View source for Quantum electrodynamics

{{Short description|Quantum field theory of electromagnetism}}
{{Quantum field theory}}
In [[particle physics]], '''quantum electrodynamics''' ('''QED''') is the [[Theory of 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 |url-access=subscription }}</ref>

In technical terms, QED can be described as a [[perturbation theory (quantum mechanics)|perturbation theory]] of the electromagnetic [[Quantum vacuum state|quantum vacuum]]. [[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 |url-access=subscription |doi-access=free }}</ref>

==History==
{{Main|History of quantum mechanics|History of quantum field theory}}
[[File:Dirac 3.jpg|upright|thumb|right|[[Paul Dirac]]]]

The first formulation of a [[quantum mechanics|quantum theory]] describing radiation and matter interaction is attributed to [[Paul Dirac]], who during the 1920s computed the coefficient of [[spontaneous emission]] of an [[atom]].<ref name=dirac>
{{cite journal
 | author=P. A. M. Dirac
 | author-link= Paul Dirac
 | year=1927
 | title=The Quantum Theory of the Emission and Absorption of Radiation
 | journal=[[Proceedings of the Royal Society of London A]]
 | volume=114 | pages=243–65
 | doi=10.1098/rspa.1927.0039
|bibcode = 1927RSPSA.114..243D
 | issue=767 | doi-access=free
}}</ref> He is 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>

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 [[Enrico Fermi]],<ref name="fermi">
{{cite journal
 | author=E. Fermi
 | author-link= Enrico Fermi
 | year=1932
 | title=Quantum Theory of Radiation
 | journal=[[Reviews of Modern Physics]]
 | volume=4 | issue= 1
 | pages=87–132
 | doi=10.1103/RevModPhys.4.87
|bibcode = 1932RvMP....4...87F }}</ref> physicists came to believe that, in principle, it was possible to perform any computation for any physical process involving photons and charged particles. However, further studies by [[Felix Bloch]] with [[Arnold Nordsieck]],<ref name="bloch">{{cite journal
 | author-link1= Felix Bloch
 | author-link2= Arnold Nordsieck
 | year=1937
 | title=Note on the Radiation Field of the Electron
 | journal=[[Physical Review]]
 | volume=52 | pages=54–59
 | doi=10.1103/PhysRev.52.54
|bibcode = 1937PhRv...52...54B
 | issue=2
 | last1= Bloch
 | first1= F.
 | last2= Nordsieck
 | first2= A.
}}</ref> and [[Victor Weisskopf]],<ref name=weisskopf>{{cite journal
 | author=V. F. Weisskopf
 | author-link= Victor Weisskopf
 | year=1939
 | title=On the Self-Energy and the Electromagnetic Field of the Electron
 | journal=[[Physical Review]]
 | volume=56 | issue= 1
 | pages=72–85
 | doi=10.1103/PhysRev.56.72
|bibcode = 1939PhRv...56...72W }}</ref>  in 1937 and 1939, revealed that such computations were reliable only at a first order of [[Perturbation theory (quantum mechanics)|perturbation theory]], a problem already pointed out by [[Robert Oppenheimer]].<ref name=oppenheimer>{{cite journal
 | author=R. Oppenheimer
 | author-link= J. Robert Oppenheimer
 | year=1930
 | title=Note on the Theory of the Interaction of Field and Matter
 | journal=[[Physical Review]]
 | volume=35 | pages=461–77
 | doi=10.1103/PhysRev.35.461
|bibcode = 1930PhRv...35..461O
 | issue=5 }}</ref> At higher orders in the series infinities emerged, making such computations meaningless and casting doubt on the theory's internal consistency. This suggested that [[special relativity]] and [[quantum mechanics]] were fundamentally incompatible.

[[File:Hans Bethe.jpg|upright|thumb|[[Hans Bethe]] ]]
Difficulties increased through the end of the 1940s. Improvements in [[microwave]] technology made it possible to take more precise measurements of the shift of the levels of a [[hydrogen atom]],<ref name=lamb>
{{cite journal
 | author-link1= Willis Lamb
 | author-link2=Robert Retherford
 | year=1947
 | title=Fine Structure of the Hydrogen Atom by a Microwave Method
 | journal=[[Physical Review]]
 | volume=72 | pages= 241–43
 | doi=10.1103/PhysRev.72.241
|bibcode = 1947PhRv...72..241L
 | issue=3
 | last1= Lamb
 | first1= Willis
 | last2= Retherford
 | first2= Robert
 | doi-access=free
 }}</ref> later known as the [[Lamb shift]] and [[magnetic moment]] of the electron.<ref name=foley>
{{cite journal
 | author-link2=Polykarp Kusch
 | author-link1=Henry M. Foley
 | year=1948
 | title=On the Intrinsic Moment of the Electron
 | journal=[[Physical Review]]
 | volume=73 | page=412
 | doi=10.1103/PhysRev.73.412
 | bibcode = 1948PhRv...73..412F
 | issue=3
 | last1= Foley
 | first1= H.M.
 | last2= Kusch
 | first2= P. }}</ref> These experiments exposed discrepancies that the theory was unable to explain.

A first indication of a possible solution was given by [[Hans Bethe]] in 1947.<ref name=bethe/><ref name=schweber>
{{cite book
 | last=Schweber
 | first=Silvan
 | author-link=Silvan Schweber
 | year=1994
 | isbn=978-0-691-03327-3
 | title=QED and the Men Who Did it: Dyson, Feynman, Schwinger, and Tomonaga
 | chapter=Chapter 5
 | page=[https://archive.org/details/qedmenwhomadeitd0000schw/page/230 230]
 | publisher=Princeton University Press
 | chapter-url=https://archive.org/details/qedmenwhomadeitd0000schw/page/230
 }}</ref>  He made the first non-relativistic computation of the shift of the lines of the hydrogen atom as measured by [[Willis Lamb]] and [[Robert Retherford]].<ref name=bethe>
{{cite journal
 | author=H. Bethe
 | author-link= Hans Bethe
 | year=1947
 | title=The Electromagnetic Shift of Energy Levels
 | journal=[[Physical Review]]
 | volume=72 | pages=339–41
 | doi=10.1103/PhysRev.72.339
|bibcode = 1947PhRv...72..339B
 | issue=4 | s2cid= 120434909
 }}</ref> Despite limitations of the computation, agreement was excellent. The idea was simply to attach infinities to corrections of mass and 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 with good experimental agreement. 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]].]]

Based on Bethe's intuition and fundamental papers on the subject by [[Shin'ichirō Tomonaga]],<ref name=tomonaga>
{{cite journal
 | author=S. Tomonaga
 | author-link= Sin-Itiro Tomonaga
 | year=1946
 | title=On a Relativistically Invariant Formulation of the Quantum Theory of Wave Fields
 | journal=[[Progress of Theoretical Physics]]
 | volume=1 | pages= 27–42
 | doi=10.1143/PTP.1.27
 | issue=2
| bibcode=1946PThPh...1...27T
 | doi-access=free
 }}</ref> [[Julian Schwinger]],<ref name=schwinger1>
{{cite journal
 | author=J. Schwinger
 | author-link= Julian Schwinger
 | year=1948
 | title=On Quantum-Electrodynamics and the Magnetic Moment of the Electron
 | journal=[[Physical Review]]
 | volume=73 | pages= 416–17
 | doi=10.1103/PhysRev.73.416
|bibcode = 1948PhRv...73..416S
 | issue=4 | doi-access=free
 }}</ref><ref name=schwinger2>
{{cite journal
 | author=J. Schwinger
 | author-link= Julian Schwinger
 | year=1948
 | title=Quantum Electrodynamics. I. A Covariant Formulation
 | journal=[[Physical Review]]
 | volume=74 | pages= 1439–61
 | doi=10.1103/PhysRev.74.1439
|bibcode = 1948PhRv...74.1439S
 | issue=10 }}</ref> [[Richard Feynman]]<ref name=feynman1>
{{cite journal
 | author=R. P. Feynman
 | author-link= Richard Feynman
 | year=1949
 | title=Space–Time Approach to Quantum Electrodynamics
 | journal=[[Physical Review]]
 | volume=76 | pages= 769–89
 | doi=10.1103/PhysRev.76.769
|bibcode = 1949PhRv...76..769F
 | issue=6 | doi-access=free
 }}</ref><ref name=feynman2>{{cite journal
 |author       = R. P. Feynman
 |author-link  = Richard Feynman
 |year         = 1949
 |title        = The Theory of Positrons
 |journal      = [[Physical Review]]
 |volume       = 76
 |pages        = 749–59
 |doi          = 10.1103/PhysRev.76.749
 |bibcode      = 1949PhRv...76..749F
 |issue        = 6
 |s2cid        = 120117564
 |url          = https://authors.library.caltech.edu/3520/
 |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
|url-access= subscription
 }}</ref><ref name=feynman3>
{{cite journal
 | author=R. P. Feynman
 | author-link= Richard Feynman
 | year=1950
 | title=Mathematical Formulation of the Quantum Theory of Electromagnetic Interaction
 | journal=[[Physical Review]]
 | volume=80 | pages= 440–57
 | doi=10.1103/PhysRev.80.440
|bibcode = 1950PhRv...80..440F
 | issue=3 | url=https://authors.library.caltech.edu/3528/1/FEYpr50.pdf
 }}</ref> and [[Freeman Dyson]],<ref name=dyson1>
{{cite journal
 | author=F. Dyson
 | author-link= Freeman Dyson
 | year=1949
 | title=The Radiation Theories of Tomonaga, Schwinger, and Feynman
 | journal=[[Physical Review]]
 | volume=75 | pages= 486–502
 | doi=10.1103/PhysRev.75.486
|bibcode = 1949PhRv...75..486D
 | issue=3 | doi-access=free
 }}</ref><ref name=dyson2>
{{cite journal
 | author=F. Dyson
 | author-link= Freeman Dyson
 | year=1949
 | title=The S Matrix in Quantum Electrodynamics
 | journal=[[Physical Review]]
 | volume=75 | pages=  1736–55
 | doi=10.1103/PhysRev.75.1736
|bibcode = 1949PhRv...75.1736D
 | issue=11 }}</ref> it was finally possible to produce fully [[Lorentz covariance|covariant]] formulations that were finite at any order in a perturbation series of quantum electrodynamics. Tomonaga, Schwinger, and Feynman were jointly awarded 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 Dyson's, 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 unlike the field-theoretic, [[Operator (physics)|operator]]-based approach of Schwinger and Tomonaga, but 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, became one of the fundamental aspects of [[quantum field theory]] and is seen as a criterion for a theory's general acceptability. Even though renormalization works well in practice, Feynman was never entirely comfortable with its mathematical validity, referring to renormalization as a "shell game" and "hocus pocus".<ref name=feynbook/>{{rp|128}}

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 is 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, developed by [[H. David Politzer]], [[Sidney Coleman]], [[David Gross]] and [[Frank Wilczek]]. Building on Schwinger's pioneering work, [[Gerald Guralnik]], [[C. R. Hagen|Dick Hagen]], and [[Tom W. B. Kibble|Tom Kibble]],<ref>
{{cite journal
 | last1=Guralnik | first1=G. S.
 | last2=Hagen | first2=C. R.
 | last3=Kibble | first3=T. W. B.
 | year=1964
 | title=Global Conservation Laws and Massless Particles
 | journal=[[Physical Review Letters]]
 | volume=13 | pages=585–87
 | doi=10.1103/PhysRevLett.13.585
 | bibcode = 1964PhRvL..13..585G
 | issue=20 | doi-access=free
 }}</ref><ref>
{{cite journal
 | last=Guralnik
 | first=G. S.
 | year=2009
 | title=The History of the Guralnik, Hagen and Kibble development of the Theory of Spontaneous Symmetry Breaking and Gauge Particles
 | journal=[[International Journal of Modern Physics A]]
 | volume=24 | pages=2601–27
 | doi=10.1142/S0217751X09045431
 | arxiv=0907.3466
|bibcode = 2009IJMPA..24.2601G
 | issue=14 | s2cid=16298371
 }}</ref> [[Peter Higgs]], [[Jeffrey Goldstone]], and others, [[Sheldon Glashow]], [[Steven Weinberg]] and [[Abdus Salam]] independently showed how the [[weak nuclear force]] and quantum electrodynamics could be merged into a single [[electroweak force]].

==Feynman's view of quantum electrodynamics==

===Introduction===
Near the end of his life, [[Richard Feynman]] gave a series of lectures on QED intended for the lay public. These lectures were transcribed and published as Feynman (1985), ''[[QED: The Strange Theory of Light and Matter]]'',<ref name=feynbook/> a classic non-mathematical exposition of QED from the point of view articulated below.

The key components of Feynman's presentation of QED are three basic actions.<ref name=feynbook/>{{rp|85}}
: A [[photon]] goes from one place and time to another place and time.
: An [[electron]] goes from one place and time to another place and time.
: An electron emits or absorbs a photon at a certain place and time.

[[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>. It 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 an electron to emit or absorb a photon Feynman calls ''j'', and 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.

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 occur via a number of ''indistinguishable'' alternative processes (a.k.a. "virtual" processes), then its probability amplitude is the '''sum''' of the probability amplitudes of the alternatives.
|If a virtual process involves a number of independent or concomitant sub-processes, then the probability amplitude of the total (compound) process is the '''product''' of the 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 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}}

An ''infinite number'' of other intermediate "virtual" processes exist in which photons are absorbed 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.

That basic scaffolding remains when one moves to a quantum description, but some conceptual changes are needed. One is that whereas we might expect in our everyday life that there would be some constraints on the points to which a particle can move, that is ''not'' true in full quantum electrodynamics. There is a nonzero probability amplitude of an electron at ''A'', or a photon at ''B'', moving as a basic action to ''any other place and time in the universe''. That includes places that could only be reached at speeds greater than that of light and also ''earlier times''. (An electron moving backwards in time can be viewed as a [[positron]] moving forward in time.)<ref name=feynbook/>{{rp|89, 98–99}}

===Probability amplitudes===
[[File:Feynmans QED probability amplitudes.gif|frame|right|Feynman replaces complex numbers with spinning arrows, which start at emission and end at detection of a particle. The sum of all resulting arrows gives a final arrow whose length squared equals the probability of the event. In this diagram, light emitted by the source '''''S''''' can reach the detector at '''''P''''' by bouncing off the mirror (in blue) at various points. Each one of the paths has an arrow associated with it (whose direction changes uniformly with the ''time'' taken for the light to traverse the path). To correctly calculate the total probability for light to reach '''''P''''' starting at '''''S''''', one needs to sum the arrows for ''all'' such paths. The graph below depicts the total time spent to traverse each of the paths above.]]

[[Quantum mechanics]] introduces an important change in the way probabilities are computed. Probabilities are still represented by the usual real numbers we use for probabilities in our everyday world, but probabilities are computed as the [[square modulus]] of [[probability amplitude]]s, which are [[complex number]]s.

Feynman avoids exposing the reader to the mathematics of complex numbers by using a simple but accurate representation of them as arrows on a piece of paper or screen. (These must not be confused with the arrows of Feynman diagrams, which are simplified representations in two dimensions of a relationship between points in three dimensions of space and one of time.) The amplitude arrows are fundamental to the description of the world given by quantum theory. They are related to our everyday ideas of probability by the simple rule that the probability of an event is the ''square'' of the length of the corresponding amplitude arrow. So, for a given process, if two probability amplitudes, '''v''' and '''w''', are involved, the probability of the process will be given either by

:<math>P = |\mathbf{v} + \mathbf{w}|^2</math>

or

:<math>P = |\mathbf{v} \, \mathbf{w}|^2.</math>

The rules as regards adding or multiplying, however, are the same as above. But where you would expect to add or multiply probabilities, instead you add or multiply probability amplitudes that now are complex numbers.

[[File:AdditionComplexes.svg|thumb|right|200px|Addition of probability amplitudes as complex numbers]]
[[File:MultiplicationComplexes.svg|thumb|right|200px|Multiplication of probability amplitudes as complex numbers]]

Addition and multiplication are common operations in the theory of complex numbers and are given in the figures. The sum is found as follows. Let the start of the second arrow be at the end of the first. The sum is then a third arrow that goes directly from the beginning of the first to the end of the second. The product of two arrows is an arrow whose length is the product of the two lengths. The direction of the product is found by adding the angles that each of the two have been turned through relative to a reference direction: that gives the angle that the product is turned relative to the reference direction.

That change, from probabilities to probability amplitudes, complicates the mathematics without changing the basic approach. But that change is still not quite enough because it fails to take into account the fact that both photons and electrons can be polarized, which is to say that their orientations in space and time have to be taken into account. Therefore, ''P''(''A'' to ''B'') consists of 16 complex numbers, or probability amplitude arrows.<ref name=feynbook/>{{rp|120–121}} There are also some minor changes to do with the quantity ''j'', which may have to be rotated by a multiple of 90° for some polarizations, which is only of interest for the detailed bookkeeping.

Associated with the fact that the electron can be polarized is another small necessary detail, which is connected with the fact that an electron is a [[fermion]] and obeys [[Fermi–Dirac statistics]]. The basic rule is that if we have the probability amplitude for a given complex process involving more than one electron, then when we include (as we always must) the complementary Feynman diagram in which we exchange two electron events, the resulting amplitude is the reverse – the negative – of the first. The simplest case would be two electrons starting at ''A'' and ''B'' ending at ''C'' and ''D''. The amplitude would be calculated as the "difference", {{nowrap|''E''(''A'' to ''D'') × ''E''(''B'' to ''C'') − ''E''(''A'' to ''C'') × ''E''(''B'' to ''D'')}}, where we would expect, from our everyday idea of probabilities, that it would be a sum.<ref name=feynbook/>{{rp|112–113}}

===Propagators===
Finally, one has to compute ''P''(''A'' to ''B'') and ''E''(''C'' to ''D'') corresponding to the probability amplitudes for the photon and the electron respectively. These are essentially the solutions of the [[Dirac equation]], which describe the behavior of the electron's probability amplitude and the [[Maxwell's equations]], which describes the behavior of the photon's probability amplitude. These are called [[Propagator|Feynman propagators]]. The translation to a notation commonly used in the standard literature is as follows:

:<math>P(A \text{ to } B) \to D_F(x_B - x_A),\quad  E(C \text{ to } D) \to S_F(x_D - x_C),</math>

where a shorthand symbol such as <math>x_A</math> stands for the four real numbers that give the time and position in three dimensions of the point labeled ''A''.

===Mass renormalization===
{{Main|Self-energy}}

[[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, saying "in mathematics one does not get rid of infinities when it does not please you".<ref name=":1" />

===Conclusions===
Within the above framework physicists were then able to calculate to a high degree of accuracy some of the properties of electrons, such as the [[anomalous magnetic dipole moment]]. However, as Feynman points out, it fails to explain why particles such as the electron have the masses they do. "There is no theory that adequately explains these numbers. We use the numbers in all our theories, but we don't understand them – what they are, or where they come from. I believe that from a fundamental point of view, this is a very interesting and serious problem."<ref name=feynbook/>{{rp|152}}

==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 [[Spin (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
|title='''QED Action'''
|indent=:
|equation = <math>S_{\text{QED}} = \int d^4x\,\left[-\frac{1}{4}F^{\mu\nu}F_{\mu\nu} + \bar\psi\,(i\gamma^\mu D_\mu - m)\,\psi\right]</math>
|border
|border colour =#50C878
|background colour = #ECFCF4
}}

where
*<math> \gamma^\mu </math> are [[Dirac matrices]].
*<math>\psi</math> a [[bispinor]] [[field (physics)|field]] of [[spin-1/2]] particles (e.g. [[electron]]–[[positron]] field).
*<math>\bar\psi\equiv\psi^\dagger\gamma^0</math>, called "psi-bar", is sometimes referred to as the [[Dirac adjoint]].
*<math>D_\mu \equiv \partial_\mu+ieA_\mu+ieB_\mu </math> is the [[gauge covariant derivative]].
**''e'' is the [[Fine-structure constant|coupling constant]], equal to the [[electric charge]] of the bispinor field.
**<math>A_\mu</math> is the [[Lorentz covariance|covariant]] [[four-potential]] of the electromagnetic field generated by the electron itself. It is also known as a gauge field or a <math>\text{U}(1)</math> connection.
**<math>B_\mu</math> is the external field imposed by external source.
*''m'' is the mass of the electron or positron.
*<math>F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu </math> is the [[electromagnetic field tensor]]. This is also known as the curvature of the gauge field.

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 - ej^\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>

=== Equations of motion ===
Expanding the covariant derivative in the Lagrangian gives

:<math>\mathcal{L} = - \frac{1}{4}F_{\mu\nu}F^{\mu\nu} + i \bar\psi \gamma^\mu \partial_\mu \psi - e\bar{\psi}\gamma^\mu A_\mu \psi -m \bar{\psi} \psi </math>
:<math> = - \frac{1}{4}F_{\mu\nu}F^{\mu\nu} + i \bar\psi \gamma^\mu \partial_\mu \psi -m \bar{\psi} \psi - ej^\mu A_\mu .</math>
For simplicity, <math>B_\mu</math> has been set to zero, with no loss of generality. Alternatively, we can absorb <math>B_\mu</math> into a new gauge field <math>A'_\mu = A_\mu + B_\mu</math> and relabel the new field as <math>A_\mu.</math>

From this Lagrangian, the equations of motion for the <math>\psi</math> and <math>A_\mu</math> fields can be obtained.

==== Equation of motion for ψ ====
These arise most straightforwardly by considering the Euler-Lagrange equation for <math>\bar\psi</math>. Since the Lagrangian contains no <math>\partial_\mu\bar\psi</math> terms, we immediately get
:<math>\frac{\partial \mathcal{L}}{\partial(\partial_\mu \bar\psi)} = 0</math>
so the equation of motion can be written
<math>(i\gamma^\mu\partial_\mu-m)\psi = e\gamma^\mu A_\mu\psi.</math>

==== Equation of motion for A<sub>μ</sub> ====
* Using the Euler–Lagrange equation for the <math>A_\mu</math> field,
{{NumBlk2||<math display="block"> \partial_\nu \left( \frac{\partial \mathcal{L}}{\partial ( \partial_\nu A_\mu )} \right) - \frac{\partial \mathcal{L}}{\partial A_\mu} = 0,</math>|3}}

the derivatives this time are
<math display="block">\partial_\nu \left( \frac{\partial \mathcal{L}}{\partial ( \partial_\nu A_\mu )} \right) = \partial_\nu \left( \partial^\mu A^\nu - \partial^\nu A^\mu \right),</math>
<math display="block">\frac{\partial \mathcal{L}}{\partial A_\mu} = -e\bar{\psi} \gamma^\mu \psi.</math>

Substituting back into ({{EquationNote|3}}) leads to

:<math>\partial_\mu F^{\mu\nu} = e\bar\psi \gamma^\nu \psi</math>
which can be written in terms of the <math>\text{U}(1)</math> current <math>j^\mu</math> as
{{Equation box 1
|indent =:
|equation = <math>\partial_\mu F^{\mu \nu} = e j^\nu.</math>
|cellpadding
|border
|border colour = #0073CF
|background colour=#F5FFFA}}

Now, if we impose the [[Lorenz gauge condition]]
<math display="block">\partial_\mu A^\mu = 0,</math>
the equations reduce to
<math display="block">\Box A^\mu = ej^\mu,</math>
which is a [[wave equation]] for the four-potential, the QED version of the classical [[Maxwell equations]] in the [[Lorenz gauge]]. (The square represents the [[wave operator]], <math>\Box = \partial_\mu \partial^\mu</math>.)

===Interaction picture===
This theory can be straightforwardly quantized by treating bosonic and fermionic sectors{{clarify|reason=Definition needed.|date=April 2015}} as free. This permits us to build a set of asymptotic states that can be used to start computation of the probability amplitudes for different processes. In order to do so, we have to compute an [[Hamiltonian (quantum mechanics)|evolution operator]], which for a given initial state <math>|i\rangle</math> will give a final state <math>\langle f|</math> in such a way to have<ref name=Peskin/>{{rp|5}}

<math display="block">S_{fi} = \langle f|U|i\rangle.</math>

This technique is also known as the [[S-matrix]], and <math>S_{fi}</math> is therefore known as the S-matrix element. The evolution operator is obtained in the [[interaction picture]], where time evolution is given by the interaction Hamiltonian, which is the integral over space of the second term in the Lagrangian density given above:<ref name=Peskin/>{{rp|123}}

<math display="block">V = e \int d^3 x\, \bar\psi \gamma^\mu \psi A_\mu,</math>

Which can also be written in terms of an integral over the interaction Hamiltonian density <math>\mathcal{H}_I = e \overline \psi \gamma^\mu \psi A_\mu</math>. Thus, one has<ref name=Peskin/>{{rp|86}}

<math display="block">U = T \exp\left[-\frac{i}{\hbar} \int_{t_0}^t dt'\, V(t')\right],</math>

where ''T'' is the [[Path-ordering|time-ordering]] operator. This evolution operator only has meaning as a series, and what we get here is a [[Perturbation theory (quantum mechanics)|perturbation series]] with the [[fine-structure constant]] as the development parameter. This series expansion of the probability amplitude <math>S_{fi}</math> is called the [[Dyson series]], and is given by:

<math display="block">
S_{fi} = \langle f | U |i\rangle 
 =\left\langle f\left|\sum _{n=0}^{\infty }{\frac {(-i)^{n}}{n!}}\int d^4x_{1} \cdots \int d^4x_{n} T \bigg\{ \mathcal{H}(x_{1})\cdots \mathcal {H}(x_{n}) \bigg \} \right|i\right\rangle
</math>

As <math>S_{fi}</math> typically contains [[Dirac delta function|delta functions]] that are not physically-measurable, it is convenient to define the '''invariant amplitude''' <math>M_{fi}</math> with:

<math display="block">
S_{fi} = \langle f|i\rangle + (iM_{fi}  \times \text{Delta function terms})
</math>

===Feynman diagrams===

Despite the conceptual clarity of the Feynman approach to QED, almost no early textbooks follow him in their presentation. When performing calculations, it is much easier to work with the [[Fourier transform]]s of the [[propagator]]s. Experimental tests of quantum electrodynamics are typically scattering experiments. In scattering theory, particles' [[Momentum|momenta]] rather than their positions are considered, and it is convenient to think of particles as being created or annihilated when they interact. Feynman diagrams then ''look'' the same, but the lines have different interpretations. The electron line represents an electron with a given energy and momentum, with a similar interpretation of the photon line. A vertex diagram represents the annihilation of one electron and the creation of another together with the absorption or creation of a photon, each having specified energies and momenta.

Using [[Wick's theorem]] on the terms of the Dyson series, all the terms of the [[S-matrix]] for quantum electrodynamics can be computed through the technique of [[Feynman diagrams]]. In this case, rules for drawing are the following<ref name=Peskin/>{{rp|801–802}}

[[Image:qed rules.jpg|488px|center]]

[[Image:qed2e.jpg|488px|center]]

To these rules we must add a further one for closed loops that implies an integration on momenta <math display="inline">\int d^4p/(2\pi)^4</math>, since these internal ("virtual") particles are not constrained to any specific energy–momentum, even that usually required by special relativity (see [[Propagator#Propagators in Feynman diagrams|Propagator]] for details). The signature of the metric <math>\eta_{\mu \nu }</math> is <math>{\rm diag}(+---)</math>.

From them, computations of [[probability amplitude]]s are straightforwardly given. An example is [[Compton scattering]], with an [[electron]] and a [[photon]] undergoing [[elastic scattering]]. Feynman diagrams are in this case<ref name=Peskin/>{{rp|158–159}}

[[Image:compton qed.jpg|300px|center]]

and so we are able to get the corresponding amplitude at the first order of a [[Perturbation theory (quantum mechanics)|perturbation series]] for the [[S-matrix]]:

<math display="block">M_{fi} = (ie)^2 \overline{u}(\vec{p}', s')\epsilon\!\!\!/\,'(\vec{k}',\lambda')^* \frac{p\!\!\!/ + k\!\!\!/ + m_e}
{(p + k)^2 - m^2_e} \epsilon\!\!\!/(\vec{k}, \lambda) u(\vec{p}, s) + (ie)^2\overline{u}(\vec{p}', s')\epsilon\!\!\!/(\vec{k},\lambda) \frac{p\!\!\!/ - k\!\!\!/' + m_e}{(p - k')^2 - m^2_e} \epsilon\!\!\!/\,'(\vec{k}', \lambda')^* u(\vec{p}, s),</math>

from which we can compute the [[Cross section (physics)|cross section]] for this scattering.

===Nonperturbative phenomena===
The predictive success of quantum electrodynamics largely rests on the use of perturbation theory, expressed in Feynman diagrams. However, quantum electrodynamics also leads to predictions beyond perturbation theory. In the presence of very strong electric fields, it predicts that electrons and positrons will be spontaneously produced, so causing the decay of the field. This process, called the [[Schwinger effect]],<ref name="Schwinger">{{cite journal | last=Schwinger | first=Julian | title=On Gauge Invariance and Vacuum Polarization | journal=Physical Review | publisher=American Physical Society (APS) | volume=82 | issue=5 | date=1951-06-01 | issn=0031-899X | doi=10.1103/physrev.82.664 | pages=664–679| bibcode=1951PhRv...82..664S }}</ref> cannot be understood in terms of any finite number of Feynman diagrams and hence is described as [[Non-perturbative|nonperturbative]]. Mathematically, it can be derived by a semiclassical approximation to the [[Path integral formulation|path integral]] of quantum electrodynamics.

==Renormalizability==
Higher-order terms can be straightforwardly computed for the evolution operator, but these terms display diagrams containing the following simpler ones<ref name=Peskin/>{{rp|ch 10}}

<gallery class="center">
Image:vacuum_polarization.svg | One-loop contribution to the [[vacuum polarization]] function <math>\Pi</math>
Image:electron_self_energy.svg | One-loop contribution to the electron [[self-energy]] function <math>\Sigma</math>
Image:vertex_correction.svg | One-loop contribution to the [[vertex function]] <math>\Gamma</math>
</gallery>

that, being closed loops, imply the presence of diverging [[integral]]s having no mathematical meaning. To overcome this difficulty, a technique called [[renormalization]] has been devised, producing finite results in very close agreement with experiments. A criterion for the theory being meaningful after renormalization is that the number of diverging diagrams is finite. In this case, the theory is said to be "renormalizable". The reason for this is that to get observables renormalized, one needs a finite number of constants to maintain the predictive value of the theory untouched. This is exactly the case of quantum electrodynamics displaying just three diverging diagrams. This procedure gives observables in very close agreement with experiment as seen e.g. for electron [[gyromagnetic ratio]].

Renormalizability has become an essential criterion for a [[quantum field theory]] to be considered as a viable one. All the theories describing [[fundamental interaction]]s, except [[gravitation]], whose quantum counterpart is only conjectural and presently under very active research, are renormalizable theories.

==Nonconvergence of series==
An argument by [[Freeman Dyson]] shows that the [[radius of convergence]] of the perturbation series in QED is zero.<ref>{{Cite web
  | last = Kinoshita
  | first = Toichiro
  | title = Quantum Electrodynamics has Zero Radius of Convergence Summarized from Toichiro Kinoshita
  | date = June 5, 1997
  | url = https://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 is 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
  | last = Espriu and Tarrach
  | title = Ambiguities in QED: Renormalons versus Triviality
  | date = Apr 30, 1996
  | arxiv = hep-ph/9604431
  | doi=10.1016/0370-2693(96)00779-4
  | volume=383
  | issue = 4
 | journal=Physics Letters B
  | pages=482–486
| bibcode=1996PhLB..383..482E| s2cid = 119095192
 }}</ref> The coupling constant runs to infinity at finite energy, signalling a [[Landau pole]]. The problem is essentially that QED appears to suffer from [[quantum triviality]] issues. This is one of the motivations for embedding QED within a [[Grand Unified Theory]].

== Electrodynamics in curved spacetime ==
{{See also |Maxwell's equations in curved spacetime |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.

==See also==
{{Portal|Physics}}
{{Div col|colwidth=20em|small=yes}}
*[[Abraham–Lorentz force]]
*[[Anomalous magnetic moment]]
*[[Bhabha scattering]]
*[[Cavity quantum electrodynamics]]
*[[Circuit quantum electrodynamics]]
*[[Compton scattering]]
*[[Euler–Heisenberg Lagrangian]]
*[[Gupta–Bleuler formalism]]
*[[Lamb shift]]
*[[Landau pole]]
*[[Moeller scattering]]
*[[Non-relativistic quantum electrodynamics]]
*[[Photon polarization]]
*[[Positronium]]
*[[Precision tests of QED]]
*[[QED vacuum]]
*''[[QED: The Strange Theory of Light and Matter]]''
*[[Quantization of the electromagnetic field]]
*[[Scalar electrodynamics]]
*[[Schrödinger equation]]
*[[Schwinger model]]
*[[Schwinger–Dyson equation]]
*[[Vacuum polarization]]
*[[Vertex function]]
*[[Wheeler–Feynman absorber theory]]
{{div col end}}

==References==
{{Reflist|30em}}

==Further reading==

===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=L. |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=Feynman |first=R. P. |author-link=Richard Feynman |title=Quantum Electrodynamics |year=1998 |publisher=Westview Press |edition=New |isbn=978-0-201-36075-2}}
* {{cite book |last1=Greiner |first1=W. |last2=Bromley |first2=D. A. |last3=Müller |first3=B. |title=Gauge Theory of Weak Interactions |year=2000 |publisher=Springer |isbn=978-3-540-67672-0}}
* {{cite book |last1=Jauch |first1=J. M. |last2=Rohrlich |first2=F. |title=The Theory of Photons and Electrons |url=https://archive.org/details/theoryofphotonse0000jauc |url-access=registration |year=1980 |publisher=Springer-Verlag |isbn=978-0-387-07295-1}}
* {{cite book |last=Kane |first=G. L. |title=Modern Elementary Particle Physics |year=1993 |publisher=Westview Press |isbn=978-0-201-62460-1}}
* {{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=P. 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=S. 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=J. |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=C. |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===
* {{cite journal | last1 = Dudley | first1 = J.M. | last2 = Kwan | first2 = A.M. | year = 1996 | title = Richard Feynman's popular lectures on quantum electrodynamics: The 1979 Robb Lectures at Auckland University | journal = American Journal of Physics | volume = 64 | issue = 6| pages = 694–98 | doi = 10.1119/1.18234 |bibcode = 1996AmJPh..64..694D }}

== External links ==
* [https://www.nobelprize.org/prizes/physics/1965/feynman/lecture/ Feynman's Nobel Prize lecture describing the evolution of QED and his role in it]
* [http://www.vega.org.uk/video/subseries/8 Feynman's New Zealand lectures on QED for non-physicists]
* [http://qed.wikina.org/ The Strange Theory of Light | Animation of Feynman pictures light by QED] – Animations demonstrating QED

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