The first formulation of aquantum theory describing radiation and matter interaction is attributed toPaul Dirac, who during the 1920s computed the coefficient ofspontaneous emission of anatom.[6] He is credited with coining the term "quantum electrodynamics".[7]
Difficulties increased through the end of the 1940s. Improvements inmicrowave technology made it possible to take more precise measurements of the shift of the levels of ahydrogen atom,[12] later known as theLamb shift andmagnetic moment of the electron.[13] These experiments exposed discrepancies that the theory was unable to explain.
A first indication of a possible solution was given byHans Bethe in 1947.[14][15] He made the first non-relativistic computation of the shift of the lines of the hydrogen atom as measured byWillis Lamb andRobert Retherford.[14] 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 namedrenormalization.
Based on Bethe's intuition and fundamental papers on the subject byShin'ichirō Tomonaga,[16]Julian Schwinger,[17][18]Richard Feynman[1][19][20] andFreeman Dyson,[21][22] it was finally possible to produce fullycovariant formulations that were finite at any order in a perturbation series of quantum electrodynamics. Tomonaga, Schwinger, and Feynman were jointly awarded the 1965Nobel Prize in Physics for their work in this area.[23] Their contributions, and Dyson's, were aboutcovariant andgauge-invariant formulations of quantum electrodynamics that allow computations of observables at any order ofperturbation theory. Feynman's mathematical technique, based on hisdiagrams, initially seemed unlike the field-theoretic,operator-based approach of Schwinger and Tomonaga, but Dyson later showed that the two approaches were equivalent.[21] Renormalization, the need to attach a physical meaning at certain divergences appearing in the theory throughintegrals, became one of the fundamental aspects ofquantum 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".[2]: 128
Neither Feynman nor Dirac were happy with that way to approach the observations made in theoretical physics, above all in quantum mechanics.[24]
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,[2] 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.[2]: 85
Aphoton goes from one place and time to another place and time.
Anelectron goes from one place and time to another place and time.
An electron emits or absorbs a photon at a certain place and time.
These actions are represented in the form of visual shorthand by the three basic elements ofdiagrams: 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 calledprobability amplitudes. The probability is the square of the absolute value of total probability amplitude,. If a photon moves from one place and time to another place and time, the associated quantity is written in Feynman's shorthand as, and it depends on only the momentum and polarization of the photon. The similar quantity for an electron moving from to is written. It depends on the momentum and polarization of the electron, in addition to a constant Feynman callsn, 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 callsj, and is sometimes called the "bare" charge of the electron: it is a constant, and is related to, but not the same as, the measuredelectron chargee.[2]: 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 toB),E(C toD) andj) acts just like our everydayprobability (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:[2]: 93
If an event can occur via a number ofindistinguishable alternative processes (a.k.a. "virtual" processes), then its probability amplitude is thesum 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 theproduct of the probability amplitudes of the sub-processes.
The indistinguishability criterion in (a) is very important: it means that there isno 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 oneis able to observe which alternative takes place, one always finds that theprobability of the event is the sum of theprobabilities 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 thephysical means for observing which alternative occurred isremoved, 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.[2]: 82
Similarly, the independence criterion in (b) is very important: it only applies to processes which are not "entangled".
Suppose we start with one electron at a certain place and time (this place and time being given the arbitrary labelA) and a photon at another place and time (given the labelB). A typical question from a physical standpoint is: "What is the probability of finding an electron atC (another place and a later time) and a photon atD (yet another place and time)?". The simplest process to achieve this end is for the electron to move fromA toC (an elementary action) and for the photon to move fromB toD (another elementary action). From a knowledge of the probability amplitudes of each of these sub-processes –E(A toC) andP(B toD) – 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]
But there are other ways in which the result could come about. The electron might move to a place and timeE, where it absorbs the photon; then move on before emitting another photon atF; then move on toC, where it is detected, while the new photon moves on toD. 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 ofE andF. We then, using rule a) above, have to add up all these probability amplitudes for all the alternatives forE andF. (This is not elementary in practice and involvesintegration.) But there is another possibility, which is that the electron first moves toG, where it emits a photon, which goes on toD, while the electron moves on toH, where it absorbs the first photon, before moving on toC. Again, we can calculate the probability amplitude of these possibilities (for all pointsG andH). 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 isCompton scattering.[citation needed]
Aninfinite 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 ofany 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 isnot true in full quantum electrodynamics. There is a nonzero probability amplitude of an electron atA, or a photon atB, moving as a basic action toany other place and time in the universe. That includes places that could only be reached at speeds greater than that of light and alsoearlier times. (An electron moving backwards in time can be viewed as apositron moving forward in time.)[2]: 89, 98–99
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 sourceS can reach the detector atP 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 thetime taken for the light to traverse the path). To correctly calculate the total probability for light to reachP starting atS, one needs to sum the arrows forall 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 thesquare modulus ofprobability amplitudes, which arecomplex numbers.
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 thesquare of the length of the corresponding amplitude arrow. So, for a given process, if two probability amplitudes,v andw, are involved, the probability of the process will be given either by
or
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.
Addition of probability amplitudes as complex numbersMultiplication 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 toB) consists of 16 complex numbers, or probability amplitude arrows.[2]: 120–121 There are also some minor changes to do with the quantityj, 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 afermion and obeysFermi–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 atA andB ending atC andD. The amplitude would be calculated as the "difference",E(A toD) ×E(B toC) −E(A toC) ×E(B toD), where we would expect, from our everyday idea of probabilities, that it would be a sum.[2]: 112–113
Finally, one has to computeP(A toB) andE(C toD) corresponding to the probability amplitudes for the photon and the electron respectively. These are essentially the solutions of theDirac equation, which describe the behavior of the electron's probability amplitude and theMaxwell's equations, which describes the behavior of the photon's probability amplitude. These are calledFeynman propagators. The translation to a notation commonly used in the standard literature is as follows:
where a shorthand symbol such as stands for the four real numbers that give the time and position in three dimensions of the point labeledA.
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 fromA toB, we must take into accountall the possible ways: all possible Feynman diagrams with those endpoints. Thus there will be a way in which the electron travels toC, emits a photon there and then absorbs it again atD before moving on toB. Or it could do this kind of thing twice, or more. In short, we have afractal-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 onad 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 toinfinite probability amplitudes. In time this problem was "fixed" by the technique ofrenormalization. However, Feynman himself remained unhappy about it, calling it a "dippy process",[2]: 128 and Dirac also criticized this procedure, saying "in mathematics one does not get rid of infinities when it does not please you".[24]
Within the above framework physicists were then able to calculate to a high degree of accuracy some of the properties of electrons, such as theanomalous 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."[2]: 152
This theory can be straightforwardly quantized by treating bosonic and fermionic sectors[clarification needed] 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 anevolution operator, which for a given initial state will give a final state in such a way to have[27]: 5
This technique is also known as theS-matrix, and is therefore known as the S-matrix element. The evolution operator is obtained in theinteraction 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:[27]: 123
Which can also be written in terms of an integral over the interaction Hamiltonian density. Thus, one has[27]: 86
whereT is thetime-ordering operator. This evolution operator only has meaning as a series, and what we get here is aperturbation series with thefine-structure constant as the development parameter. This series expansion of the probability amplitude is called theDyson series, and is given by:
As typically containsdelta functions that are not physically-measurable, it is convenient to define theinvariant amplitude with:
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 theFourier transforms of thepropagators. Experimental tests of quantum electrodynamics are typically scattering experiments. In scattering theory, particles'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 thenlook 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.
UsingWick's theorem on the terms of the Dyson series, all the terms of theS-matrix for quantum electrodynamics can be computed through the technique ofFeynman diagrams. In this case, rules for drawing are the following[27]: 801–802
To these rules we must add a further one for closed loops that implies an integration on momenta, since these internal ("virtual") particles are not constrained to any specific energy–momentum, even that usually required by special relativity (seePropagator for details). The signature of the metric is.
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 theSchwinger effect,[28] cannot be understood in terms of any finite number of Feynman diagrams and hence is described asnonperturbative. Mathematically, it can be derived by a semiclassical approximation to thepath integral of quantum electrodynamics.
Higher-order terms can be straightforwardly computed for the evolution operator, but these terms display diagrams containing the following simpler ones[27]: ch 10
that, being closed loops, imply the presence of divergingintegrals having no mathematical meaning. To overcome this difficulty, a technique calledrenormalization 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 electrongyromagnetic ratio.
Renormalizability has become an essential criterion for aquantum field theory to be considered as a viable one. All the theories describingfundamental interactions, exceptgravitation, whose quantum counterpart is only conjectural and presently under very active research, are renormalizable theories.
An argument byFreeman Dyson shows that theradius of convergence of the perturbation series in QED is zero.[29] The basic argument goes as follows: if thecoupling constant were negative, this would be equivalent to theCoulomb force constant being negative. This would "reverse" the electromagnetic interaction so thatlike charges wouldattract andunlike charges wouldrepel. 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 anasymptotic series.
From a modern perspective, we say that QED is not well defined as a quantum field theory to arbitrarily high energy.[30] The coupling constant runs to infinity at finite energy, signalling aLandau pole. The problem is essentially that QED appears to suffer fromquantum triviality issues. This is one of the motivations for embedding QED within aGrand Unified Theory.
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.
Dudley, J.M.; Kwan, A.M. (1996). "Richard Feynman's popular lectures on quantum electrodynamics: The 1979 Robb Lectures at Auckland University".American Journal of Physics.64 (6):694–98.Bibcode:1996AmJPh..64..694D.doi:10.1119/1.18234.
