physikalisch-mathematische Klasse |year=1925 |isbn=978-3-527-60895-9 |volume=1925 |pages=3–14 |language=de |doi=10.1002/3527608958.ch28 |author-link=Albert Einstein}}</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 (bosons) 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 |last1=Anderson |first1=M. H. |last2=Ensher |first2=J. R. |last3=Matthews |first3=M. R. |last4=Wieman |first4=Carl E. |author4-link=Carl Wieman |last5=Cornell |first5=Eric Allin |author5-link=Eric Allin Cornell |year=1995 |title=Observation of Bose–Einstein Condensation in a Dilute Atomic Vapor |journal=[[Science (journal)|Science]] |volume=269 |issue=5221 |pages=198–201 |bibcode=1995Sci...269..198A |doi=10.1126/science.269.5221.198 |jstor=2888436 |pmid=17789847 |s2cid=540834 |doi-access=}}</ref> It was later used by [[Lene Hau]] to slow, and then completely stop, light in 1999<ref>{{Cite web |last=Cuneo |first=Michael |date=1999-02-18 |title=Physicists Slow Speed of Light |url=https://news.harvard.edu/gazette/story/1999/02/physicists-slow-speed-of-light/ |access-date=2023-12-07 |website=Harvard Gazette |language=en-US}}</ref> and 2001.<ref>{{Cite web |title=Light Changed to Matter, Then Stopped and Moved |url=https://www.photonics.com/Articles/Light_Changed_to_Matter_Then_Stopped_and_Moved/a28520 |access-date=2023-12-07 |website=www.photonics.com}}</ref>

The modern view on this is that photons are, by virtue of their integer spin, bosons (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 |last1=Streater |first1=R. F. |title=PCT, Spin and Statistics, and All That |last2=Wightman |first2=A. S. |publisher=Addison-Wesley |year=1989 |isbn=978-0-201-09410-7 |language=en}}</ref>

==Stimulated and spontaneous emission==
{{Main|Stimulated emission|Laser}}
[[File:Stimulatedemission.png|thumb|upright=1.75|[[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, Albert Einstein showed that Planck's radiation law could be derived from a semi-classical, statistical treatment of photons and atoms, which implies a link between the rates at which atoms emit and absorb photons. The condition follows from the assumption that functions of the emission and absorption of radiation by the atoms are independent of each other, and that thermal equilibrium is made by way of the radiation's interaction with the atoms. Consider a cavity in [[thermal equilibrium]] with all parts of itself and filled with [[electromagnetic radiation]] and that the atoms 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 at which they are ''absorbed''.<ref name="Einstein1916a">{{cite journal |last=Einstein |first=Albert |author-link=Albert Einstein |year=1916 |title=Strahlungs-emission und -absorption nach der Quantentheorie |journal=[[Verhandlungen der Deutschen Physikalischen Gesellschaft]] |language=de |volume=18 |pages=318–323 |bibcode=1916DPhyG..18..318E}}</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 of 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, or the emission of a photon initiated by the interaction of the atom with a passing photon and the return of the atom to the lower-energy state. 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 emissions in response to ambient photons ([[stimulated emission|induced or stimulated emission]]). In thermodynamic equilibrium, the number of atoms in state <math>i</math> and those in state <math>j</math> 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</math> and <math>g_j</math> are the [[degenerate energy level|degeneracy]] of the state <math>i</math> and that of <math>j</math>, respectively, <math>E_i</math> and <math>E_j</math> their energies, <math>k</math> the [[Boltzmann constant]] and <math>T</math> 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 <math>A_{ij}</math> and <math>B_{ij}</math> are collectively known as the ''Einstein coefficients''.<ref>{{cite book |last1=Wilson |first1=J. |title=Lasers: Principles and Applications |last2=Hawkes |first2=F. J. B. |publisher=Prentice Hall |year=1987 |isbn=978-0-13-523705-2 |location=New York |at=Section 1.4 |language=en-us}}</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>{{cite journal |last=Einstein |first=Albert |author-link=Albert Einstein |year=1916 |title=Strahlungs-emission und -absorption nach der Quantentheorie |journal=[[Verhandlungen der Deutschen Physikalischen Gesellschaft]] |language=de |volume=18 |pages=318–323 |bibcode=1916DPhyG..18..318E |quote=p. 322: Die Konstanten <math>A^n_m</math> and <math>B^n_m</math> würden sich direkt berechnen lassen, wenn wir im Besitz einer im Sinne der Quantenhypothese modifizierten Elektrodynamik und Mechanik wären."}}</ref> Not long thereafter, in 1926, [[Paul Dirac]] derived the <math>B_{ij}</math> rate constants by using a semiclassical approach,<ref name="Dirac1926">{{cite journal |last=Dirac |first=Paul A. M. |author-link=Paul Dirac |year=1926 |title=On the Theory of Quantum Mechanics |journal=Proceedings of the Royal Society A |volume=112 |issue=762 |pages=661–677 |bibcode=1926RSPSA.112..661D |doi=10.1098/rspa.1926.0133 |doi-access=free}}</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=Paul A. M. |author-link=Paul Dirac |year=1927 |title=The Quantum Theory of the Emission and Absorption of Radiation |journal=Proceedings of the Royal Society A |volume=114 |issue=767 |pages=243–265 |bibcode=1927RSPSA.114..243D |doi=10.1098/rspa.1927.0039 |doi-access=free}}</ref><ref name="Dirac1927b">
{{cite journal |last=Dirac |first=Paul A. M. |author-link=Paul Dirac |year=1927b |title=The Quantum Theory of Dispersion |journal=Proceedings of the Royal Society A |volume=114 |issue=769 |pages=710–728 |bibcode=1927RSPSA.114..710D |doi=10.1098/rspa.1927.0071 |doi-access=free}}</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 |last1=Heisenberg |first1=Werner |author-link=Werner Heisenberg |last2=Pauli |first2=Wolfgang |author-link2=Wolfgang Pauli |year=1929 |title=Zur Quantentheorie der Wellenfelder |journal=[[European Physical Journal|Zeitschrift für Physik]] |language=de |volume=56 |issue=1–2 |page=1 |bibcode=1929ZPhy...56....1H |doi=10.1007/BF01340129 |s2cid=121928597}}</ref><ref name="Heisenberg1930">
{{cite journal |last1=Heisenberg |first1=Werner |author-link=Werner Heisenberg |last2=Pauli |first2=Wolfgang |author-link2=Wolfgang Pauli |year=1930 |title=Zur Quantentheorie der Wellenfelder |journal=[[European Physical Journal|Zeitschrift für Physik]] |language=de |volume=59 |issue=3–4 |page=139 |bibcode=1930ZPhy...59..168H |doi=10.1007/BF01341423 |s2cid=186219228}}</ref><ref name="Fermi1932">{{cite journal |last=Fermi |first=Enrico |author-link=Enrico Fermi |year=1932 |title=Quantum Theory of Radiation |journal=[[Reviews of Modern Physics]] |volume=4 |issue=1 |page=87 |bibcode=1932RvMP....4...87F |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 of the two paths a single photon would take.<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=Max |author-link=Max Born |year=1926 |title=Zur Quantenmechanik der Stossvorgänge |journal=[[European Physical Journal|Zeitschrift für Physik]] |language=de |volume=37 |issue=12 |pages=863–867 |bibcode=1926ZPhy...37..863B |doi=10.1007/BF01397477 |s2cid=119896026}}</ref><ref name="Born1926b">{{cite journal |last=Born |first=Max |author-link=Max Born |year=1926 |title=Quantenmechanik der Stossvorgänge |journal=[[European Physical Journal|Zeitschrift für Physik]] |language=de |volume=38 |issue=11–12 |page=803 |bibcode=1926ZPhy...38..803B |doi=10.1007/BF01397184 |s2cid=126244962}}</ref> was inspired by Einstein's later work searching for a more complete theory.<ref name="ghost_field">{{cite book|last=Pais|first=A.|author-link=Abraham Pais|year=1986|url={{google books |plainurl=y |id=mREnwpAqz-YC|page=260}}|page=260|title=Inward Bound: Of Matter and Forces in the Physical World|publisher=Oxford University Press|isbn=978-0-19-851997-3}} 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>

==Quantum field theory==
===Quantization of the electromagnetic field===
{{Main|Quantum field theory}}
[[File:VisibleEmrWavelengths.svg|thumb|upright=1.2|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''&nbsp;=&nbsp;''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=Peter |author-link=Peter Debye |year=1910 |title=Der Wahrscheinlichkeitsbegriff in der Theorie der Strahlung |url=https://zenodo.org/record/1424189 |journal=[[Annalen der Physik]] |language=de |volume=33 |issue=16 |pages=1427–1434 |bibcode=1910AnP...338.1427D |doi=10.1002/andp.19103381617}}</ref> He 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 black-body 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 |last1=Born |first1=Max |author-link=Max Born |last2=Heisenberg |first2=Werner |author2-link=Werner Heisenberg |last3=Jordan |first3=Pascual |author3-link=Pascual Jordan |year=1925 |title=Quantenmechanik II |journal=[[European Physical Journal|Zeitschrift für Physik]] |language=de |volume=35 |issue=8–9 |pages=557–615 |bibcode=1926ZPhy...35..557B |doi=10.1007/BF01379806 |s2cid=186237037}}</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.
[[File:Electron-scattering.svg|left|thumb|[[Feynman diagram]] of two electrons interacting by exchange of a virtual photon.]]
[[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.<ref>{{cite journal|last1=Jaeger|first1=Gregg|title=Are virtual particles less real?|journal=Entropy |volume=21 |issue=2|page=141|date=2019|doi=10.3390/e21020141|pmid=33266857|pmc=7514619|bibcode=2019Entrp..21..141J|url=http://philsci-archive.pitt.edu/15858/1/Jaeger%20Are%20Virtual%20Particles%20Less%20Real_%20entropy-21-00141-v3.pdf|doi-access=free}}</ref>

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]].<ref>{{Cite book |last=Zee |first=Anthony |title=[[Quantum Field Theory in a Nutshell]] |date=2003 |publisher=[[Princeton University Press]] |isbn=0-691-01019-6 |location=Princeton, New Jersey |language=en-us |oclc=50479292 |author-link=Anthony Zee}}</ref>

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>{{Cite book |last1=Itzykson |first1=C. |url=https://archive.org/details/quantumfieldtheo0000itzy |title=Quantum Field Theory |last2=Zuber |first2=J.-B. |publisher=McGraw-Hill |year=1980 |isbn=978-0-07-032071-0 |at=Photon–photon-scattering section 7–3–1, renormalization chapter 8–2 |url-access=registration}}</ref> Such photon–photon scattering (see [[two-photon physics]]), 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|arxiv = 0711.3003|s2cid=118517359}}</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.

===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. |url={{google books |plainurl=y |id=nnuW_kVJ500C}} |title=Quantum field theory |publisher=Cambridge University Press |year=1996 |isbn=978-0-521-47814-4 |edition=2nd |location=England |language=en-uk}}</ref> For the [[electromagnetic field]], this gauge symmetry is the [[Abelian group|Abelian]] [[unitary group|U(1) symmetry]] of [[complex number]]s of absolute value 1, which reflects the ability to vary the [[complex geometry|phase]] of a complex field without affecting [[observable]]s or [[real number|real valued functions]] made from it, such as the [[energy]] or the [[Lagrangian (field theory)|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 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.<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=978-0-521-26092-3|url=https://archive.org/details/elementarypartic00hugh}}</ref>

===Hadronic properties===
{{main|Photon structure function}}
Measurements of the interaction between energetic photons and [[hadron]]s 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>{{Cite journal |last1=Bauer |first1=T. H. |last2=Spital |first2=R. D. |last3=Yennie |first3=D. R. |last4=Pipkin |first4=F. M. |year=1978 |title=The hadronic properties of the photon in high-energy interactions |journal=Reviews of Modern Physics |volume=50 |issue=2 |pages=261 |bibcode=1978RvMP...50..261B |doi=10.1103/RevModPhys.50.261}}</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 mesons.<ref>{{cite journal |last1=Sakurai |first1=J. J. |year=1960 |title=Theory of strong interactions |journal=Annals of Physics |volume=11 |issue=1 |pages=1–48 |bibcode=1960AnPhy..11....1S |doi=10.1016/0003-4916(60)90126-3}}</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>{{Cite journal |doi=10.1016/0370-2693(73)90520-0 |title=Two-photon processes in the parton model |journal=[[Physics Letters B]] |volume=44 |issue=2 |pages=195 |year=1973 |last1=Walsh |first1=T.F. |last2=Zerwas |first2=P. |bibcode=1973PhLB...44..195W}}</ref><ref>{{Cite journal |doi=10.1016/0550-3213(77)90038-4 |title=Anomalous cross section for photon–photon scattering in gauge theories |journal=Nuclear Physics B |volume=120 |issue=2 |pages=189–202 |year=1977 |last1=Witten |first1=E. |bibcode=1977NuPhB.120..189W}}</ref> A comprehensive comparison of data with theoretical predictions was presented in a review in 2000.<ref>{{Cite journal |doi=10.1016/S0370-1573(99)00115-5 |title=The photon structure from deep inelastic electron–photon scattering |journal=Physics Reports |volume=332 |issue=4–6 |pages=165–317| year = 2000 |last1=Nisius |first1=R. |bibcode=2000PhR...332..165N |arxiv=hep-ex/9912049|s2cid=119437227 }}</ref>

===Contributions to the mass of a system===
{{See also|Mass in special relativity|Mass in 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=978-0-534-39294-9 |language=en}}</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 correction]]s" 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 |last1=Itzykson |first1=C. |url=https://archive.org/details/quantumfieldtheo0000itzy |title=Quantum Field Theory |last2=Zuber |first2=J.-B. |publisher=McGraw-Hill |year=1980 |isbn=978-0-07-032071-0 |url-access=registration}}</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 lens]]ing, 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|last1=Stephani|first1=H.|url={{google books |plainurl=y |id=V04_vLQvstcC|page=86}}|last2=Stewart|first2=J.|pages=86 ff, 108 ff|title=General Relativity: An Introduction to the Theory of Gravitational Field|isbn=978-0-521-37941-0|publisher=Cambridge University Press|year=1990}}</ref>

==In matter==
{{See also|Refractive index|Group velocity|Photochemistry}}

Light that travels through transparent matter does so at a lower speed than ''c'', the speed of light in vacuum. 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 that 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 to produce [[quasi-particle]]s known as [[polariton]] (see [[List of quasiparticles|this list]] for some other quasi-particles); this polariton has a nonzero [[effective mass (solid-state physics)|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]] (not to be confused with scattering). 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 |last1=Patterson |first1=J. D. |title=Solid-State Physics: Introduction to the Theory |last2=Bailey |first2=B. C. |publisher=[[Springer Science+Business Media|Springer]] |year=2007 |isbn=978-3-540-24115-7 |language=en}}</ref>

Photons can be scattered by matter. For example, photons engage in so many collisions on the way from the [[Solar core|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|first=R.|last=Naeye|publisher=CRC Press|year=1998|isbn=978-0-7503-0484-9|url={{google books |plainurl=y |id=06_9B7S_q_YC|page=16}}|oclc=40180195}}</ref> however, once in open space, a photon takes only 8.3 minutes to reach Earth.<ref>{{Cite book |last1 = Koupelis |first1 = Theo |last2 = Kuhn |first2 = Karl F. |year = 2007 | url=https://books.google.com/books?id=6rTttN4ZdyoC&pg=PA102 |title = In Quest of the Universe |page=102 |publisher=Jones and Bartlett Canada |isbn = 9780763743871}}</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 [[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 |last1=Pine |first1=S. H. |title=Organic Chemistry |last2=Hendrickson |first2=J. B. |last3=Cram |first3=D. J. |last4=Hammond |first4=G. S. |publisher=McGraw-Hill |year=1980 |isbn=978-0-07-050115-7 |edition=4th |language=en}}</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>

==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 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 of sufficient energy strikes a metal plate and knocks free an electron, initiating an ever-amplifying avalanche of electrons. [[Semiconductor]] [[charge-coupled device]] chips use a similar effect: 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 contained in the device, causing a detectable change of [[Electrical conductivity|conductivity]] of the gas.<ref>Photomultiplier section 1.1.10, CCDs section 1.1.8, Geiger counters section 1.3.2.1 in {{cite book |last=Kitchin |first=C. R. |title=Astrophysical Techniques |publisher=CRC Press |year=2008 |isbn=978-1-4200-8243-2 |location=Boca Raton, Florida |language=en-us}}</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 determine the frequency of the light emitted from a given photon emission. For example, the [[emission spectrum]] of a [[gas-discharge lamp]] can be altered by filling it with (mixtures of) gases with different electronic [[energy level]] configurations.<ref>{{cite book |last=Waymouth |first=John |url=https://archive.org/details/electricdischarg00waym |title=Electric Discharge Lamps |date=1971 |publisher=The M.I.T. Press |isbn=978-0-262-23048-3 |location=Cambridge, Massachusetts |language=en-us |url-access=registration}}</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 spectrum 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|author1=Denk, W. |author1-link=Winfried Denk|author2-link=Karel Svoboda (scientist)|author2=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|s2cid=2414593 |doi-access=free}}</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 |last=Lakowicz |first=J. R. |url={{google books |plainurl=y |id=-PSybuLNxcAC|page=529}} |title=Principles of Fluorescence Spectroscopy |publisher=Springer |year=2006 |isbn=978-0-387-31278-1 |pages=529 ff |language=en}}</ref>

Several different kinds of [[hardware random number generator]]s 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|first1=T.|last1=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 |s2cid=13118587}}</ref><ref>{{Cite journal|first1=A.|last1=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>

==Quantum optics and computation==
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>
Introductory-level material on the various sub-fields of quantum optics can be found in {{cite book |last=Fox |first=M. |url={{google books |id=Q-4dIthPuL4C |plainurl=y}} |title=Quantum Optics: An introduction |publisher=Oxford University Press |year=2006 |isbn=978-0-19-856673-1 |via=Google Books}}
</ref>

[[Two-photon physics]] studies interactions between photons, which are rare. In 2018, Massachusetts Institute of Technology researchers announced the discovery of bound photon triplets, which may involve [[polariton]]s.<ref name="NW-20180216">{{cite news |last=Hignett |first=Katherine |date=16 February 2018 |title=Physics creates new form of light that could drive the quantum computing revolution |magazine=[[Newsweek]] |url=http://www.newsweek.com/photons-light-physics-808862 |access-date=17 February 2018 }}</ref><ref name="SCI-20180216">{{cite journal |last1=Liang |first1=Qi-Yu |display-authors=etal |date=16 February 2018 |title=Observation of three-photon bound states in a quantum nonlinear medium |journal=[[Science (journal)|Science]] |volume=359 |issue=6377 |pages=783–786 |doi=10.1126/science.aao7293 |pmid=29449489 |pmc=6467536 |arxiv=1709.01478 |bibcode=2018Sci...359..783L }}</ref>

==See also==
{{Portal|Physics}}
{{div col |colwidth=15em |content=

* [[Advanced Photon Source]] at&nbsp;Argonne National Laboratory
* [[Ballistic photon]]
* [[Dirac equation]]
* [[Doppler effect]]
* [[EPR paradox]]
* [[High energy X-ray imaging technology]]
* [[Luminiferous aether]]
* [[Medipix]]
* [[Phonon]]
* [[Photography]]
* [[Photon counting]]
* [[Photon epoch]]
* [[Photonic molecule]]
* [[Photonics]]
* [[Single-photon source]]
* [[Static forces and virtual-particle exchange]]
* [[Variable speed of light]]

}} <!-- end "content=" -->

==Notes==
{{notelist}}
{{reflist|group="Note"}}

==References==
{{reflist|25em}}

==Further reading==
<!-- Ordered by date published; two general histories cited at end -->
{{refbegin}}

;By date of publication:
* {{cite book |last1=Alonso |first1=M. |last2=Finn |first2=E. J. |year=1968 |title=Fundamental University Physics |volume=III: Quantum and Statistical Physics |language=en |publisher=Addison-Wesley |isbn=978-0-201-00262-1}}
* {{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 |issue=4 |pages=853–860 |doi=10.1103/PhysRevD.9.853 |bibcode=1974PhRvD...9..853C |s2cid=118320287 |url=http://www.escholarship.org/uc/item/3wm0v847}}
* {{cite book |last=Pais |first=Abraham |author-link=Abraham Pais |year=1982 |language=en |title=[[Subtle is the Lord: The Science and the Life of Albert Einstein]] |publisher=Oxford University Press}}
* {{cite book |last=Feynman |first=Richard |author-link=Richard Feynman |year=1985 |isbn=978-0-691-12575-6 |language=en-us |title=QED: The Strange Theory of Light and Matter |publisher=Princeton University Press |title-link=QED: The Strange Theory of Light and Matter}}
* {{cite journal |last1=Grangier |first1=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 |issue=4 |pages=173–179 |doi=10.1209/0295-5075/1/4/004 |bibcode=1986EL......1..173G |citeseerx=10.1.1.178.4356 |s2cid=250837011}}
* {{cite journal |last=Lamb |first=Willis E. |author-link=Willis Lamb |year=1995 |title=Anti-photon |journal=[[Applied Physics B]] |volume=60 |issue=2–3 |pages=77–84 |doi=10.1007/BF01135846 |bibcode=1995ApPhB..60...77L|s2cid=263785760 }}
* {{cite magazine |title=Special supplemental issue |magazine=Optics and Photonics News |volume=14 |date=October 2003 |url=http://www.sheffield.ac.uk/polopoly_fs/1.14183!/file/photon.pdf}}
** {{cite magazine |last1=Roychoudhuri |first1=C. |last2=Rajarshi |first2=R. |year=2003 |title=The nature of light: What is a photon? |magazine=[[Optics and Photonics News]] |volume=14 |pages=S1 (Supplement)}}
** {{cite magazine |last=Zajonc |first=A. |year=2003 |title=Light reconsidered |magazine=[[Optics and Photonics News]] |volume=14 |pages=S2–S5 (Supplement)}}
** {{cite magazine |last=Loudon |first=R. |year=2003 |title=What is a photon? |magazine=[[Optics and Photonics News]] |volume=14 |pages=S6–S11 (Supplement)}}
** {{cite magazine |last=Finkelstein |first=D. |year=2003 |title=What is a photon? |magazine=[[Optics and Photonics News]] |volume=14 |pages=S12–S17 (Supplement)}}
** {{cite magazine |last1=Muthukrishnan |first1=A. |last2=Scully |first2=M. O. |last3=Zubairy |first3=M. S. |year=2003 |title=The concept of the photon – revisited |magazine=[[Optics and Photonics News]] |volume=14 |pages=S18–S27 (Supplement)}}
** {{cite magazine |last1=Mack |first1=H. |last2=Schleich |author-link2=Wolfgang P. Schleich |first2=Wolfgang P. |year=2003 |title=A photon viewed from Wigner phase space |magazine=[[Optics and Photonics News]] |volume=14 |pages=S28–S35 (Supplement)}}
* {{cite web |last=Glauber |first=R. |year=2005 |title=One Hundred Years of Light Quanta |series=Physics Lecture |website=Nobel Prize |url=http://nobelprize.org/nobel_prizes/physics/laureates/2005/glauber-lecture.pdf |access-date=2009-06-29 |archive-url=https://web.archive.org/web/20080723150609/http://nobelprize.org/nobel_prizes/physics/laureates/2005/glauber-lecture.pdf |archive-date=2008-07-23 |url-status=dead}}
* {{Cite journal |last=Hentschel |first=K. |year=2007 |title=Light quanta: The maturing of a concept by the stepwise accretion of meaning |url=https://eldorado.tu-dortmund.de/handle/2003/24257 |journal=Physics and Philosophy |volume=1 |issue=2 |pages=1–20}}
;Education with single photons:
* {{cite journal |last1=Thorn |first1=J. J. |last2=Neel |first2=M. S. |last3=Donato |first3=V. W. |last4=Bergreen |first4=G. S. |last5=Davies |first5=R. E. |last6=Beck |first6=M. |year=2004 |title=Observing the quantum behavior of light in an undergraduate laboratory |journal=[[American Journal of Physics]] |volume=72 |issue=9 |pages=1210–1219 |doi=10.1119/1.1737397 |bibcode=2004AmJPh..72.1210T |url=http://people.whitman.edu/~beckmk/QM/grangier/Thorn_ajp.pdf}}
* {{cite journal |last1=Bronner |first1=P. |last2=Strunz |first2=Andreas |last3=Silberhorn |first3=Christine |last4=Meyn |first4=Jan-Peter |year=2009 |title=Interactive screen experiments with single photons |journal=[[European Journal of Physics]] |volume=30 |issue=2 |pages=345–353 |doi=10.1088/0143-0807/30/2/014 |bibcode=2009EJPh...30..345B |s2cid=38626417 |url=http://www.QuantumLab.de}}

{{refend}}

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