Editing Photon (section)

=== 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.