quantum electrodynamics

Monday, 13 May 2013

development of quantum electrodynamics-sin itiro tomonaga

Development of Quantum Electrodynamics

Personal recollections

(1) In 1932, when I started my research career as an assistant to Nishina, Dirac published a paper in the Proceedings of the Royal Society, London¹. In this paper, he discussed the formulation of relativistic quantum mechanics, especially that of electrons interacting with the electromagnetic field. At that time a comprehensive theory of this interaction had been formally completed by Heisenberg and Pauli ², but Dirac was not satisfied with this theory and tried to construct a new theory from a different point of view. Heisenberg and Pauli regarded the (electromagnetic) field itself as a dynamical system amenable to the Hamiltonian treatment; its interaction with particles could be described by an interaction energy, so that the usual method of Hamiltonian quantum mechanics could be applied. On the other hand, Dirac thought that the field and the particles should play essentially different roles. That is to say, according to him, "the role of the field is to provide a means for making observations of a system of particles" and therefore "we cannot suppose the field to be a dynamical system on the same footing as the particles and thus be something to be observed in the same way as the particles".

Based on such a philosophy, Dirac proposed a new theory, the so-called many-time theory, which, besides being a concrete example of his philosophy was of much more satisfactory and beautiful form than other theories presented up to then. In fact, from the relativistic point of view, these other theories had a common defect which was inherent in their Hamiltonian formalism. The Hamiltonian dynamics was developed on the basis of non-relativistic concepts which make a sharp distinction between time and space. It formulates a physical law by describing how the state of a dynamical system changes with time. Speaking quantum-mechanically, it is a formalism to describe how the probability amplitude changes with time t. Now, as an example, let us consider a system composed of N particles, and let the coordinates of each particle be r₁, r₂,..., r_N. Then the probability amplitude of the system is a function of the N variables r₁, r₂,..., r_N, and in addition, of the time t to which the amplitude is referred. Thus this function contains only one time variable in contrast to N space variables. In the theory of relativity, however, time and space must be treated on an entirely equal footing so that the above imbalance is not satisfactory. On the other hand, in Dirac's theory which does not use the Hamiltonian formalism, it becomes possible to consider different time variables for each particle, so that the probability amplitude can be expressed as a function of r₁, t₁, r₂t₂,...,r_N t_N. Accordingly, the theory satisfies the requirement of the principle of relativity that time and space be treated with complete equality. The reason why the theory is called the many-time theory is because N distinct time variables are used in this way.

This paper of Dirac's attracted my interest because of the novelty of its philosophy and the beauty of its form. Nishina also showed a great interest in this paper and suggested that I investigate the possibility of predicting some new phenomena by this theory. Then I started computations to see whether the Klein-Nishina formula could be derived from this theory or whether any modification of the formula might result. I found out immediately however, without performing the calculation through to the end, that it would yield the same answer as the previous theory. This new theory of Dirac's was in fact mathematically equivalent to the older Heisenberg-Pauli theory, and I realized during the calculation that one could pass from one to the other by a unitary transformation. The equivalence of these two theories was also discovered by Rosenfeld³ and by Dirac-Fock-Podolsky⁴ and was soon published in their papers.

Though Dirac's many-time formalism turned out to be equivalent to the Heisenberg-Pauli theory, it had the advantage that it gave us the possibility of generalizing the former interpretation of the probability amplitude. Namely, while one could calculate the probability of finding particles at points with coordinates r₁, r₂,..., r_N, all at the time t according to the previous theory, one could now compute more generally the probability that the first particle is at r₁ at time t₁, the second at r₂ at time t₂, … and the N-th at rN at time tN. This was first discussed by Bloch⁵ in 1934.

(2) In this many-time theory developed by Dirac, electrons were treated according to the particle picture. Alternatively, in quantum theory, any particle should be able to be treated according to the wave picture. As a matter of fact, electrons were also treated as waves in the Heisenberg-Pauli theory, and it was well known that this wave treatment was frequently more convenient than the particle treatment. So the question arose as to whether one could reformulate the Heisenberg-Pauli theory in a way which would be more satisfactory relativistically, when electrons were treated as waves as well as the electromagnetic field.

As Dirac already pointed out, the Heisenberg-Pauli theory is built upon the Hamiltonian formalism and therefore the probability amplitude contains only one time variable. That is to say, the probability amplitude is given as a function of the field strength at different space points and of one common time variable. However, the concept of a common time at different space points does not have a relativistically covariant meaning.

Around 1942, Yukawa⁶ wrote a paper emphasizing this unsatisfactory aspect of the quantum field theory. He thought it necessary to use the idea of the g.t.f. (generalized transformation function) proposed by Dirac⁷ to correct this defect of the theory. Here I shall omit talking about the g.t.f., but, briefly, Yukawa's idea was to introduce as the basis of a new theory a concept which generalized the conventional conception of the probability amplitude. However, as pointed out also by Yukawa, we encounter the difficulty that, in doing this, cause and effect can not be clearly separated from each other. According to Yukawa, the inseparability of cause and effect would be an essential feature of quantum field theory, and without abandoning the causal way of thinking which strictly separates cause and effect, it would not be possible to solve various difficulties appearing in quantum field theory about which I will talk later. I thought however, that it might be possible (without introducing such a drastic change as Yukawa and Dirac tried to do) to remedy the unsatisfactory, unpleasant aspect of the Heisenberg-Pauli theory of having a common time at different space points. In other words, it should be possible, I thought, to define a relativistically meaningful probability amplitude which would be manifestly covariant, without being forced to give up the causal way of thinking. In having this expectation I was recalling Dirac's many-time theory which had enchanted me 10 years before.

When there are N particles in Dirac's many-time theory, we assign a time t₁ to the first particle,t₂ to the second, and so on, thus introducing N different times, t₁, t₂, ...,t_N, instead of the one common time t. Similarly, I tried in quantum field theory to see whether it was possible to assign different times, instead of one common time, to each space point. And in fact I was able to show that this was possible⁸.

As there are an infinite number of space points in field theory in contrast to the finite number of particles in particle theory, the number of time variables appearing in the probability amplitude became infinite. But it turned out that no essential difficulty appeared. An interpretation quite analogous to the one discussed by Bloch in connection with Dirac's many-time theory could be given to our probability amplitude containing an infinite number of time variables. Further, it was found that the theory thus formulated was completely covariant and that this covariant formulation was equivalent in its whole content to the Heisenberg-Pauli theory: it was shown, just as in the case of the many-time theory, that we could pass from one to the other by a unitary transformation. I began this work about 1942, and completed it in 1946.

(3) As I mentioned a little while ago, there are many difficulties in the quantum mechanics of fields. In particular, infinite quantities always arise which are associated with the presence of field reactions in various processes. The first phenomenon which attracted our attention as a manifestation of field reactions was the electromagnetic mass of the electron. The electron, having a charge, produces an electromagnetic field around itself. In turn, this field, the so-called self-field of the electron, interacts with the electron. We call this interaction the field reaction. Because of the field reaction the apparent mass of the electron differs from the original mass. The excess mass due to this field reaction is called the electromagnetic mass of the electron and the experimentally observed mass is the sum of the original mass and this electromagnetic mass. The concept of the electromagnetic mass had already appeared in the classical theory of the electron by Lorentz, who computed the electromagnetic mass by applying the classical theory and obtained the result that the mass becomes infinite for the point (zero size) electron. On the other hand, the electromagnetic mass was computed in quantum theory by various people, and here I mention particularly the work of Weisskopf⁹. According to him, the quantum-mechanical electromagnetic mass turned out to be infinite, and although the order of the divergence was much weaker than in the case of the Lorentz theory, the observed mass, which included this additional mass, would be infinite. This would be, of course, contrary to experiment.

In order to overcome the difficulty of an infinitely large electromagnetic mass, Lorentz considered the electron not to be point-like but to have a finite size. It is very difficult, however, to incorporate a finite sized electron into the framework of relativistic quantum theory. Many people tried various means to overcome this problem of infinite quanties, but nobody succeeded.

In connection with field reactions, the next problem which attracted the attention of physicists was determining what kind of influence the field reaction exerts in electron-scattering processes. Let us consider, as a concrete example, a problem in which an electron is scattered by an external field. In the ordinary treatment, we neglect the effect of field reactions on the scattered electron, assuming that it is negligibly small. Then the behavior of the scattering obtained by calculation (e.g. the Rutherford formula) fits very well with experiment. But what will happen if the influence of field reaction is taken into account? This theoretical problem was examined non-relativistically by Braunbeck-Weinmann¹⁰ and Pauli-Fierz¹¹ and relativistically by Dancoff¹².

While Dancoff applied an approximation method, the perturbation method, in his relativistic calculation, Pauli and Fierz treated the problem in such a way that the most important part of the field reaction was first separated out exactly by employing a contact transformation method which was similar to the one which Bloch-Nordsieck¹³ had published a year before. Since Pauli and Fierz adopted a non-relativistic model, and further simplified the problem by using the so-called dipole approximation, their calculation was especially transparent. At any rate, both non-relativistic and relativistic calculations exhibited several infinities in the scattering processes*.

The conclusions of these people were fatal to the theory. That is, the influence of the field reaction becomes infinite in this problem. The effect of field reaction on a quantity called the scattering cross section, which expresses quantitatively the behavior of the scattering, rather than becoming negligibly small, becomes infinitely large. This does not, of course, agree with experiment.

This discouraging state of affairs generated in many people a strong distrust of quantum field theory. There were even those with the extreme view that the concept of field reaction itself had nothing to do at all with the true law of nature.

On the other hand, there was also the view that the field reaction might not be altogether meaningless but would play an essential role in the scattering processes, though the appearance of divergences revealed a defect of the theory. Heisenberg¹⁴, in his paper published in 1949, emphasized that the field reaction would be crucial in meson-nucleon scattering. Just at that time I was studying at Leipzig, and I still remember vividly how Heisenberg enthusiastically explained this idea to me and handed me galley proofs of his forthcoming paper. Influenced by Heisenberg, I came to believe that the problem of field reactions far from being meaningless was one which required a frontal attack.

Thus, after coming back to Japan from Leipzig, I began to examine the nature of the infinities appearing in scattering processes at the same time that I was engaged in the above-mentioned work of formulating a covariant field theory. What I wanted to know was what kind of relationship exists between the infinity associated with the scattering process and that associated with the mass. If you read the above-mentioned papers of Bloch-Nordsieck and Pauli-Fierz, you will see that one of the terms containing infinite quantities is first separated out by a contact transformation and this term turns out to be just the term modifying the mass. Besides this kind of infinity there appeared, according to Pauli-Fierz, another kind of infinity characteristic of the scattering process. I further investigated a couple of simple models which were not realistic, but could be solved exactly. What was understood from these models, was that the most strongly divergent terms in the scattering process had the same form as the expression giving the modification of the particle mass due to field reactions, and therefore both should be manifestations of the same effect. In other words, at least a portion of the infinities appearing in the scattering process could be amalgamated into the infinity associated with the particle mass, leaving infinities proper to the scattering process alone. These turned out to be more weakly divergent than the infinity associated with the mass.

Since these conclusions were derived from non-relativistic or unrealistic models, it was still doubtful whether the same thing would occur in the case of relativistic electrons interacting with the electromagnetic field. Dancoff tried to answer this question. He calculated relativistically the infinities appearing in the scattering process and determined which of them could be amalgamated into the mass and which remained as infinities proper to the scattering process alone. He found that there remained, in the latter group of infinite terms, one which was at least as divergent as the infinity of the mass, a finding which differed from the conclusion based on fictitious models.

Actually, there are two kinds of field reactions in the case of the relativistic electron and electromagnetic field. One of them ought to be called "of mass type" and the other "of vacuum polarization type". The field reaction of mass type changes the apparent electronic mass from its original value by the amount of the electromagnetic mass as was calculated by Weisskopf. On the other hand, the field reaction of vacuum polarization type changes the apparent electronic charge from its original value. As was discussed in further papers by Weisskopf15 and others, infinite terms appear in the apparent electronic charge if the effect of vacuum polarization is taken into account. However, in this talk, for simplicity, I will mention only briefly the divergence of the vacuum polarization type.

(4) In the meantime, in 1946, Sakata¹⁶ proposed a promising method of eliminating the divergence of the electron mass by introducing the idea of a field of cohesive force. It was the idea that there exists unknown field, of the type of the meson field which interacts with the electron in addition to the electromagnetic field. Sakata named this field the cohesive force field, because the apparent electronic mass due to the interaction of this field and the electron, though infinite, is negative and therefore the existence of this field could stabilize the electron in some sense. Sakata pointed out the possibility that the electromagnetic mass and the negative new mass cancel each other and that the infinity could be eliminated by suitably choosing the coupling constant between this field and the electron. Thus the difficulty which had troubled people for a long time seemed to disappear insofar as the mass was concerned. (It was found later that Pais¹⁷ proposed the same idea in the U.S. independently of Sakata.) Then what concerned me most was whether the infinities appearing in the electron-scattering process could also be removed by the idea of a plus-minus cancellation.

An example of a computation of how the field reaction influences the scattering process was already given by Dancoff. What we had to do was just to replace the electromagnetic field by the cohesive force field in Dancoff's calculation. I mobilized young people around me and we performed the computation together¹⁸ Infinities with negative sign actually appeared in the scattering cross-section as was expected. However, when we compared these with the infinities with positive sign which Dancoff calculated for the electromagnetic field, the two infinities did not cancel each other completely. That is to say, according to our result, the Sakata theory led to the cancellation of infinities for the mass but not for the scattering process. It was also known that the infinity of vacuum polarization type was not cancelled by the introduction of the cohesive force field.

Unfortunately, Dancoff did not publish the detailed calculations in his paper, and while we were engaged in the above considerations, we felt it necessary to do Dancoff's calculation over again for ourselves in parallel with the computation of the influence of the cohesive force field. At the same time I happened to discover a simpler method of calculation.

This new method of calculation was to use the technique of contact transformations based on the previously mentioned formalism of the covariant field theory and was in a sense a relativistic generalization of the Pauli-Fierz method. This method had the advantage of separating the electromagnetic mass from the beginning, just as was shown in their paper.

Our new method of calculation was not at all different in its contents from Dancoff's perturbation method, but had the advantage of making the calculation more clear. In fact, what took a few months in the Dancoff type of calculation could be done in a few weeks. And it was by this method that a mistake was discovered in Dancoff's calculation ; we had also made the same mistake in the beginning. Owing to this new, more lucid method, we noticed that, among the various terms appearing in both Dancoff's and our previous calculation, one term had been overlooked. There was only one missing term, but it was crucial to the final conclusion. Indeed, if we corrected this eror, the infinities appearing in the scattering process of an electron due to the electro-magnetic and cohesive force fields cancelled completely, except for the divergence of vacuum polarization type.

(5) When this unfortunate error of Dancoff's was discovered, we had to reexamine his conclusions concerning the relation between the divergence of the scattering process and the divergence of the mass, in particular, the conclusion that there remained a portion of the infinities of the scattering process which could not be amalgamated into the modification of the mass. In fact, it turned out that after correcting the error, the infinity of mass type appearing in the scattering process could be reduced completely to the modification of the mass, and the remaining field reaction belonging to the scattering proper was not divergent¹⁹. In other words, the highest divergence part of the infinities appearing in the scattering process, in the relativistic as well as in the non-relativistic case, could be attributed to the infinity of mass. The reason why the remaining part became finite in the relativistic case was due to the fact that the order of the highest divergence was only log co, and after amalgamating the divergence into the mass term, the remainder was convergent. The great value of this method of contact transformations was that once the infinity of the mass was separated out, we obtained a divergence-free theoretical framework.

In this way the nature of various infinities became fairly clear. Though I did not describe here the infinity of vacuum polarization type, this too appears in the scattering process, as mentioned earlier. However, Dancoff had already discovered that this infinity could be amalgamated into an apparent change in the electronic charge. To state the conclusion, therefore, all infinities appearing in the scattering process can be attributed either to the infinity of the electromagnetic mass or to the infinity appearing in the electronic charge - there are no other divergences in the theory.

It is a very pleasant thing that no divergence is involved in the theory except for the two infinities of the electronic mass and charge. We cannot say that we have no divergences in the theory, since the mass and charge are in fact infinite. It is to be noticed, however, that if we reduce the infinities appearing in the scattering process to modifications of mass and charge, the remaining terms all become finite. Further, if we examine the structure of the theory, after the infinities are amalgamated into the mass and charge terms, we see that the only mass and charge appearing in the theory are the values modified by field reactions - the original values and excess ones due to field reactions never appear separately.

This situation gives rise to the following possibility. The theory does not of course yield a resolution of the infinities. That is, since those parts of the modified mass and charge due to field reactions contain divergence, it is impossible to calculate them by the theory. However, the mass and charge observed in experiments are not the original mass and charge but the mass and charge as modified by field reactions, and they are finite. On the other hand, the mass and charge appearing in the theory are, as I mentioned above, after all the values modified by field reactions. Since this is so, and particularly since the theory is unable to calculate the modified mass and charge, we may adopt the procedure of substituting experimental values for them phenomenologically. When a theory is incompetent in part, it is a common procedure to rely on experiment for that part. This procedure is called the renormalization of mass and charge, and our method has brought the possibility that the theory will lead to finite results by the renormalization even if it contains defects.

The idea of renormalization is far from new. Many people used explicitly or implicitly this idea, and we find the word renormalization already in Dancoff's paper. In his calculation it appeared, because of an error that there still remained a divergence in the scattering even after the renormalization of the electron mass. This error was very unfortunate; if he had performed the calculation correctly, the history of renormalization theory would have been completely different.

(6) This period, around 1946-1948, was soon after the second world war, and it was quite difficult in Japan to obtain information from abroad. But soon we got the news that in the U.S., Lewis and Epstein²⁰ found Dancoff's mistake and gave the same conclusions as ours, Schwinger²¹ constructed a covariant field theory similar to ours, and he was probably performing various calculations making use of it. In particular, little by little news arrived that the so-called Lamb-shift was discovered²² as a manifestation of the electromagnetic field reaction and that Bethe²³ was calculating it theoretically. The first information concerning the Lamb-shift was obtained not through the Physical Review, but through the popular science column of a weekly U.S. magazine. This information about the Lamb-shift prompted us to begin a calculation more exactly than Bethe's tentative one.

The Lamb-shift is a phenomenon in which the energy levels of a hydrogen atom show some shifts from the levels given by the Dirac theory. Bethe thought that the field reactions were primarily responsible for this shift. According to his calculation, field reactions give rise to an infinite level shift, but he thought that it should be possible to make it finite by a mass renormalization and a tentative calculation yielded a value almost in agreement with experiments.

This problem of the level shift is different from the scattering process, but it was conceivable that the renormalization which was effective in avoiding infinities in the scattering process would be workable in this case as well. In fact, the contact transformation method of Pauli and Fierz devised to solve the scattering problem could be applied to this case, clarifying Bethe's calculation and justifying his idea. Therefore the method of covariant contact transformations, by which we did Dancoff's calculation over again would also be useful for the problem of performing the relativistic calculation for the Lamb-shift. This was our prediction.

The calculation of the Lamb-shift was done by many people in the U.S.²⁴. Among others, Schwinger, commanding powerful mathematical techniques, and by making thorough use of the method of covariant contact transformations, very skilfully calculated not only the Lamb-shift but other quantities such as the anomalous magnetic moment of the electron. After long, laborious calculations, less skilful than Schwinger's, we²⁵ obtained a result for the Lambshift which was in agreement with Americans'. Furthermore, Feynman²⁶ devised a convenient method based on an ingenious idea which could be used to extend the approximation of Schwinger and ours to higher orders, and Dyson²⁷ showed that all infinities appearing in quantum electrodynamics could be treated by the renormalization procedure to an arbitrarily high order of approximation. Furthermore, this method devised by Feynman and developed by Dyson was shown by many people to be applicable not only to quantum electrodynamics, but to statistical mechanics and solid-state physics as well, and provided a new, powerful method in these fields. However, these matters will probably be discussed by Schwinger and Feynman themselves and need not be explained by me. So far I have told you the story of how I played a tiny, partial role in the recent development of quantum electrodynamics, and here I would like to end my talk.

The given extract is the lecture of noble prize winning physicist sin itiro tomonaga.

introduction of muons in quntum electrodynamics

QED + μ: introducing the muon

It’s time to return to our ongoing exploration of the structure of the Standard Model. Our primary tools are Feynman diagrams, which we introduced in previous posts (part 1, part 2). By now we’ve already familiarized ourselves with quantum electrodynamics (QED): the theory of electrons, positrons, and photons. Now we’re going to start adding on pieces to build up the Standard Model. We’ll start with the muon, portrayed below by Los Angeles artist Julie Peasley. (These handmade plushes can be found at her website, The Particle Zoo.)

We’re all familiar with the electron. Allow me to introduce its heavier cousin, the muon (μ). Where did this muon come from? Or, as Nobel Prize winner I. I. Rabi once asked, “Who ordered that?” (This is still an unanswered question!) Besides its mass, the muon has the same fundamental properties as the electron: it has the same charge, feels the same forces, and—like the electron—has an anti-particle partner.

Feynman rules for QED+μ

This makes it really easy to extend our Feynman rules. We’ll call our theory “QED+μ,” quantum electrodynamics with an additional particle. We just have to write the rules for two copies of QED:

Let’s recall how to interpret this. The three lines tell us that we have three kinds or particles in the theory: electrons (e), muons (μ), and photons (γ). Recall that the matter particles, the ones whose lines have an arrow, also have antiparticles. We indicate antiparticles by arrows pointing in the wrong direction when we read the diagrams from left-to-right. The vertex rules tell us that we have two kinds of interactions: a photon can either interact with two electrons or two muons.

It’s important to note that we cannot have photon couplings that mix electrons and muons. In terms of conservation laws, we say that electron and muon number are each conserved. For example, in the theory we’ve developed so far, you cannot have a muon decay into an electron and a photon. (We’ll introduce these sorts of interactions next time when we discuss electroweak theory.)

Exercise: Is the following diagram allowed in QED + μ?

Answer: Yes! But doesn’t this violate conservation of electron and muon number? You start out with two e‘s on the left and end up with two μ’s. Hint: what are the arrows telling you?

Once you’ve convinced yourself that the above diagram doesn’t violate electron or muon conservation, let me remark that this is an easy way to produce muons at low energy electron colliders. You just smash an electron against a positron and sometimes you’ll end up with a muon-antimuon pair which you can detect experimentally.

Exercise: when we previously did electron-positron to electron-positron scattering, we had to include two diagrams. Why is there only one diagram for eμ to eμ? Hint: draw the two diagrams for ee to ee and check if the Feynman rules still allow both diagrams if we convert the final states to muons.

Detecting muons, some collider physics

If you think about this a little, you might wonder: if electrons and muons are so similar, how can experimentalists distinguish between them at a collider? Seth and Mike might scold me for skipping over some information about the interaction of charged particles through matter, but one simple way to distinguish muons from electrons is to measure their energy and momenta. We know that (away from a potential) a particle’s energy is the sum of its kinetic energy plus it’s mass energy added in quadrature E²=m²c⁴+p²c² (this is the “real” version of E=mc²). Since muons are heavier than electrons, we can just check the mass of the particle by plugging in the measured energy and momentum.

Actually, this is an oversimplified picture. In order not to annoy the other US/LHC bloggers, I’d better provide a slightly less oversimplified “cartoon.” Electrons are light, so let’s imagine that they’re ping pong balls. On the other hand, muons are heavy, so let’s imagine them as bowling balls. As you probably know, the LHC detectors are big and full of stuff… by that I mean atoms, which in turn are made up of a nucleus and a cloud of electrons. We can thus imagine a sea of ping-pong balls (think of a Chuck-E-Cheese ball pit). When electrons hit this ball pit, they end up distributing all of their energy into the other balls. This happens in the electromagnetic calorimeter, or ECAL. “Calor” is Latin for heat, so you can guess that the ECAL is really just a big fancy thermometer that measures the energy that the electron dissipates. Muons on the other hand, are bowling balls that are so massive that they just barrel straight through the ball pit to get to the other side. Here’s a very scientific illustration:

I hope we don’t get any comments saying, “oh man, muons are jerks.” In fact, they’re quite the opposite: muons are the only Standard Model particles that make it all the way to the outside of the detector, making it easy for us to identify them. In fact, the big distinctive toroidal magnets on the ATLAS detector below are there to bend the path of muons to help the outermost detector determine the muon momentum by measuring the curvature of their trail.

Exercise: [for those who want to do some actual calculations, requires a high school physics background] Convince yourself that this heuristic picture is correct by calculating the final momenta of a ball colliding elastically with (a) a ball of the same mass and (b) a ball of much lighter mass.

ATLAS toroidal magnets. Image from the Interactions.org Image Bank

Neat things that muons can do

Let me make a few more semi-historical remarks: our QED+μ model is just a theoretical toy. Historically, scientists knew immediately that something was weird about the muon: unlike electrons, it decayed into other particles and seemed to interact with mesons in unusual ways. In fact, for a while people thought that muons were a kind of meson. These differences ended up being a harbinger of something more interesting: the weak force.

Exercise: convince yourself that our Feynman rules for QED+μ do not allow muon decay, i.e. μ turning into non-μ stuff.

Muons are generated in the sky when cosmic rays hit atoms of the upper atmosphere. These rain down onto the Earth and force us to put our dark matter experiments deep underground to avoid their ‘noise.’ What’s really neat, however, is that the fact that muons make it to the surface of the Earth is a rousing experimental check of relativity. We know that muons at rest decay in microseconds. In this time, it seems like there’s no way for them to traverse the kilometers (about 4 km) between the Earth and its upper atmosphere; even if they were traveling at the speed of light! (c ~ 3. 10⁸ m/s). What’s happening is the phenomenon of time dilation!

Introducing the tau (via the Socratic method)

Exercise: the Standard Model actually has another cousin of the electron, the tau (τ), leading to three charged leptons in total. Write down the Feynman rules for the theory QED+μ+τ, i.e. the theory of electrons, muons, and taus interacting via photons. Make sure that electron, muon, and tau number are all conserved. Draw the diagram for tau production in an electron-positron collider.

Exercise: Above we argued that muons are special because they barrel right through our detectors like bowling balls through an array of ping pong balls. Taus are even heavier, shouldn’t they also make it to the outside of the detector?

Answer: This was a bit of a trick question. The logic is correct that sufficiently energetic taus should make it all the way to the outside of the detector in our QED+μ+τ theory. However, this is not the full story for electrons, muons, and taus (collectively known as leptons) in the Standard Model. Like muons, taus are unstable and will decay. In fact, they decay much more quickly than muons because they have more mass and can decay into stuff (they have more “phase space”). While muons are like bowling balls barreling through the detector, taus are more like grenades that burst into hadronic “shrapnel” inside the calorimeters. They are usually very difficult to reconstruct from the data.

A preview of things to come:

Now we’re very familiar with putting together multiple copies of QED. For now, there are only three copies we have to worry about. It is an open question why this is the case. The existence of at least three copies, however, turns out to be significant for the imbalance of matter and anti-matter in the universe. In the next post we’ll introduce the weak force and really see what we can do with these leptons.

I’m currently in the middle of my “Advancement to Candidacy” exam, so my posts might be a little more delayed than usual this month. By the end of it, however, I hope to be blogging as an official PhD candidate.

Erratum: virtual particles

I wanted to correct a misleading statement I made in my previous QED post: I discussed the visualization of virtual particles as balls that two kids toss back and forth while standing on frictionless ice. Conservation of momentum causes the two kids to slide apart as they throw and catch the ball, generating what we observe macroscopically as a repulsive force. We mentioned that it’s more difficult to see how this could give rise to an attractive force. I suggested that this is a phenomenon coming from the accumulated effect of many quantum exchanges. While this is true, there is a simpler way to understand this: pretend the ball has negative momentum! Since the particle is virtual, it is inherently quantum mechanical and needn’t have ‘on-shell’ (physical) momentum. Thus one could imagine tossing the ball with negative momentum, causing one to be deflected in the same direction as the ball was tossed. Similarly, catching the ball with negative momentum would push one in thedirection that the ball came from.

Does it make sense classically? No! But that’s okay because they’re virtual particles.

That’s all for now, folks!

quantum electrodynamics - an overview

quantum electrodynamics (QED), quantum field theory of the interactions of charged particles with the electromagnetic field. It describes mathematically not only all interactions of light with matter but also those of charged particles with one another. QED is a relativistic theory in that Albert Einstein’stheory of special relativity is built into each of its equations. Because the behaviour of atoms and molecules is primarily electromagnetic in nature, all of atomic physics can be considered a test laboratory for the theory. Some of the most precise tests of QED have been experiments dealing with the properties of subatomic particles known as muons. The magnetic moment of this type of particle has been shown to agree with the theory to nine significant digits. Agreement of such high accuracy makes QED one of the most successful physical theories so far devised.

In 1928 the English physicist P.A.M. Dirac laid the foundations for QED with his discovery of a wave equation that described the motion and spin of electrons and incorporated both quantum mechanicsand the theory of special relativity. The QED theory was refined and fully developed in the late 1940s by Richard P. Feynman, Julian S. Schwinger, and Tomonaga Shin’ichirō, independently of one another. QED rests on the idea that charged particles (e.g., electrons and positrons) interact by emitting and absorbing photons, the particles that transmit electromagnetic forces. These photons are “virtual”; that is, they cannot be seen or detected in any way because their existence violates the conservation of energy and momentum. The photon exchange is merely the “force” of the interaction, because interacting particles change their speed and direction of travel as they release or absorb the energy of a photon. Photons also can be emitted in a free state, in which case they may be observed as light or other forms of electromagnetic radiation.

The interaction of two charged particles occurs in a series of processes of increasing complexity. In the simplest, only one virtual photon is involved; in a second-order process, there are two; and so forth. The processes correspond to all the possible ways in which the particles can interact by the exchange of virtual photons, and each of them can be represented graphically by means of the so-called Feynman diagrams. Besides furnishing an intuitive picture of the process being considered, this type of diagram prescribes precisely how to calculate the variable involved. Each subatomic process becomes computationally more difficult than the previous one, and there are an infinite number of processes. The QED theory, however, states that the more complex the process—that is, the greater the number of virtual photons exchanged in the process—the smaller the probability of its occurrence. For each level of complexity, the contribution of the process decreases by an amount given by α²—where α is a dimensionless quantity called the fine-structure constant, with a numerical value equal to (¹/₁₃₇). Thus, after a few levels the contribution is negligible. In a more-fundamental way the factor α serves as a measure of the strength of the electromagnetic interaction. It equals^e²/_{4πε_o[planck]c}, where e is the electron charge, [planck] is Planck’s constant divided by 2π, c is the speed of light, and ε_o is the permittivity of free space.

QED is often called a perturbation theory because of the smallness of the fine-structure constant and the resultant decreasing size of higher-order contributions. This relative simplicity and the success of QED have made it a model for other quantum field theories. Finally, the picture of electromagnetic interactions as the exchange of virtual particles has been carried over to the theories of the otherfundamental interactions of matter, the strong force, the weak force, and the gravitational force.

reference from britannica encyclopedia.

feymann model of quantum electrodynamics

Near the end of his life, Richard P. 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,^[1]^[20] 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.

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.

Feynman diagram elements

These actions are represented in a form of visual shorthand by the three basic elements of Feynman 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.

It is important not to over-interpret these diagrams. Nothing is implied about how a particle gets from one point to another. The diagrams do not imply that the particles are moving in straight or curved lines. They do not imply that the particles are moving with fixed speeds. The fact that the photon is often represented, by convention, by a wavy line and not a straight one does not imply that it is thought that it is more wavelike than is an electron. The images are just symbols to represent the actions above: photons and electrons do, somehow, move from point to point and electrons, somehow, emit and absorb photons. We do not know how these things happen, but the theory tells us about the probabilities of these things happening.

As well as the visual shorthand for the actions Feynman introduces another kind of shorthand for the numerical quantities which tell us about the probabilities. If a photon moves from one place and time—in shorthand, A—to another place and time—in shorthand, B—the associated quantity is written in Feynman's shorthand as P(A to B). The similar quantity for an electron moving from C to D is written E(C to D). The quantity which tells us about the probability for the emission or absorption of a photon he calls 'j'. This is related to, but not the same as, the measured electron charge 'e'.

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 quantities to calculate the probability of any such complex interaction. It turns out that the basic idea of QED can be communicated while making the assumption that the quantities mentioned above are just our everyday probabilities. (A simplification of Feynman's book.) Later on this will be corrected to include specifically quantum mathematics, following Feynman.

The basic rules of probabilities that will be used are that a) if an event can happen in a variety of different ways then its probability is the sum of the probabilities of the possible ways and b) if a process involves a number of independent subprocesses then its probability is the product of the component probabilities.

Basic constructions [edit]

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 that the photon moves from B to D (another elementary action). From a knowledge of the probabilities of each of these subprocesses – E(A to C) and P(B to D) – then we would expect to calculate the probability of both happening by multiplying them, using rule b) above. This gives a simple estimated answer to our question.

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 probabilities 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 by multiplying the probabilities 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 probabilities for all the alternatives for E and F. (This is not elementary in practice, and involvesintegration.) But there is another possibility: that 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 of these possibilities (for all points G and H). We then have a better estimation for the total probability by adding the probabilities 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.

There are an infinite number of other intermediate processes in which more and more photons are absorbed and/or emitted. For each of these possibilities there is a Feynman diagram describing it. This implies a complex computation for the resulting probabilities, 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.

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 possibility 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.)

Probability amplitudes [edit]

Feynman replaces complex numbers with spinning arrows, which start at emission and ends at detection of a particle. The sum of all resulting arrows represents the total probability of the event. In this diagram, light emitted by the source S bounces off a few segments of the mirror (in blue) before reaching the detector at P. The sum of all paths must be taken into account. The graph below depicts the total time spent to traverse each of the paths above.

Quantum mechanics introduces an important change on the way probabilities are computed. It has been found that the quantities which we have to use to represent the probabilities are not the usual real numbers we use for probabilities in our everyday world, but complex numbers which are called probability amplitudes.

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 actually 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. No satisfactory reason has been given for why they are needed. But pragmatically we have to accept that they are an essential part of our description of all quantum phenomena. 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

$P=|\mathbf{v}+\mathbf{w}|^2$

$P=|\mathbf{v}\times\mathbf{w}|^2.$

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 numbers

Multiplication of probability amplitudes as complex numbers

Addition and multiplication are familiar 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 start 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) actually consists of 16 complex numbers, or probability amplitude arrows. 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 just 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", E(A to B) × E(C to D) − E(A to C) × E(B to D), where we would expect, from our everyday idea of probabilities, that it would be a sum.

Propagators [edit]

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 describes the behavior of the electron's probability amplitude and the Klein–Gordon equation 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:

$P(\mbox{A to B}) \rightarrow D_F(x_B-x_A),\quad E(\mbox{C to D}) \rightarrow S_F(x_D-x_C)$

where a shorthand symbol such as $x_A$ stands for the four real numbers which give the time and position in three dimensions of the point labeled A.

Mass renormalization [edit]

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 end points. 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 very difficult 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 (see below and the article on mass renormalization). However, Feynman himself remained unhappy about it, calling it a "dippy process".^[20]

Conclusions [edit]

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 totally 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."^[23]

Mathematics [edit]

Mathematically, QED is an abelian gauge theory with the symmetry group U(1). The gauge field, which mediates the interaction between the charged spin-1/2 fields, is the electromagnetic field. The QED Lagrangian for a spin-1/2 field interacting with the electromagnetic field is given by the real part of

$\mathcal{L}=\bar\psi(i\gamma^\mu D_\mu-m)\psi -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$

where

$\gamma^\mu$ are Dirac matrices;

$\psi$ a bispinor field of spin-1/2 particles (e.g. electron–positron field);

$\bar\psi\equiv\psi^\dagger\gamma_0$ , called "psi-bar", is sometimes referred to as Dirac adjoint;

$D_\mu \equiv \partial_\mu+ieA_\mu+ieB_\mu \,\!$ is the gauge covariant derivative;

e is the coupling constant, equal to the electric charge of the bispinor field;

A_μ is the covariant four-potential of the electromagnetic field generated by the electron itself;

B_μ is the external field imposed by external source;

$F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu \,\!$ is the electromagnetic field tensor.

Equations of motion [edit]

To begin, substituting the definition of D into the Lagrangian gives us

$\mathcal{L} = i \bar\psi \gamma^\mu \partial_\mu \psi - e\bar{\psi}\gamma_\mu (A^\mu+B^\mu) \psi -m \bar{\psi} \psi - \frac{1}{4}F_{\mu\nu}F^{\mu\nu}. \,$

Next, we can substitute this Lagrangian into the Euler–Lagrange equation of motion for a field:

$\partial_\mu \left( \frac{\partial \mathcal{L}}{\partial ( \partial_\mu \psi )} \right) - \frac{\partial \mathcal{L}}{\partial \psi} = 0 \,$

(2)

to find the field equations for QED.

The two terms from this Lagrangian are then

$\partial_\mu \left( \frac{\partial \mathcal{L}}{\partial ( \partial_\mu \psi )} \right) = \partial_\mu \left( i \bar{\psi} \gamma^\mu \right), \,$

$\frac{\partial \mathcal{L}}{\partial \psi} = -e\bar{\psi}\gamma_\mu (A^\mu+B^\mu) - m \bar{\psi}. \,$

Substituting these two back into the Euler–Lagrange equation (2) results in

$i \partial_\mu \bar{\psi} \gamma^\mu + e\bar{\psi}\gamma_\mu (A^\mu+B^\mu) + m \bar{\psi} = 0 \,$

with complex conjugate

$i \gamma^\mu \partial_\mu \psi - e \gamma_\mu (A^\mu+B^\mu) \psi - m \psi = 0. \,$

Bringing the middle term to the right-hand side transforms this second equation into

$i \gamma^\mu \partial_\mu \psi - m \psi = e \gamma_\mu (A^\mu+B^\mu) \psi \,$

The left-hand side is like the original Dirac equation and the right-hand side is the interaction with the electromagnetic field.

One further important equation can be found by substituting the Lagrangian into another Euler–Lagrange equation, this time for the field, A^μ:

$\partial_\nu \left( \frac{\partial \mathcal{L}}{\partial ( \partial_\nu A_\mu )} \right) - \frac{\partial \mathcal{L}}{\partial A_\mu} = 0\,.$

(3)

The two terms this time are

$\partial_\nu \left( \frac{\partial \mathcal{L}}{\partial ( \partial_\nu A_\mu )} \right) = \partial_\nu \left( \partial^\mu A^\nu - \partial^\nu A^\mu \right), \,$

$\frac{\partial \mathcal{L}}{\partial A_\mu} = -e\bar{\psi} \gamma^\mu \psi \,$

and these two terms, when substituted back into (3) give us

$\partial_\nu F^{\nu \mu} = e \bar{\psi} \gamma^\mu \psi \,$

Now, if we impose the Lorenz gauge condition, that the divergence of the four potential vanishes

$\partial_{\mu} A^\mu = 0$

then we get

$\Box A^{\mu}=e\bar{\psi} \gamma^{\mu} \psi\,,$

which is a wave equation for the four potential, the QED version of the classical Maxwell equations in the Lorenz gauge.

Interaction picture [edit]

This theory can be straightforwardly quantized by treating bosonic and fermionic sectors as free. This permits us to build a set of asymptotic states which can be used to start a computation of the probability amplitudes for different processes. In order to do so, we have to compute an evolution operator that, for a given initial state $|i\rangle$ , will give a final state $\langle f|$ in such a way to have

$M_{fi}=\langle f|U|i\rangle.$

This technique is also known as the S-Matrix. 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:

$V=e\int d^3x\bar\psi\gamma^\mu\psi A_\mu$

and so, one has

$U=T\exp\left[-\frac{i}{\hbar}\int_{t_0}^tdt'V(t')\right]$

where T is the time ordering operator. This evolution operator only has meaning as a series, and what we get here is a perturbation series with the fine structure constant as the development parameter. This series is called the Dyson series.

Feynman diagrams [edit]

Despite the conceptual clarity of this 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 transforms of the propagators. Quantum physics considers particle's momenta rather than their positions, 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 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

To these rules we must add a further one for closed loops that implies an integration on momenta $\int d^4p/(2\pi)^4$ , since these internal ("virtual") particles are not constrained to any specific energy–momentum – even that usually required by special relativity (see this article for details). From them, computations of probability amplitudes are straightforwardly given. An example is Compton scattering, with an electron and a photon undergoing elastic scattering. Feynman diagrams are in this case

and so we are able to get the corresponding amplitude at the first order of a perturbation series for the S-matrix:

$M_{fi}=(ie)^{2}\overline{u}(\vec{p}\,',s')\epsilon\!\!\!/\,'(\vec{k}\,',\lambda')^{*}{p\!\!\!/+k\!\!\!/+m_{e} \over (p+k)^{2}-m^{2}_{e}}\epsilon\!\!\!/(\vec{k},\lambda)u(\vec{p},s)+(ie)^{2}\overline{u}(\vec{p}\,',s')\epsilon\!\!\!/(\vec{k},\lambda){p\!\!\!/-k\!\!\!/'+m_{e}\over (p-k')^{2}-m^{2}_{e}}\epsilon\!\!\!/\,'(\vec{k}\,',\lambda')^{*}u(\vec{p},s)$

from which we are able to compute the cross section for this scattering.

Renormalizability [edit]

Higher order terms can be straightforwardly computed for the evolution operator but these terms display diagrams containing the following simpler ones

One-loop contribution to the vacuum polarizationfunction $\Pi\,$
One-loop contribution to the electron self-energyfunction $\Sigma \,$
One-loop contribution to the vertex function $\Gamma\,$

that, being closed loops, imply the presence of diverging integrals having no mathematical meaning. To overcome this difficulty, a technique like renormalization has been devised, producing finite results in very close agreement with experiments. It is important to note that a criterion for theory being meaningful after renormalization is that the number of diverging diagrams is finite. In this case the theory is said 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 interactions, except gravitation whose quantum counterpart is presently under very active research, are renormalizable theories.

Nonconvergence of series [edit]

An argument by Freeman Dyson shows that the radius of convergence of the perturbation series in QED is zero.^[24] The basic argument goes as follows: if the 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 do not converge, but are an asymptotic series.

The above article is a reference from the wikipedia