Editing Photon (section)

== 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 |access-date=2019-08-25 |archive-date=2020-03-14 |archive-url=https://web.archive.org/web/20200314211718/https://zenodo.org/record/1424189 |url-status=live }}</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|access-date=2021-05-19|archive-date=2023-06-11|archive-url=https://web.archive.org/web/20230611010352/http://philsci-archive.pitt.edu/15858/1/Jaeger%20Are%20Virtual%20Particles%20Less%20Real_%20entropy-21-00141-v3.pdf|url-status=live}}</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] {{Webarchive|url=https://web.archive.org/web/20080418033045/http://nobelprize.org/nobel_prizes/physics/laureates/1979/glashow-lecture.html |date=2008-04-18 }}, delivered 8 December 1979.</ref><ref name="Salam">[http://nobelprize.org/nobel_prizes/physics/laureates/1979/salam-lecture.html Abdus Salam Nobel lecture] {{Webarchive|url=https://web.archive.org/web/20080418033106/http://nobelprize.org/nobel_prizes/physics/laureates/1979/salam-lecture.html |date=2008-04-18 }}, delivered 8 December 1979.</ref><ref name="Weinberg">[http://nobelprize.org/nobel_prizes/physics/laureates/1979/weinberg-lecture.html Steven Weinberg Nobel lecture] {{Webarchive|url=https://web.archive.org/web/20080418033111/http://nobelprize.org/nobel_prizes/physics/laureates/1979/weinberg-lecture.html |date=2008-04-18 }}, 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 |page=261 |bibcode=1978RvMP...50..261B |doi=10.1103/RevModPhys.50.261 }}</ref> in spite of the fact that the electrical 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, which mediate the residual [[nuclear force]].<ref>{{cite journal |last=Sakurai |first=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 appears to have as components a charge-neutral flux of quarks and gluons, quasi-free according to asymptotic freedom in [[quantum chromodynamics|QCD]]. That flux is described by the [[Photon Structure Function|''photon structure function'']].<ref>{{cite journal |last1=Walsh |first1=T.F. |last2=Zerwas |first2=P. |year=1973 |title=Two-photon processes in the parton model |journal=[[Physics Letters B]] |volume=44 |issue=2 |page=195 |bibcode=1973PhLB...44..195W |doi=10.1016/0370-2693(73)90520-0 }}</ref><ref>{{cite journal |last=Witten |first=E. |year=1977 |title=Anomalous cross section for photon–photon scattering in gauge theories |journal=[[Nuclear Physics B]] |volume=120 |issue=2 |pages=189–202 |bibcode=1977NuPhB.120..189W |doi=10.1016/0550-3213(77)90038-4 }}</ref> A review by {{harvp|Nisius|2000}} presented a comprehensive comparison of data with theoretical predictions.<ref>{{cite journal |last=Nisius |first=R. |year=2000 |title=The photon structure from deep inelastic electron–photon scattering |journal=[[Physics Reports]] |volume=332 |issue=4–6 |pages=165–317 |bibcode=2000PhR...332..165N |arxiv=hep-ex/9912049 |s2cid=119437227 |doi=10.1016/S0370-1573(99)00115-5 }}</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>