Quantum electrodynamics: Difference between revisions - Wikipedia

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{{Short description|Quantum field theory of electromagnetism}}
{{Quantum field theory}}

In [[particle physics]], '''quantum electrodynamics''' ('''QED''') is the [[relativityTheory theoryof relativity|relativistic]] [[quantum field theory]] of [[electrodynamics]].<ref name="feynman1" /><ref name="feynbook" /><ref name=":0" /> In essence, it describes how [[light]] and [[matter]] interact and is the first theory where full agreement between [[quantum mechanics]] and [[special relativity]] is achieved.<ref name="feynbook" /> QED mathematically describes all [[phenomenon|phenomena]] involving [[electric charge|electrically charged]] particles interacting by means of exchange of [[photon]]s and represents the [[quantum mechanics|quantum]] counterpart of [[classical electromagnetism]] giving a complete account of matter and light interaction.<ref name="feynbook" /><ref name=":0">{{Cite journal |last=Feynman |first=R. P. |date=1950 |title=Mathematical Formulation of the Quantum Theory of Electromagnetic Interaction |journal=Physical Review |volume=80 |issue=3 |pages=440–457 |doi=10.1103/PhysRev.80.440 |bibcode=1950PhRv...80..440F |url=https://authors.library.caltech.edu/3528/ |access-date=2019-09-23 |archive-date=2020-09-14 |archive-url=https://web.archive.org/web/20200914231627/https://authors.library.caltech.edu/3528/ |url-status=dead }}</ref>

In technical terms, QED can be described as a [[perturbationvery theoryaccurate (quantumway mechanics)|perturbationto theory]]calculate the probability of the electromagneticposition and movement of particles, even those massless such as photons, and the quantity depending on position (field) of those particles, and described light and matter beyond the [[VacuumWave–particle stateduality|quantumwave-particle vacuumduality]]. proposed by [[Albert Einstein]] in 1905. [[Richard Feynman]] called it "the jewel of physics" for its [[precision tests of QED|extremely accurate predictions]] of quantities like the [[anomalous magnetic moment]] of the electron and the [[Lamb shift]] of the [[energy level]]s of [[hydrogen]].<ref name=feynbook>{{cite book |last=Feynman |first=Richard |author-link=Richard Feynman |year=1985 |isbn=978-0-691-12575-6 |title=QED: The Strange Theory of Light and Matter |publisher=Princeton University Press}}</ref>{{rp|Ch1}} It is the most precise and stringently tested theory in physics.<ref>{{Cite book |last=Venkataraman |first=Ganeshan |title=Quantum Revolution II — QED: The Jewel of Physics |year=1994 |publisher=Universities Press |isbn=978-8173710032 |language=en |author-link=Ganeshan Venkataraman}}</ref><ref>{{Cite journal |date=2023-10-05 |title=Testing the limits of the standard model of particle physics with a heavy, highly charged ion |url=https://www.nature.com/articles/d41586-023-02620-7 |access-date=2023-10-23 |journal=Nature|doi=10.1038/d41586-023-02620-7 |pmid=37794145 |s2cid=263670732 }}</ref>

==History==

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|bibcode = 1927RSPSA.114..243D

| issue=767 | doi-access=free

}}</ref> He is also credited with coining the term "quantum electrodynamics".<ref>{{Cite web |title=Quantum Field Theory > The History of QFT |url=https://plato.stanford.edu/entries/quantum-field-theory/qft-history.html |access-date=2023-10-22 |website=Stanford Encyclopedia of Philosophy |first1=Meinard |last1=Kuhlmann |date=Aug 10, 2020 |orig-date=Jun 22, 2006 |url-status=live |archive-url=https://archive.today/20240616034116/https://plato.stanford.edu/entries/quantum-field-theory/qft-history.html |archive-date= 16 Jun 2024 }}</ref>

}}</ref>

Dirac described the quantization of the [[electromagnetic field]] as an ensemble of [[harmonic oscillator]]s with the introduction of the concept of [[creation and annihilation operators]] of particles. In the following years, with contributions from [[Wolfgang Pauli]], [[Eugene Wigner]], [[Pascual Jordan]], [[Werner Heisenberg]] and an elegant formulation of quantum electrodynamics by [[Enrico Fermi]],<ref name=fermi>

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| doi=10.1103/PhysRev.72.339

|bibcode = 1947PhRv...72..339B

| issue=4 | s2cid= 120434909

| issue=4 }}</ref> Despite the limitations of the computation, agreement was excellent. The idea was simply to attach infinities to corrections of [[mass]] and [[charge (physics)|charge]] that were actually fixed to a finite value by experiments. In this way, the infinities get absorbed in those constants and yield a finite result in good agreement with experiments. This procedure was named [[renormalization]].

[[File:Feynman and Oppenheimer at Los Alamos.jpg|thumb|right|[[Richard Feynman|Feynman]] (center) and [[J. Robert Oppenheimer|Oppenheimer]] (right) at [[Los Alamos National Laboratory|Los Alamos]].]]

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| issue=2

| bibcode=1946PThPh...1...27T

| doi-access=free

}}</ref> [[Julian Schwinger]],<ref name=schwinger1>

{{cite journal

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|bibcode = 1949PhRv...76..769F

| issue=6 | doi-access=free

}}</ref><ref name=feynman2>{{cite journal

| author-link = RichardR. P. Feynman▼

{{cite journal

| author-link =R. P.Richard Feynman

| year = 1949▼

▲ | author-link= Richard Feynman

| title = The Theory of Positrons▼

▲ | year=1949

| journal = [[Physical Review]]▼

▲ | title=The Theory of Positrons

|volume = 76

▲ | journal=[[Physical Review]]

|pages volume=76 | pages= 749–59

| doi = 10.1103/PhysRev.76.749

|bibcode = 1949PhRv...76..749F

| issue=6 | s2cid = 1201175646

|s2cid = 120117564

| url = https://authors.library.caltech.edu/3520/

}}</ref><ref name=feynman3>▼

|access-date = 2021-11-19

|archive-date = 2022-08-09

|archive-url = https://web.archive.org/web/20220809030941/https://authors.library.caltech.edu/3520/

|url-status = dead

▲ }}</ref><ref name=feynman3>

{{cite journal

| author=R. P. Feynman

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|bibcode = 1949PhRv...75.1736D

| issue=11 }}</ref> it was finally possible to get fully [[Lorentz covariance|covariant]] formulations that were finite at any order in a perturbation series of quantum electrodynamics. Shin'ichirō Tomonaga, Julian Schwinger and Richard Feynman were jointly awarded with the 1965 [[Nobel Prize in Physics]] for their work in this area.<ref name=nobel65>{{cite web | title = The Nobel Prize in Physics 1965 | publisher = Nobel Foundation | url = http://nobelprize.org/nobel_prizes/physics/laureates/1965/index.html|access-date=2008-10-09}}</ref> Their contributions, and those of [[Freeman Dyson]], were about [[Lorentz covariance|covariant]] and [[gauge-invariant]] formulations of quantum electrodynamics that allow computations of observables at any order of [[Perturbation theory (quantum mechanics)|perturbation theory]]. Feynman's mathematical technique, based on his [[Feynman diagram|diagrams]], initially seemed very different from the field-theoretic, [[Operator (physics)|operator]]-based approach of Schwinger and Tomonaga, but [[Freeman Dyson]] later showed that the two approaches were equivalent.<ref name="dyson1"/> [[Renormalization]], the need to attach a physical meaning at certain divergences appearing in the theory through [[integral]]s, has subsequently become one of the fundamental aspects of [[quantum field theory]] and has come to be seen as a criterion for a theory's general acceptability. Even though renormalization works very well in practice, Feynman was never entirely comfortable with its mathematical validity, even referring to renormalization as a "shell game" and "hocus pocus".<ref name=feynbook/>{{rp|128}}

Thence, neither Feynman nor Dirac were happy with that way to approach the observations made in theoretical physics, above all in quantum mechanics.<ref name=":1">{{Citation |title=The story of the positron - Paul Dirac (1975) |url=https://www.youtube.com/watch?v=Ci86Aps7CMo |access-date=2023-07-19 |language=en}}</ref>

QED has served as the model and template for all subsequent quantum field theories. One such subsequent theory is [[quantum chromodynamics]], which began in the early 1960s and attained its present form in the 1970s work by [[H. David Politzer]], [[Sidney Coleman]], [[David Gross]] and [[Frank Wilczek]]. Building on the pioneering work of [[Julian Schwinger|Schwinger]], [[Gerald Guralnik]], [[C. R. Hagen|Dick Hagen]], and [[Tom W. B. Kibble|Tom Kibble]],<ref>

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[[File:Feynman Diagram Components.svg|thumb|right|300px|[[Feynman diagram]] elements]]

These actions are represented in the form of visual shorthand by the three basic elements of [[Feynman diagram|diagrams]]: a wavy line for the photon, a straight line for the electron and a junction of two straight lines and a wavy one for a vertex representing emission or absorption of a photon by an electron. These can all be seen in the adjacent diagram.

As well as the visual shorthand for the actions, Feynman introduces another kind of shorthand for the numerical quantities called [[Quantum electrodynamics#Probability amplitudes|probability amplitudes]]. The probability is the square of the absolute value of total probability amplitude, <math>\text{probability} = | f(\text{amplitude}) |^2</math>. If a photon moves from one place and time <math>A</math> to another place and time <math>B</math>, the associated quantity is written in Feynman's shorthand as <math>P(A \text{ to } B)</math>, and it depends on only the momentum and polarization of the photon. The similar quantity for an electron moving from <math>C</math> to <math>D</math> is written <math>E(C \text{ to } D)</math>. TheIt depends on the momentum and polarization of the electron, in addition to a constant Feynman calls ''n'', sometimes called the "bare" mass of the electron: it is related to, but not the same as, the measured electron mass. Finally, the quantity that tells us about the probability amplitude for thean emissionelectron orto absorptionemit ofor absorb a photon heFeynman calls ''j''., Thisand is sometimes called the "bare" charge of the electron: it is a constant, and is related to, but not the same as, the measured [[Elementary charge|electron charge]] ''e''.<ref name=feynbook/>{{rp|91}}

QED is based on the assumption that complex interactions of many electrons and photons can be represented by fitting together a suitable collection of the above three building blocks and then using the probability amplitudes to calculate the probability of any such complex interaction. It turns out that the basic idea of QED can be communicated while assuming that the square of the total of the probability amplitudes mentioned above (''P''(''A'' to ''B''), ''E''(''C'' to ''D'') and ''j'') acts just like our everyday [[probability]] (a simplification made in Feynman's book). Later on, this will be corrected to include specifically quantum-style mathematics, following Feynman.

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The basic rules of probability amplitudes that will be used are:<ref name=feynbook/>{{rp|93}}

{{Ordered list|list_style_type = lower-alpha

|If an event can happenoccur invia a varietynumber of different''indistinguishable'' alternative processes (a.k.a. "virtual" waysprocesses), then its probability amplitude is the '''sum''' of the probability amplitudes of the possible waysalternatives.

|If a virtual process involves a number of independent or concomitant sub-processes, then itsthe probability amplitude of the total (compound) process is the '''product''' of the component probability amplitudes of the sub-processes.

}}

The indistinguishability criterion in (a) is very important: it means that there is ''no observable feature present in the given system'' that in any way "reveals" which alternative is taken. In such a case, one cannot observe which alternative actually takes place without changing the experimental setup in some way (e.g. by introducing a new apparatus into the system). Whenever one ''is'' able to observe which alternative takes place, one always finds that the ''probability'' of the event is the sum of the ''probabilities'' of the alternatives. Indeed, if this were not the case, the very term "alternatives" to describe these processes would be inappropriate. What (a) says is that once the ''physical means'' for observing which alternative occurred is ''removed'', one cannot still say that the event is occurring through "exactly one of the alternatives" in the sense of adding probabilities; one must add the amplitudes instead.<ref name=feynbook/>{{rp|82}}

Similarly, the independence criterion in (b) is very important: it only applies to processes which are not "entangled".

===Basic constructions===

Suppose, we start with one electron at a certain place and time (this place and time being given the arbitrary label ''A'') and a photon at another place and time (given the label ''B''). A typical question from a physical standpoint is: "What is the probability of finding an electron at ''C'' (another place and a later time) and a photon at ''D'' (yet another place and time)?". The simplest process to achieve this end is for the electron to move from ''A'' to ''C'' (an elementary action) and for the photon to move from ''B'' to ''D'' (another elementary action). From a knowledge of the probability amplitudes of each of these sub-processes – ''E''(''A'' to ''C'') and ''P''(''B'' to ''D'') – we would expect to calculate the probability amplitude of both happening together by multiplying them, using rule b) above. This gives a simple estimated overall probability amplitude, which is squared to give an estimated probability.{{Citation needed|date=September 2020}}

[[File:Compton Scattering.svg|thumb|left|200px|[[Compton scattering]] ]]

But there are other ways in which the end result could come about. The electron might move to a place and time ''E'', where it absorbs the photon; then move on before emitting another photon at ''F''; then move on to ''C'', where it is detected, while the new photon moves on to ''D''. The probability of this complex process can again be calculated by knowing the probability amplitudes of each of the individual actions: three electron actions, two photon actions and two vertexes – one emission and one absorption. We would expect to find the total probability amplitude by multiplying the probability amplitudes of each of the actions, for any chosen positions of ''E'' and ''F''. We then, using rule a) above, have to add up all these probability amplitudes for all the alternatives for ''E'' and ''F''. (This is not elementary in practice and involves [[Integral|integration]].) But there is another possibility, which is that the electron first moves to ''G'', where it emits a photon, which goes on to ''D'', while the electron moves on to ''H'', where it absorbs the first photon, before moving on to ''C''. Again, we can calculate the probability amplitude of these possibilities (for all points ''G'' and ''H''). We then have a better estimation for the total probability amplitude by adding the probability amplitudes of these two possibilities to our original simple estimate. Incidentally, the name given to this process of a photon interacting with an electron in this way is [[Compton scattering]].{{Citation needed|date=September 2020}}

There is an ''infinite number'' of other intermediate "virtual" processes in which more and more photons are absorbed and/or emitted. For each of these processes, a Feynman diagram could be drawn describing it. This implies a complex computation for the resulting probability amplitudes, but provided it is the case that the more complicated the diagram, the less it contributes to the result, it is only a matter of time and effort to find as accurate an answer as one wants to the original question. This is the basic approach of QED. To calculate the probability of ''any'' interactive process between electrons and photons, it is a matter of first noting, with Feynman diagrams, all the possible ways in which the process can be constructed from the three basic elements. Each diagram involves some calculation involving definite rules to find the associated probability amplitude.

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[[File:Electron self energy loop.svg|thumb|right|200px|[[Electron self-energy]] loop]]

A problem arose historically which held up progress for twenty years: although we start with the assumption of three basic "simple" actions, the rules of the game say that if we want to calculate the probability amplitude for an electron to get from ''A'' to ''B'', we must take into account ''all'' the possible ways: all possible Feynman diagrams with those endpoints. Thus there will be a way in which the electron travels to ''C'', emits a photon there and then absorbs it again at ''D'' before moving on to ''B''. Or it could do this kind of thing twice, or more. In short, we have a [[fractal]]-like situation in which if we look closely at a line, it breaks up into a collection of "simple" lines, each of which, if looked at closely, are in turn composed of "simple" lines, and so on ''ad infinitum''. This is a challenging situation to handle. If adding that detail only altered things slightly, then it would not have been too bad, but disaster struck when it was found that the simple correction mentioned above led to ''infinite'' probability amplitudes. In time this problem was "fixed" by the technique of [[renormalization]]. However, Feynman himself remained unhappy about it, calling it a "dippy process".,<ref name=feynbook/>{{rp|128}} and Dirac also criticized this procedure as "in mathematics one does not get rid of infinities when it does not please you".<ref name=":1" />

===Conclusions===

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==Mathematical formulation==

=== QED action ===

Mathematically, QED is an [[abelian group|abelian]] [[gauge theory]] with the symmetry group [[U(1)]], defined on [[Minkowski space]] (flat spacetime). The [[gauge field]], which mediates the interaction between the charged [[spinSpin (physics)|spin-1/2]] [[field (physics)|field]]s, is the [[electromagnetic field]].

The QED [[Lagrangian (field theory)|Lagrangian]] for a spin-1/2 field interacting with the electromagnetic field in natural units gives rise to the action<ref name=Peskin>{{cite book | last1 =Peskin | first1 =Michael | last2 =Schroeder | first2 =Daniel | title =An introduction to quantum field theory | publisher =Westview Press | edition =Reprint | date =1995 | isbn =978-0201503975 | url-access =registration | url =https://archive.org/details/introductiontoqu0000pesk }}</ref>{{rp|78}}

{{Equation box 1

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Expanding the covariant derivative reveals a second useful form of the Lagrangian (external field <math>B_\mu</math> set to zero for simplicity)

:<math>\mathcal{L} = - \frac{1}{4}F_{\mu\nu}F^{\mu\nu} + \bar\psi(i\gamma^\mu \partial_\mu - m)\psi - jej^\mu A_\mu</math>

where <math>j^\mu</math> is the conserved <math>\text{U}(1)</math> current arising from Noether's theorem. It is written

:<math>j^\mu = \bar\psi\gamma^\mu\psi.</math>

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| url = http://www.lassp.cornell.edu/sethna/Cracks/QED.html

| access-date = May 6, 2017

}}</ref> The basic argument goes as follows: if the [[fine-structure constant|coupling constant]] were negative, this would be equivalent to the [[Coulomb force constant]] being negative. This would "reverse" the electromagnetic interaction so that ''like'' charges would ''attract'' and ''unlike'' charges would ''repel''. This would render the vacuum unstable against decay into a cluster of electrons on one side of the universe and a cluster of positrons on the other side of the universe. Because the theory is "sick" for any negative value of the coupling constant, the series does not converge but areis at best an [[asymptotic series]].

From a modern perspective, we say that QED is not well defined as a quantum field theory to arbitrarily high energy.<ref>{{cite journal

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== Electrodynamics in curved spacetime ==

{{See also |Maxwell's equations in curved spacetime |Dirac equation in curved spacetime}}

{{See also |Dirac equation in curved spacetime}}

This theory can be extended, at least as a classical field theory, to curved spacetime. This arises similarly to the flat spacetime case, from coupling a free electromagnetic theory to a free fermion theory and including an interaction which promotes the partial derivative in the fermion theory to a gauge-covariant derivative.

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===Books===

* {{cite book |last1=Berestetskii |first1=V. B. |last2=Lifshitz |first2=E. M. |author2-link=Evgeny Lifshitz |last3=Pitaevskii |first3=L. P. |author3-link=Lev Pitaevskii |title=Course of Theoretical Physics, Volume 4: Quantum Electrodynamics |title-link=Course of Theoretical Physics |year=1982 |publisher=Elsevier |edition=2 |isbn=978-0-7506-3371-0}}

* {{cite book |last=De Broglie |first=Louis |author-link=Louis de Broglie |title=Recherches sur la theorie des quanta [Research on quantum theory] |year=1925 |publisher=Wiley-Interscience |location=France}}

* {{cite book |last=FeynmanDe Broglie |first=Richard PhillipsL. |author-link=RichardLouis Feynmande Broglie |title=QuantumRecherches Electrodynamicssur la theorie des quanta [Research on quantum theory] |year=19981925 |publisher=Westview PressWiley-Interscience |editionlocation=New |isbn=978-0-201-36075-2France}}

* {{cite book |last1last=JauchFeynman |first1first=JR.M P. |last2author-link=RohrlichRichard |first2=F.Feynman |title=TheQuantum Theory of Photons and ElectronsElectrodynamics |urlyear=https://archive.org/details/theoryofphotonse0000jauc1998 |url-accesspublisher=registrationWestview |year=1980Press |publisheredition=Springer-VerlagNew |isbn=978-0-387201-0729536075-12}}

* {{cite book |last1=Greiner |first1=WalterW. |last2=Bromley |first2=D. A. |last3=Müller |first3=BerndtB. |title=Gauge Theory of Weak Interactions |year=2000 |publisher=Springer |isbn=978-3-540-67672-0}}

* {{cite book |lastlast1=KaneJauch |firstfirst1=Gordon,J. LM. |last2=Rohrlich |first2=F. |title=ModernThe ElementaryTheory Particleof PhysicsPhotons and Electrons |url=https://archive.org/details/theoryofphotonse0000jauc |url-access=registration |year=19931980 |publisher=Westview PressSpringer-Verlag |isbn=978-0-201387-6246007295-1}}

* {{cite book |last=MillerKane |first=ArthurG. IL. |title=EarlyModern QuantumElementary Electrodynamics:Particle A SourcebookPhysics |year=19951993 |publisher=Cambridge UniversityWestview Press |isbn=978-0-521201-5689162460-31}}

* {{cite book |last=Miller |first=A. I. |title=Early Quantum Electrodynamics: A Sourcebook |year=1995 |publisher=Cambridge University Press |isbn=978-0-521-56891-3}}
* {{cite book|last1=Milonni|first1=PeterP. W.|author-link1=Peter W. Milonni|title=The Quantum Vacuum: An Introduction to Quantum Electrodynamics|date=1994|publisher=Academic Press|location=Boston|isbn=0124980805|url=https://books.google.com/books?id=uPHJCgAAQBAJ|lccn=93029780|oclc=422797902}}

* {{cite book |last=Schweber |first=SilvanS. S. |title=QED and the Men Who Made It |url=https://archive.org/details/qedmenwhomadeitd0000schw |url-access=registration |year=1994 |publisher=Princeton University Press |isbn=978-0-691-03327-3}}

* {{cite book |last=Schwinger |first=JulianJ. |author-link=Julian Schwinger |title=Selected Papers on Quantum Electrodynamics |year=1958 |publisher=Dover Publications |isbn=978-0-486-60444-2}}

* {{cite book |last1=Tannoudji-Cohen |first1=ClaudeC. |author-link1=Claude Cohen-Tannoudji |last2=Dupont-Roc |first2=Jacques |last3=Grynberg |first3=Gilbert |title=Photons and Atoms: Introduction to Quantum Electrodynamics |year=1997 |publisher=Wiley-Interscience |isbn=978-0-471-18433-1}}

===Journals===

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* http://qed.wikina.org/ – Animations demonstrating QED

{{-Clear}}

{{QED}}

{{Quantum field theories}}