# Loose ends…

Pre-scriptum (dated 26 June 2020): My views on the true nature of light, matter and the force or forces that act on them have evolved significantly as part of my explorations of a more realist (classical) explanation of quantum mechanics. If you are reading this, then you are probably looking for not-to-difficult reading. In that case, I would suggest you read my re-write of Feynman’s introductory lecture to QM. If you want something shorter, you can also read my paper on what I believe to be the true Principles of Physics. Having said that, I still think there are a few good quotes and thoughts in this post too. đ

Original post:

It looks like I am getting ready for my nextÂ plungeÂ into Roger Penrose’sÂ Road to Reality. I still need to learn more about those Hamiltonian operators and all that, but I can sort of ‘see’ what they are supposed to do now. However, before I venture off on another series of posts on math instead of physics, I thought I’d briefly present what Feynman identified as ‘loose ends’ in his 1985 Lectures on Quantum Electrodynamics –Â a few years before his untimely death – and then see if any of those ‘loose ends’ appears less loose today, i.e. some thirty years later.

The three-forces model and coupling constants

All three forces in the Standard Model (the electromagnetic force, the weak force and the strong force) are mediated by force carrying particles:Â bosons. [Let me talk about the Higgs field later and – of course – I leave out the gravitational force, for which we do not have a quantum field theory.]

Indeed, the electromagnetic force is mediated by the photon; the strong force is mediated by gluons; and the weak force is mediated by W and/or Z bosons. The mechanism is more or less the same for all. There is a so-calledÂ coupling (or a junction)Â between a matter particle (i.e. aÂ fermion) and a force-carrying particleÂ (i.e. the boson), and theÂ amplitudeÂ for this coupling to happen is given by a number that is related toÂ a so-calledÂ coupling constant.Â

Let’s give an example straight away – and let’s do it for the electromagnetic force, which is the only force we have been talking about so far. The illustration below shows three possible ways for two electrons moving in spacetime to exchange a photon. This involves two couplings: one emission, and one absorption. The amplitude for an emission or an absorption is the same: it’s âj. So the amplitude here will be (âj)(âj) =Â j2. Note that the two electronsÂ repelÂ each other as they exchange a photon, which reflects the electromagnetic force between them from a quantum-mechanical point of view !

We will have a number like this for all three forces. Feynman writes the coupling constant for the electromagnetic force asÂ Â jÂ and the coupling constant for the strong force (i.e. the amplitude for a gluon to be emitted or absorbed by a quark)Â asÂ g. [As for the weak force, he is rather short on that and actually doesn’t bother to introduce a symbol for it. I’ll come back on that later.]

The coupling constant is a dimensionless number and one can interpret it as the unit of ‘charge’ for the electromagnetic and strong force respectively. So the ‘charge’ qÂ of a particle should be read as qÂ times the coupling constant. Of course, we can argue about that unit. The elementary charge for electromagnetism was or is – historically – the charge of the proton (q = +1), but now the proton is no longer elementary: it consists of quarks with charge â1/3 and +2/3 (for the d and u quark) respectively (a proton consists of two u quarks and one d quark, so you can write it as uud). So what’sÂ j then? Feynman doesn’t give its precise value but uses an approximate value of â0.1. It is an amplitude so it should be interpreted as a complex number to be added or multiplied with other complex numbers representing amplitudes – so â0.1 is “a shrink to about one-tenth, and half a turn.” [In these 1985Â Lectures on QED, which he wrote for a lay audience, he calls amplitudes ‘arrows’, to be combined with other ‘arrows.’ In complex notation,Â â0.1 = 0.1eiÏÂ = 0.1(cosÏ + isinÏ).]

Let me give a precise number. The coupling constant for the electromagnetic force is the so-called fine-structure constant, and it’s usually denoted by the alpha symbol (Î±). There is a remarkably easy formula for Î±, which becomes even easier if we fiddle with units to simplify the matter even more. Let me paraphrase Wikipedia on Î±Â here, because I have no better way of summarizing it (the summary is also nice as it shows how changing units – replacing the SI units by so-called naturalÂ units – can simplify equations):

1. There are three equivalent definitions ofÂ Î±Â in terms of other fundamental physical constants:

$\alpha = \frac{k_\mathrm{e} e^2}{\hbar c} = \frac{1}{(4 \pi \varepsilon_0)} \frac{e^2}{\hbar c} = \frac{e^2 c \mu_0}{2 h}$
whereÂ eÂ is the elementary charge (so that’s the electric charge of the proton);Â Ä§Â =Â h/2Ï is the reduced Planck constant;Â cÂ is the speed of light (in vacuum);Â Î”0Â is the electric constant (i.e. the so-calledÂ permittivity of free space);Â Â”0Â is the magnetic constant (i.e. the so-called permeability of free space); andÂ keÂ is the Coulomb constant.

2. In the old centimeter-gram-secondÂ variant of the metric system (cgs), the unit of electric charge is chosenÂ such thatÂ the Coulomb constant (or the permittivity factor) equals 1.Â Then the expression of the fine-structure constant just becomes:

$\alpha = \frac{e^2}{\hbar c}$

3. When using so-calledÂ natural units, we equateÂ Î”0Â ,Â cÂ andÂ Ä§Â to 1. [That does not mean they are the same, but they just become the unit for measurement for whatever is measured in them. :-)] The value of the fine-structure constant then becomes:

$\alpha = \frac{e^2}{4 \pi}.$

Of course, then it just becomes a matter of choosing a value for e. Indeed, we still haven’t answered the question as to what we should choose as ‘elementary’: 1 or 1/3? If we take 1, then Î± is just a bit smaller than 0.08 (around 0.0795775 to be somewhatÂ more precise). If we take 1/3 (the value for a quark), then we get a much smaller value: about 0.008842 (I won’t bother too much about the rest of the decimals here). Feynman’s (very) rough approximation of â0.1 obviously uses the historic proton charge, so e = +1.

The coupling constant for the strong force isÂ muchÂ bigger. In fact, if we use the SI units (i.e. one of the three formulas for Î± under point 1 above), then we get an alpha equal to some 7.297Ă10â3. In fact, its value will usually be written as 1/Î±, and so we get a value of (roughly) 1/137. In this scheme of things, the coupling constant for the strongÂ force is 1, so that’s 137 times bigger.

Coupling constants, interactions, and Feynman diagrams

So how does it work? The Wikipedia article on coupling constants makes an extremely useful distinction between the kinetic partÂ and the properÂ interaction partÂ of an ‘interaction’. Indeed, before we just blindly associateÂ qubitsÂ with particles, it’s probably useful toÂ not onlyÂ look at how photon absorption and/or emission works, but also at how a process as common as photon scattering works (so we’re talking ComptonÂ scatteringÂ here – discovered in 1923, and it earned Compton a Nobel Prize !).

The illustration below separates theÂ kineticÂ andÂ interactionÂ part properly: the photon and the electron are bothÂ deflected (i.e. the magnitude and/or direction of their momentum (p) changes) – that’s the kinetic part – but, in addition,Â the frequency of the photon (and, hence, its energy – cf. E = hÎœ) is also affected – so that’s the interaction part I’d say.

With an absorption or an emission, the situation is different, but it also involves frequencies (and, hence, energy levels), as show below: an electron absorbing a higher-energy photon will jump two or more levels as it absorbs the energy by moving to a higher energy level (i.e. a so-calledÂ excitedÂ state), and when it re-emits the energy, the emitted photon will have higher energy and, hence, higher frequency.

This business of frequencies and energy levels may not be so obvious when looking at those Feynman diagrams, but I should add that these Feynman diagrams are not just sketchy drawings: the time and space axis is precisely defined (time and distance are measured in equivalent units) and so the direction of travel of particles (photons, electrons, or whatever particle is depicted) does reflect the direction of travel and, hence, conveys precious information about both the direction as well as the magnitude of the momentum of those particles. That being said, a Feynman diagram does not care about a photon’s frequency and, hence, its energy (its velocity will always be c, and it has no mass, so we can’t get any information from its trajectory).

Let’s look at these Feynman diagrams now, and the underlying force model, which I refer to as the boson exchange model.

The boson exchange model

The quantum field model – for all forces – is aÂ boson exchangeÂ model. In this model, electrons, for example, are kept in orbit through the continuous exchange of (virtual) photons between the proton and the electron, as shown below.

Now, I should say a few words about these ‘virtual’ photons. The most important thing is that you should look at them as being ‘real’. They may be derided as being only temporary disturbances of the electromagnetic field but they are very real force carriers in the quantum field theory of electromagnetism. They may carry very low energy as compared to ‘real’ photons, but they do conserve energy and momentum – in quite a strange way obviously: while it is easy to imagine a photon pushing an electron away, it is a bit more difficult to imagine it pulling it closer, which is what it does here. Nevertheless, that’s how forces are being mediated byÂ virtual particles in quantum mechanics: we have matter particles carryingÂ charge but neutralÂ bosons taking care of the exchange between those charges.

In fact, note how Feynman actually cares about the possibility of one of those ‘virtual’ photons briefly disintegrating into an electron-positron pair, which underscores the ‘reality’ of photons mediating the electromagnetic force between a proton and an electron, Â thereby keeping them close together. There is probably no better illustration to explain the difference between quantum field theory and the classical view of forces, such as the classical view on gravity: there are noÂ gravitonsÂ doing for gravity what photons are doing for electromagnetic attraction (or repulsion).

Pandora’s Box

I cannot resist a small digression here. The ‘Box of Pandora’ to which Feynman refers in the caption of the illustration above is the problem of calculating the coupling constants. Indeed, j is the coupling constant for an ‘ideal’ electron to couple with some kind of ‘ideal’ photon, but how do we calculate that when we actually know that all possible paths in spacetime have to be considered and that we have all of these ‘virtual’ mess going on? Indeed, in experiments, we can only observe probabilities for realÂ electrons to couple with real photons.

In the ‘Chapter 4’ to which the caption makes a reference, he briefly explains the mathematical procedure, which he invented and for which he got a Nobel Prize. He calls it a ‘shell game’. It’s basically an application of ‘perturbation theory’, which I haven’t studied yet. However, he does so with skepticism about its mathematical consistency – skepticism which I mentioned and explored somewhat in previous posts, so I won’t repeat that here. Here, I’ll just note that the issue of ‘mathematical consistency’ is much more of an issueÂ for the strong force, because the coupling constant is so big.

Indeed, terms with j2, j3,Â j4Â etcetera (i.e. the terms involved in adding amplitudes for all possible paths and all possible ways in which an event can happen) quickly become veryÂ small as the exponent increases, but terms withÂ g2, g3,Â g4Â etcetera do not become negligibly small. In fact, they don’t become irrelevant at all. Indeed, if we wrote Î± for the electromagnetic force asÂ 7.297Ă10â3, then the Î± for the strong force is one, and so none of these terms becomes vanishingly small. I won’t dwell on this, but just quote Wikipedia’s very succinct appraisal of the situation: “If Î±Â is much less than 1 [in a quantum field theory with a dimensionless coupling constant Î±], then the theory is said to beÂ weakly coupled. In this case it is well described by an expansion in powers of Î±Â called perturbation theory. [However] If the coupling constant is of order one or larger, the theory is said to beÂ strongly coupled. An example of the latter [the onlyÂ example as far as I am aware: we don’t have like a dozen different forces out there !]Â is the hadronic theory of strong interactions,Â which is why it is called strong in the first place. [Hadrons is just a difficult word for particles composed of quarks – so don’t worry about it: you understand what is being said here.] In such a case non-perturbative methods have to be used to investigate the theory.”

Hmm… If Feynman thought his technique for calculating weak coupling constants was fishy, then his skepticism about whether or not physicists actually know what they are doing when calculating stuff using the strong coupling constant is probably justified. But let’s come back on that later. With all that we know here, we’re ready to present a picture of the ‘first-generation world’.

The first-generation world

The first-generation isÂ ourÂ world, excluding all that goes inÂ those particle accelerators, where they discovered so-called second- and third-generation matter – but I’ll come back to that. Our world consists of only four matterÂ particles, collectively referred to as (first-generation) fermions: two quarks (a u and a d type), the electron, and the neutrino. This is what is shown below.

Indeed, u and d quarks make up protons and neutrons (a proton consists of two u quarks and one d quark, and a neutron must be neutral, so it’s two d quarks and one u quark), and then there’s electrons circling around them and so that’s our atoms. And from atoms, we make molecules and then you know the rest of the story.Â Genesis !Â

Oh… But why do we need the neutrino? [Damn – you’re smart ! You see everything, don’t you? :-)] Well… There’s something referred to asÂ beta decay: this allows a neutron to become a proton (and vice versa).Â Beta decay explains why carbon-14 will spontaneously decay into nitrogen-14. Indeed, carbon-12 is the (very) stable isotope, whileÂ carbon-14 has a life-time of 5,730 Â± 40 years âonlyâ and, hence, measuring how much carbon-14 is left in some organic substance allows us to date it (thatâs what (radio)carbon-dating is about). Now, aÂ betaÂ particleÂ can refer to an electronÂ orÂ a positron, so we can haveÂ ÎČâÂ decay (e.g. the above-mentioned carbon-14 decay) orÂ ÎČ+Â decay (e.g. magnesium-23 into sodium-23). If we have ÎČâÂ decay, then some electron will be flying out in order to make sure the atom as a whole stays electrically neutral. If it’s ÎČ+Â decay, then emitting a positron will do the job (I forgot to mention that each of the particles above also has a anti-matter counterpart – but don’t think I tried to hide anything else: the fermionÂ picture above is pretty complete). That being said, Wolfgang Pauli, one of those geniuses who invented quantum theory, noted, in 1930 already, that some momentum and energy was missing, and so he predicted the emission of this mysterious neutrinos as well. Guess what? These things are very spooky (relatively high-energy neutrinos produced by stars (our Sun in the first place) are going through your and myÂ my body,Â right now and right here,Â at a rate of someÂ hundred trillion per second) but, because they are so hard to detect, the first actual traceÂ of their existence was found in 1956 only. [Neutrino detection is fairly standard business now, however.] But back to quarks now.

Quarks are held together by gluons – as you probably know. Quarks come in flavors (u and d), but gluons come in ‘colors’. It’s a bit of a stupid name but the analogy works great. Quarks exchange gluons all of the time and so that’s what ‘glues’ them so strongly together. Indeed, the so-called ‘mass’ that gets converted into energy when a nuclear bomb explodes is not the mass of quarks (their mass is only 2.4 and 4.8 MeV/c2. Nuclear power is binding energy between quarks that gets converted into heat and radiation and kinetic energy and whatever else a nuclear explosion unleashes. That binding energy is reflected in the difference between the mass of a proton (or a neutron) – around 938 MeV/c2Â – and the mass figure you get when you add two u‘s and oneÂ d, which isÂ them 9.6 MeV/c2Â only. This ratio – a factor of one hundredÂ –Â illustrates once again the strength of theÂ strong force: 99% of the ‘mass’ of a proton or an electron is due to the strong force.Â Â  Â

But I am digressing too much, and I haven’t even started to talk about the bosonsÂ associated with the weak force. Well… I won’t just now. I’ll just move on the second- and third-generation world.

Second- and third-generation matter

When physicists started to look for those quarks in their particle accelerators, Nature had already confused them by producing lots of other particles in these accelerators: in the 1960s, there were more than four hundred of them. Yes. Too much. But they couldn’t get them back in the box. đ

Now, all these ‘other particles’Â areÂ unstableÂ but they survive long enough â aÂ muon, for example,Â disintegrates afterÂ 2.2 millionths of a secondÂ (on average) â to deserve the âparticleâ title, as opposed to a âresonanceâ, whose lifetime can be as short asÂ a billionth of a trillionth of a second. And so, yes, the physicists had to explain them too. So theÂ guys who devised the quark-gluon model (the model is usually associated with Murray Gell-Mann but – as usual with great ideas – some others worked hard on it as well) had already included heavier versions of their quarks to explain (some of) these other particles. And so we do not only have heavier quarks, but also a heavier version of the electron (that’s the muonÂ I mentioned) as well as a heavier version of the neutrino (the so-calledÂ muonÂ neutrino). The two new ‘flavors’ of quarks were called s and c. [Feynman hates these names but let me give them: u stands for up, d for down, s for strange andÂ c for charm. Why? Well… According to Feynman: “For no reason whatsoever.”]

Traces of the second-generation sÂ andÂ cÂ quarks were found in experiments in 1968 and 1974 respectively (it took six years to boost the particle accelerators sufficiently), and the third-generation bÂ quark (forÂ beauty orÂ bottom – whatever) popped up in Fermilab‘s particle accelerator in 1978. To be fully complete, it then took 17 years to detect the super-heavyÂ t quark – which stands forÂ truth. Â [Of all the quarks, this name is probably the nicest: “If beauty, thenÂ truth” – as Lederman and Schramm write in their 1989 history of all of this.]

What’s next? Will there be a fourth or even fifth generation? Back in 1985, Feynman didn’t exclude it (and actually seemed to expect it), but current assessments are more prosaic. Indeed, Wikipedia writes that,Â According to the results of the statistical analysis by researchers from CERN and the Humboldt University of Berlin,Â the existence of further fermions can be excluded with a probability of 99.99999% (5.3 sigma).” If you want to know why… Well… Read the rest of the Wikipedia article. It’s got to do with the Higgs particle.

So theÂ completeÂ model of reality is the one I already inserted in a previous post and, if you find it complicated, remember that the first generation of matter is the one that matters and, among the bosons, it’s the photons and gluons. If you focus on these only, it’s not complicated at all – and surely a huge improvement over those 400+ particles no one understood in the 1960s.

As for the interactions, quarks stick together – and rather firmly so – by interchanging gluons. They thereby ‘change color’ (which is the same as saying there is some exchange of ‘charge’). I copy Feynman’s original illustration hereunder (not because there’s no better illustration: the stuff you can find on Wikipedia has actual colors !) but just because it’s reflects the other illustrations above (and, perhaps, maybe I also want to make sure – with this black-and-white thing – that you don’t think there’s something like ‘real’ color inside of a nucleus).

So whatÂ areÂ the loose ends then? The problem of ‘mathematical consistency’ associated with the techniques used to calculate (or estimate) these coupling constants – which Feynman identifies as a key defectÂ in 1985 – is is a form of skepticism about the Standard Model that isÂ notÂ shared by others. It’s more about the other forces. So let’s now talk about these.

The weak force as theÂ weirdÂ force: about symmetry breaking

I included the weak force in the title of one of the sub-sections above (“The three-forces model”) and then talked about the other two forces only. The W+Â , WâÂ and Z bosons – usually referred to, as a group, as the W bosons, or the ‘intermediate vector bosons’ – are an odd bunch. First, note that they are the only ones that do not only have a (rest) mass (and not just a little bit: they’re almost 100 times heavier than a the proton or neutron – or a hydrogen atom !) but, on top of that, they also have electric charge (except for the Z boson). They are reallyÂ the odd ones out. Â Feynman does not doubt their existence (a Fermilab team produced them in 1983, and they got a Nobel Prize for it, so little room for doubts here !), but it is obvious he finds the weak force interaction model rather weird.

He’s not the only one: in a wonderful publication designed to make a case for more powerful particle accelerators (probably successful, because theÂ Large Hadron ColliderÂ came through – and discovered credible traces of the Higgs field, which is involved in the story that is about to follow),Â Leon LedermanÂ andÂ David SchrammÂ look at the asymmety involved in having massive W bosons and massless photons and gluons, as just one of the many asymmetries associated with the weak force. Let me develop this point.

We like symmetries. They are aesthetic. But so I am talking something else here: in classical physics, characterized by strict causality and determinism,Â we can – in theory – reverse the arrow of time. In practice, we can’t – because of entropy – but, in theory, so-called reversible machines are not a problem. However, in quantum mechanics we cannot reverse time for reasons that have nothing to do with thermodynamics. In fact, there are several types of symmetries in physics:

1. Parity (P) symmetry revolves around the notion that Nature should not distinguish between right- and left-handedness, so everything that works inÂ ourÂ world, should also work in theÂ mirrorÂ world. Now, theÂ weak forceÂ does not respect P symmetry. That was shown by experiments on the decay of pions, muons and radioactive cobalt-60 in 1956 and 1957 already.
2. Charge conjugation or charge (C) symmetry revolves around the notion that a world in which we reverse all (electric) charge signs (so protons would have minus one as charge, and electrons have plus one) would also just work the same. The same 1957 experiments showed that the weak force does alsoÂ notÂ respect C symmetry.
3. Initially, smart theorists noted that theÂ combined operation of CP was respected by these 1957 experiments (hence, the principle of P and C symmetry could be substituted by a combined CP symmetry principle) but, then, in 1964, Val Fitch and James Cronin, proved that the spontaneous decay of neutral kaons (don’t worry if you don’t know what particle this is: you can look it up) into pairs of pions did not respect CP symmetry. In other words, it was – again – the weak forceÂ notÂ respecting symmetry. [Fitch and Cronin got a Nobel Prize for this, so you can imagine it did mean something !]
4. We mentioned time reversal (T) symmetry: how is that being broken? In theory, we can imagine a film being made of those eventsÂ notÂ respecting P, C or CP symmetry and then just pressing the ‘reverse’ button, can’t we? Well… I must admit I do not master the details of what I am going to write now, but let me just quote Lederman (another Nobel Prize physicist) and Schramm (an astrophysicist): “Years before this, [Wolfgang] Pauli [Remember him from his neutrino prediction?] had pointed out that a sequence of operations like CPT could be imagined and studied; that is, in sequence, change all particles to antiparticles, reflect the system in a mirror, and change the sign of time. Pauli’s theorem was that all nature respected the CPT operation and, in fact, that this was closely connected to the relativistic invariance of Einstein’s equations. There is a consensus that CPT invariance cannot be broken –Â at least not at energy scales below 1019Â GeV [i.e. the Planck scale]. However, if CPT is a valid symmetry, then, when Fitch and Cronin showed that CP is a broken symmetry, they also showed that T symmetry must be similarly broken.” (Lederman and Schramm, 1989, From Quarks to the Cosmos,Â p. 122-123)

So the weak force doesn’t care about symmetries. Not at all. That being said, there is an obvious difference between the asymmetries mentioned above, and the asymmetry involved in W bosons having mass and other bosons not having mass. That’s true. Especially because now we have that Higgs field to explain why W bosons have mass – and not only W bosons but also the matter particles (i.e. the three generations of leptons and quarks discussed above). The diagram shows what interacts with what.

But so the Higgs field doesÂ notÂ interact with photons and gluons. Why? Well… I am not sure. Let me copy the Wikipedia explanation: “The Higgs field consists of four components, two neutral ones and two charged component fields. Both of the charged components and one of the neutral fields are Goldstone bosons, which act as the longitudinal third-polarization components of the massive W+, Wâ and Z bosons. The quantum of the remaining neutral component corresponds to (and is theoretically realized as) the massive Higgs boson.”

Huh? […]Â This ‘answer’ probably doesn’t answer your question. What I understand from the explanation above, is that the Higgs field only interacts with W bosons because its (theoretical) structure is such that it only interacts with W bosons. Now, you’ll remember Feynman’s oft-quoted criticism of string theory:Â I donât like that for anything that disagrees with an experiment, they cook up an explanationâa fix-up to say.”Â Is the Higgs theory such cooked-up explanation? No. ThatÂ kind of criticism would not apply here, in light of the fact that – some 50 yearsÂ afterÂ the theory – there is (some) experimental confirmation at least !

But you’ll admit itÂ doesÂ all look ‘somewhat ugly.’ However, while that’s a ‘loose end’ of the Standard Model, it’s not a fundamental defect or so. The argument is more about aesthetics, but then different people have different views on aesthetics – especially when it comes to mathematical attractiveness or unattractiveness.

So… NoÂ realÂ loose end here I’d say.

Gravity

The other ‘loose end’ that Feynman mentions in his 1985 summary is obviously still very relevant today (much more than his worries about the weak force I’d say). It is the lack of a quantum theory of gravity. There is none. Of course, the obvious question is: why would we need one? We’ve got Einstein’s theory, don’t we? What’s wrong with it?

The short answer to the last question is: nothing’s wrong with it – on the contrary ! It’s just that it is – well… – classical physics. No uncertainty. As such, the formalism of quantum field theory cannot be applied to gravity. That’s it. What’s Feynman’s take on this? [Sorry I refer to him all the time, but I made it clear in the introduction of this post that I would be discussing ‘his’ loose ends indeed.] Well… He makes two points – a practical one and a theoretical one:

1. “Because the gravitation force is so much weaker than any of the other interactions, it is impossible at the present time to make any experiment that is sufficiently delicate to measure any effect that requires the precision of a quantum theory to explain it.”

Feynman is surely right about gravity being ‘so much weaker’. Indeed, you should note that, at a scale of 10â13Â cm (that’s the picometer scale – so that’s the relevant scale indeed at the sub-atomic level), the coupling constants compare as follows: if the coupling constant of the strong force is 1, the coupling constant of the electromagnetic force is approximately 1/137, so that’s a factor of 10â2Â approximately. The strength of the weak force asÂ measured by the coupling constant would be smaller with a factorÂ 10â13Â (so that’s 1/10000000000000 smaller). Incredibly small, but so we do have a quantum field theory for the weak force ! However, the coupling constant for the gravitational force involves a factor 10â38. Let’s face it: this isÂ unimaginablyÂ small.

However, Feynman wrote this in 1985 (i.e. thirtyÂ years ago) andÂ scientists wouldn’t be scientists if they would not at least tryÂ to set up some kind of experiment. So there it is: LIGO. Let me quote Wikipedia on it:

LIGO,Â which stands for theÂ Laser Interferometer Gravitational-Wave Observatory, is a large-scale physics experiment aiming to directly detect gravitation waves. […]Â At the cost of \$365 million (in 2002 USD), it is the largest and most ambitious project ever funded by the NSF. Observations at LIGO began in 2002 and ended in 2010; no unambiguous detections of gravitational waves have been reported. The original detectors were disassembled and are currently being replaced by improved versions known as “Advanced LIGO”.

So, let’s see what comes out of that. I won’t put my money on it just yet. đ Let’s go to the theoretical problem now.

2. “Even though there is no way to test them, there are, nevertheless, quantum theories of gravity that involve ‘gravitons’ (which would appear under a new category of polarizations, called spin “2”) and other fundamental particles (some with spin 3/2). The best of these theories is not able to include the particles that we do find, and invents a lot of particles that we don’t find. [In addition] The quantum theories of gravity also have infinities in the terms with couplings [Feynman does not refer to a coupling constant but to a factor n appearing in the so-called propagator for an electron – don’t worry about it: just note it’s a problem with one of those constants actually being larger than one !], but the “dippy process” that is successful in getting rid of the infinities in quantum electrodynamics doesn’t get rid of them in gravitation. So not only have we no experiments with which to check a quantum theory of gravitation, we also have no reasonable theory.”

Phew !Â After reading that, you wouldn’t apply for a job at that LIGO facility, would you? That being said, the fact that there is a LIGO experiment would seem to undermine Feynman’s practicalÂ argument. But then is his theoreticalÂ criticism still relevant today? I am not an expert, but it would seem to be the case according to Wikipedia’s update on it:

“Although a quantum theory of gravity is needed in order to reconcile general relativity with the principles of quantum mechanics, difficulties arise when one attempts to apply the usual prescriptions of quantum field theory. From a technical point of view, the problem is that the theory one gets in this way is not renormalizable and therefore cannot be used to make meaningful physical predictions. As a result, theorists have taken up more radical approaches to the problem of quantum gravity, the most popular approaches being string theory and loop quantum gravity.”

Hmm… String theory and loop quantum gravity? That’s the stuff that Penrose is exploring. However, I’d suspect that for these (string theory and loop quantum gravity), Feynman’s criticism probably still rings true – to some extent at least:Â

âI donât like that theyâre not calculating anything. I donât like that they donât check their ideas. I donât like that for anything that disagrees with an experiment, they cook up an explanationâa fix-up to say, âWell, it might be true.â For example, the theory requires ten dimensions. Well, maybe thereâs a way of wrapping up six of the dimensions. Yes, thatâs all possible mathematically, but why not seven? When they write their equation, the equation should decide how many of these things get wrapped up, not the desire to agree with experiment. In other words, thereâs no reason whatsoever in superstring theory that it isnât eight out of the ten dimensions that get wrapped up and that the result is only two dimensions, which would be completely in disagreement with experience. So the fact that it might disagree with experience is very tenuous, it doesnât produce anything; it has to be excused most of the time. It doesnât look right.â

What to say by way of conclusion? Not sure. I thinkÂ my personal “researchÂ agenda” is reasonably simple: I just want to try to understand all of the above somewhat better and then, perhaps, I might be able to understand some of what Roger Penrose is writing. đ

# Light and matter

Pre-scriptum (dated 26 June 2020): This post does not seem to have suffered from the attack by the dark force. However, my views on the nature of light and matter have evolved as part of my explorations of a more realist (classical) explanation of quantum mechanics. If you are reading this, then you are probably looking for not-to-difficult reading. In that case, I would suggest you read my re-write of Feynman’s introductory lecture to QM. If you want something shorter, you can also read my paper on what I believe to be the true Principles of Physics.

Original post:

In my previous post, I discussed theÂ de BroglieÂ wave of a photon. It’s usually referred to as ‘the’ wave function (or the psi function) but, as I explained, for every psi – i.e. the position-space wave function Îš(x ,t) – there is also aÂ phiÂ – i.e. the momentum-space wave function ÎŠ(p, t).

In that post, I alsoÂ compared itÂ – without much formalism – to theÂ de BroglieÂ wave of ‘matter particles’. Indeed, in physics, we look at ‘stuff’ as being made of particles and, while the taxonomy of theÂ particle zoo of the Standard ModelÂ of physics is rather complicated, one ‘taxonomic’ principle stands out: particles are eitherÂ matter particles (known as fermions) or force carriers (known as bosons). It’s a strict separation: either/or. No split personalities.

A quick overview before we start…

Wikipedia’s overview of particles in the Standard Model (including the latest addition: the Higgs boson) illustrates this fundamental dichotomy in nature: we have the matter particles (quarks and leptons) on one side, and the bosons (i.e. the force carriers) on the other side.

Don’t be put off by my remark on theÂ particle zoo: it’s a term coined in the 1960s, when the situation was quite confusing indeed (like more than 400 ‘particles’). However, the picture is quite orderly now. In fact, the Standard ModelÂ put an end to the discovery of ‘new’ particles, and it’s been stable since the 1970s, as experiments confirmed the reality of quarks. Indeed, all resistance to Gell-Man’s quarks and his flavor and color concepts – which are just words to describe new types of ‘charge’ – similar to electric charge but with more variety), ended when experiments by Stanford’s Linear Accelerator Laboratory (SLAC) in November 1974Â confirmed the existence of the (second-generation and, hence, heavy and unstable)Â ‘charm’ quark (again, the names suggest some frivolity but it’s serious physical research).

As for the Higgs boson, its existence of the Higgs boson had also been predicted, since 1964 to be precise, but it took fifty years to confirm it experimentally because only something like theÂ Large Hadron ColliderÂ could produce the required energy to find it in these particle smashing experiments – a rather crude way of analyzing matter, you may think, but so be it. [In case you harbor doubts on the Higgs particle, please note that, while CERN is the first to admit further confirmation is needed, the Nobel Prize CommitteeÂ apparently found the evidence ‘evidence enough’ to finally award Higgs and others a Nobel Prize for their ‘discovery’ fifty years ago – and, as you know, the Nobel Prize committee members are usually rather conservative in their judgment. So you would have to come up with a rather complex conspiracy theory to denyÂ its existence.]

Also note that theÂ particle zooÂ is actually less complicated than it looks at first sight: the (composite) particles that areÂ stableÂ in our world – this world –Â consist of three quarks only: a proton consists of two up quarks and one down quark and, hence, is written as uud., and a neutron is two down quarks and one up quark: udd. Hence, for all practical purposes (i.e. for our discussion how light interacts with matter), only the so-called first generation of matter-particles – so that’s the first column in the overview above – are relevant.

All the particles in the second and third column are unstable. That being said, they survive long enough – a muon disintegrates after 2.2 millionths of a second (on average) – to deserve the ‘particle’ title, as opposed to a ‘resonance’, whose lifetime can be as short as a billionth of a trillionth of a second – but we’ve gone through these numbers before and so I won’t repeat that here. Why do we need them? Well… We don’t, but they are a by-product of our world view (i.e. the Standard Model) and, for some reason, we find everything what this Standard Model says should exist, even if most of the stuff (all second- and third-generation matter particles, and all these resonances, vanish rather quickly – but so that also seems to be consistent with the model). [As for a possible fourth (or higher) generation, Feynman didn’t exclude it when he wrote his 1985 Lectures on quantum electrodynamics, but, checking on Wikipedia, I find the following: “According to the results of the statistical analysis by researchers from CERN and the Humboldt University of Berlin,Â the existence of further fermions can be excluded with a probability of 99.99999% (5.3 sigma).” If you want to know why… Well… Read the rest of the Wikipedia article. It’s got to do with the Higgs particle.]

As for the (first-generation) neutrino in the table – the only one which you may not be familiar with – these are very spooky things but – I don’t want to scare you – relatively high-energy neutrinos are going through your and myÂ my body,Â right now and here,Â at a rate of some hundred trillion per second. They are produced by stars (stars are huge nuclear fusion reactors, remember?), and also as a by-product of these high-energy collisions in particle accelerators of course. But they are very hard to detect: the first trace of their existence was found in 1956 only – 26 years after their existence had been postulated: the fact that Wolfgang Pauli proposed their existence in 1930 to explain how beta decay could conserve energy, momentum and spin (angular momentum) demonstrates not only the genius but also the confidence of these early theoretical quantum physicists. Most neutrinos passing through Earth are produced by our Sun. Now they are being analyzed more routinely. The largestÂ neutrino detectorÂ on Earth is calledÂ IceCube. It sits on the South Pole â orÂ underÂ it, as itâs suspendedÂ underÂ the Antarctic ice, and it regularly capturesÂ high-energy neutrinos in the range of 1 to 10 TeV.Â

Let me – to conclude this introduction – just quickly list and explain the bosonsÂ (i.e the force carriers) in the table above:

1. Of all of the bosons, the photonÂ (i.e. the topic of this post), is the most straightforward: there is onlyÂ typeÂ of photon, even if it comes in different possible states of polarization.

[…]

I should probably do a quick note on polarization here – even if all of the stuff that follows will make abstraction of it. Indeed, the discussion on photons that follows (largely adapted from Feynman’s 1985 Lectures on Quantum Electrodynamics) assumes that there is no such thing as polarization – because it would make everything even more complicated.Â The concept of polarization (linear, circular or elliptical) has a direct physical interpretation in classical mechanics (i.e. light as an electromagnetic wave). In quantum mechanics, however, polarization becomes a so-calledÂ qubit (quantum bit): leaving aside so-called virtual photons (these are short-range disturbances going between a proton and an electron in an atom – effectively mediating the electromagnetic force between them), the property of polarization comes in two basis states (0 and 1, or left and right), but these two basis states can be superposed. In ket notation: if ÂŠ0âȘ and ÂŠ1âȘ are the basis states, then any linear combination Î±Â·ÂŠ0âȘ + ĂÂ·ÂŠ1âȘ is also a valid state providedâÎ±â2 +Â âÎČâ2Â = 1, in line with the need to get probabilities that add up to one.

In case you wonder why I am introducing these kets, there is no reason for it, except that I will be introducing some other tools in this post – such as Feynman diagrams – and so that’s all. In order to wrap this up, I need to note that ketsÂ are used in conjunction withÂ bras. So we have a bra-ket notation: the ket gives the starting condition, and the bra – denoted as â©Â ÂŠ – gives the final condition. They are combined in statements such as â© particle arrives at xÂŠparticle leaves from sâȘ or – in short – â© xÂŠsâȘ and, while x and s would have some real-number value, â©Â xÂŠsâȘ would denote the (complex-valued) probability amplitude associated wit the event consisting of these two conditions (i.e the starting and final condition).

But don’t worry about it. This digression is just what it is: a digression. Oh… Just make a mental note that the so-calledÂ virtual photons (the mediators that are supposed to keep the electron in touch with the proton) have four possible states of polarization – instead of two. They are related to the fourÂ directions of space (x, y and z) and time (t). đ

2. Gluons, the exchange particles for the strong force, are more complicated: they come in eight so-calledÂ colors.Â In practice, one should think of these colors as different charges, but so we have more elementary chargesÂ in this caseÂ than just plus or minus one (Â±1) – as we have for the electric charge. So it’s just another type ofÂ qubitÂ in quantum mechanics.

[Note that the so-called elementary Â±1 values for electric charge areÂ notÂ really elementary: it’s â1/3 (for the downÂ quark,Â and for the second- and third-generation strange and bottom quarks as well) and +2/3 (for the up quark as well as for the second- and third-generation charm and top quarks). That being said, electric charge takes two values only, and the Â±1 value is easily found from a linear combination of the â1/3 and +2/3 values.]

3. Z and W bosons carry the so-called weak force, aka as Fermiâs interaction: they explain how one type of quark can change into another, thereby explaining phenomena such asÂ betaÂ decay. Beta decay explains why carbon-14 will, after a very long time (as compared to the ‘unstable’ particles mentioned above),Â spontaneously decay into nitrogen-14. Indeed, carbon-12 is the (very) stable isotope, whileÂ carbon-14 has a life-time of 5,730 Â± 40 years ‘only’Â  (so one can’t call carbon-12 ‘unstable’: perhaps ‘less stable’ will do)Â and, hence, measuring how much carbon-14 is left in some organic substance allows us to date it (that’s what (radio)carbon-dating is about). As for the name, aÂ betaÂ particle can refer to an electronÂ orÂ a positron, so we can haveÂ ÎČâÂ decay (e.g. the above-mentioned carbon-14 decay) as well as ÎČ+Â decay (e.g. magnesium-23 into sodium-23). There’s also alpha and gamma decay but that involves different things.Â

As you can see from the table, WÂ±Â and Z0Â bosons are veryÂ heavy (157,000 and 178,000 times heavier than a electron!), and WÂ± carry the (positive or negative) electric charge. So why don’t we see them? Well… They are so short-lived that we can only see a tiny decay width, just a very tiny little trace,Â so they resemble resonances in experiments. That’s also the reason why we see little or nothing of the weak force in real-life: the force-carrying particles mediating this force don’t get anywhere.

4. Finally, as mentioned above, theÂ Higgs particleÂ – and, hence, of the associated Higgs field â had been predicted since 1964 already but its existence was only (tentatively)Â experimentallyÂ confirmed last year. The Higgs field gives fermions, and also the WÂ and ZÂ bosons, mass (but not photons and gluons), and – as mentioned above – thatâs why the weak force has such short range as compared to the electromagnetic and strong forces. Note, however, that theÂ Higgs particle does actuallyÂ notÂ explain the gravitational force, so itâsÂ notÂ the (theoretical) graviton and there is no quantum field theory for the gravitational force as yet. JustÂ GoogleÂ it and you’ll quickly find out why: there’s theoretical as well as practical (experimental) reasons for that.

The Higgs field stands out from the other force fields because it’s a scalarÂ field (as opposed to a vector field). However, I have no idea how this so-called Higgs mechanismÂ (i.e. the interaction with matter particles (i.e. with the quarks and leptons, but not directly with neutrinos it would seem from the diagram below), with W and Z bosons, and with itself – but not with the massless photons and gluons) actually works. But then I still have aÂ veryÂ long way to go on thisÂ Road to Reality.

In any case…Â The topic of this post is to discuss light and its interaction with matter – not the weak or strong force, nor the Higgs field.

Let’s go for it.

Amplitudes, probabilities and observable properties

Being born a boson orÂ a fermionÂ makes a big difference. That being said, both fermions and bosons areÂ waviclesÂ described by a complex-valuedÂ psiÂ function, colloquially known as the wave function. To be precise, there will be several wave functions, and the square of their modulus (sorry for the jargon) will give you the probability of someÂ observable property having a value in some relevant range, usually denoted by Î. [I also explained (in my post on Bose and Fermi) how the rules forÂ combiningÂ amplitudes differ for bosons versus fermions, and how that explains why they are what they are: matter particles occupy space, while photons not onlyÂ canÂ but alsoÂ likeÂ to crowd together in, for example, a powerful laser beam. I’ll come back on that.]

For all practical purposes, relevant usually means ‘small enough to be meaningful’. For example, we may want to calculate the probability of detecting an electron in some tiny spacetime interval (Îx,Â Ît). [Again, ‘tiny’ in this context means small enough to be relevant: if we are looking at a hydrogen atom (whose size is a few nanometer), then Îx is likely to be a cube or a sphere with an edge or a radius of a few picometer only (a picometer is aÂ thousandthÂ of a nanometer, so it’s a millionth of a millionth of a meter); and, noting that the electron’s speed is approximately 2200 km per second… Well… I will letÂ youÂ calculate a relevant Ît. :-)]

If we want to do that, then we will need to square the modulus of the correspondingÂ wave function Îš(x, t). To be precise, we will have to do a summationÂ of all the values âÎš(x, t)â2Â over the interval and, because x and t are real (and, hence, continuous) numbers, that means doing some integral (because an integral is the continuous version of a sum).

But that’s only one example of an observable property: position. There are others. For example, we may not be interested in the particle’sÂ exact position but only in its momentum or energy. Well, we have another wave function for that: the momentum wave function ÎŠ(x ,t). In fact, if you looked at my previous posts, you’ll remember the two are related because they are conjugate variables: Fourier transforms duals of one another. A less formal way of expressing that is to refer to the uncertainty principle. But this is not the time to repeat things.

The bottom line is that all particles travel through spacetime with a backpack full of complex-valued wave functions. We don’t know who and where these particles are exactly, and so weÂ can’t talk to them – but we can e-mail God and He’ll send us the wave function that we need to calculate some probability we are interested in because we want to check – in all kinds of experiments designed to fool them – if it matches with reality.

As mentioned above, I highlighted the main difference between bosons and fermions in myÂ Bose and FermiÂ post, so I won’t repeat that here. Just note that, when it comes to working with those probabilityÂ amplitudesÂ (that’s just another word for these psi and phiÂ functions), it makes aÂ hugeÂ difference: fermions and bosons interactÂ veryÂ differently. Bosons are party particles: they like to crowd and will always welcome an extra one. Fermions, on the other hand, will exclude each other: that’s why there’s something referred to as the Fermi exclusion principleÂ in quantum mechanics. That’s why fermions make matter (matter needs space) and bosons are force carriers (they’ll just call friends to help when the load gets heavier).

Light versus matter: Quantum Electrodynamics

OK. Let’s get down to business. This post is about light, or about light-matterÂ interaction. Indeed, in my previous post (on Light), IÂ promised to say something about the amplitude of a photon to go from point A to B (because – as I wrote in my previous post – that’s more ‘relevant’, when it comes to explaining stuff, than the amplitude of a photon to actually beÂ at point x at time t), and so that’s what I will do now.

In his 1985 Lectures on Quantum ElectrodynamicsÂ (which are lectures for the lay audience), Feynman writes the amplitude of a photon to go from point A to B as P(A to B) – and the P stands for photon obviously, not for probability. [I am tired of repeating that you need to square the modulus of an amplitude to get a probability but – here you are – I have said it once more.] That’s in line with the other fundamental wave function in quantum electrodynamics (QED): the amplitude of an electron to go from A to B, which is written as E(A to B). [You got it: E just stands for electron, not for our electric field vector.]

I also talked about the thirdÂ fundamental amplitude in my previous post: the amplitude of an electron to absorb or emit a photon. So let’s have a look at these three. As Feynman says: “âOut of these three amplitudes, we can make theÂ whole world, aside from what goes on in nuclei, and gravitation, as always!âÂ

Well… Thank you, Mr Feynman:Â I’ve always wanted to understand the World (especially if you made it).

The photon-electron coupling constant j

Let’s start with the last of those three amplitudes (or wave functions): the amplitude of an electron to absorbÂ orÂ emit a photon. Indeed, absorbing or emitting makes no difference: we have the same complex number for both. It’s a constant – denoted by jÂ (for junctionÂ number)Â – equal to â0.1 (a bit less actually but it’s good enough as an approximation in the context of this blog).

Huh?Â Minus 0.1?Â That’s not a complex number, is it? It is. Real numbers are complex numbers too: â0.1 is 0.1eiÏÂ in polar coordinates. As Feynman puts it: it’s “a shrink to about one-tenth, and half a turn.”Â The ‘shrink’ is the 0.1 magnitude of this vector (or arrow), and the ‘half-turn’ is the angle of Ï (i.e. 180 degrees). He obviously refers to multiplying (no adding here)Â j with other amplitudes, e.g. P(A, C) and E(B, C) if the coupling is to happen at or near C. And, as you’ll remember, multiplying complex numbers amounts to adding their phases, and multiplying their modulus (so that’s adding the angles and multiplying lengths).

Let’s introduce a Feynman diagram at this point – drawn by Feynman himself – which shows three possible ways of two electrons exchanging a photon. We actually have two couplings here, and so the combined amplitude will involve two j‘s. In fact, if we label the starting point of the two lines representing our electrons as 1 and 2 respectively, and their end points as 3 and 4, then the amplitude for these events will be given by:

E(1 to 5)Â·jÂ·E(5 to 3)Â·E(2 to 6)Â·jÂ·E(6 to 3)

Â As for how thatÂ jÂ factor works,Â please do read the caption of the illustration below: the same jÂ describes both emission as well as absorption. It’s just that we haveÂ bothÂ an emissionÂ as well asÂ an as absorption here, so we have aÂ j2 factor here, which is less than 0.1Â·0.1 = 0.01. At this point, it’s worth noting that it’s obvious that the amplitudes we’re talking about here – i.e. for one possible wayÂ of an exchange like the one below happening – are very tiny. They only become significant when we add many of these amplitudes, which – as explained below – is what has to happen: one has to consider all possible paths, calculate the amplitudes for them (through multiplication), and then add all these amplitudes, to then – finally – square the modulus of the combined ‘arrow’ (or amplitude) to get some probability of something actually happening. [Again, that’s the best we can do: calculate probabilities that correspond to experimentally measured occurrences. We cannot predictÂ anything in the classical sense of the word.]

A Feynman diagram is not just some sketchy drawing. For example, we have to care about scales: the distance and time units are equivalent (so distance would be measured in light-seconds or, else, time would be measured in units equivalent to the time needed for light to travel one meter). Hence, particles traveling through time (and space) – from the bottom of the graph to the top – will usuallyÂ notÂ Â be traveling at an angle of more than 45 degrees (as measured from the time axis) but, from the graph above, it is clear that photons do. [Note that electrons moving through spacetime are represented by plain straight lines, while photons are represented by wavy lines. It’s just a matter of convention.]

More importantly, a Feynman diagram is a pictorial device showing what needs to be calculated and how. Indeed, with all the complexities involved, it is easy to lose track of what should be added and what should be multiplied, especially when it comes to much more complicated situations like the one described above (e.g. making sense of a scattering event). So, while the coupling constant j (aka as the ‘charge’ of a particle – but it’s obviously not the electric charge) is just a number, calculating an actual E(A to B) amplitudes is not easy – not only because there are many different possible routes (paths) but because (almost) anything can happen. Let’s have a closer look at it.

E(A to B)

As Feynman explains in his 1985 QED Lectures: “E(A to B) can be represented as a giant sum of a lot of different ways an electron can go from point A to B in spacetime: the electron can take a ‘one-hop flight’, going directly from point A to B; it could take a ‘two-hop flight’, stopping at an intermediate point C; it could take a ‘three-hop flight’ stopping at points D and E, and so on.”

Fortunately, the calculation re-uses known values: the amplitude for each ‘hop’ – from C to D, for example – is P(F to G) – so that’s the amplitude of aÂ photon (!)Â to go from F to G – even if we are talking an electron here. But there’s a difference: we alsoÂ have to multiply the amplitudes for each ‘hop’ with the amplitude for each ‘stop’, and that’s represented by another number – not j but n2. So we have an infinite series of terms for E(A to B): P(A to B) + P(A to C)Â·n2Â·P(C to B)Â + P(A to D)Â·n2Â·P(D to E)Â·n2Â·P(E to B) + …Â for all possible intermediate points C, D, E, and so on, as per the illustration below.

You’ll immediately ask: what’s the value of n? It’s quite important to know it, because we want to know how big these n2,Â n4Â etcetera terms are. I’ll be honest: I have not come to terms with that yet. According to Feynman (QED, p. 125), it is the ‘rest mass’ of an ‘ideal’ electron: an ‘ideal’ electron is an electron that doesn’t know Feynman’s amplitude theory and just goes from point to point in spacetime using only the direct path. đ Hence, it’s not a probability amplitude like j: a properÂ probability amplitude will always have a modulus less than 1, and so when we see exponential terms like j2, j4,… we know we should not be all that worried – because these sort of vanish (go to zero) for sufficiently large exponents. For E(A to B), we do notÂ have such vanishing terms. IÂ will not dwell on this rightÂ here, but I promise to discuss it in the Post ScriptumÂ of this post. The frightening possibility is that n might beÂ a number larger than one.

[As we’re freewheeling a bit anyway here, just a quick note on conventions: I should not be writingÂ jÂ in bold-face, because it’s a (complex- or real-valued) number and symbols representing numbers are usually notÂ written in bold-face: vectors are written in bold-face. So, while you can look at a complex number as a vector, well… It’s just one of these inconsistencies I guess. The problem with using bold-face letters to represent complex numbers (like amplitudes) is that they suggest that the ‘dot’ in a product (e.g. jÂ·j) is an actual dot projectÂ (aka as a scalar product or an innerÂ product) of two vectors. That’s not the case. We’re multiplying complex numbers here, and so we’re just using the standard definition of a product of complex numbers. This subtlety probably explains why Feynman prefers to write the above product as P(A to B) + P(A to C)*n2*P(C to B)Â + P(A to D)*n2*P(D to E)*n2*P(E to B) + … But then I find that using that asterisk to represent multiplication is a bit funny (although it’s a pretty common thing in complex math) and so I am not using it. Just be aware that a dot in a product may not always mean the same type of multiplication: multiplying complex numbers and multiplying vectors is not the same. […] And I won’t write j in bold-face anymore.]

P(A to B)

Regardless of the value for n,Â it’s obvious we need a functional form for P(A to B), because that’s the other thing (other than n) that we need to calculate E(A to B). So what’s the amplitude of a photon to go from point A to B?

Well… The function describing P(A to B) is obviously some wave function – so that’s a complex-valued function of x and t. It’s referred to as a (Feynman) propagator: a propagator functionÂ gives the probability amplitude for a particle to travel from one place to another in a given time, or to travel with a certain energy and momentum. [So our function for E(A to B) will be a propagator as well.] You can check out the details on it on Wikipedia. Indeed, IÂ couldÂ insert the formula here, but believe me if I say it would only confuse you. The points to note is that:

1. The propagator is also derived from the wave equation describing the system, so that’s some kind of differential equation which incorporates the relevant rules and constraints that apply to the system. For electrons, that’s the SchrĂ¶dinger equation I presented in my previous post. For photons… Well… As I mentioned in my previous post, there is ‘something similar’ for photons – there mustÂ be –Â but I have not seen anything that’s equally ‘simple’ as the SchrĂ¶dinger equation for photons. [I haveÂ GoogledÂ a bit but it’s obvious we’re talking pretty advanced quantum mechanics here – so it’s not the QM-101 course that I am currently trying to make sense of.]Â
2. The most important thing (in this context at least) is that the key variable in this propagator (i.e. the Feynman propagator for the photon) is I: thatÂ spacetime interval which I mentioned in my previous post already:

I = Îr2Â â Ît2Â = Â (z2– z1)2Â +Â (y2– y1)2Â +Â (x2– x1)2Â âÂ (t2–Â t1)2

In this equation, we need to measure the time and spatial distance between two points in spacetime in equivalent unitsÂ (these ‘points’ are usually referred to as four-vectors), so weâd use light-seconds for the unit of distance or, for the unit of time, the time it takes for light to travel one meter.Â [If we don’t want to transform time or distance scales, then we have to writeÂ IÂ as I =Â c2Ît2Â âÂ Îr2.] Now, there are three types of intervals:

1. For time-like intervals, we have a negativeÂ value for I, so Ît2Â > Îr2. For two events separated by a time-like interval, enough time passes between them so there could be a causeâeffect relationship between the two events. In a Feynman diagram, the angle between the time axis and the line between the two events will be less than 45 degrees from the vertical axis. The traveling electrons in the Feynman diagrams above are an example.
2. For space-like intervals, we have a positive value for I, so Ît2Â < Îr2. Events separated by space-like intervals cannot possibly be causally connected. The photons traveling between point 5 and 6 in the first Feynman diagram are an example, but then photonsÂ do have amplitudes to travel faster than light.
3. Finally, for light-like intervals, I = 0, orÂ Ît2 = Îr2. The points connected by the 45-degree lines in the illustration below (which Feynman uses to introduce his Feynman diagrams) are an example of points connected by light-like intervals.

[Note that we are using the so-called space-like convention (+++â) here for I. There’s also a time-like convention, i.e. withÂ +âââ as signs: I =Â Ît2Â âÂ Îr2Â so just check when you would consult other sources on this (which I recommend) and if you’d feel I am not getting the signs right.]

Now, what’s the relevance of this? To calculate P(A to B), we have toÂ addÂ the amplitudesÂ for all possible pathsÂ that the photon can take, and not in space, but in spacetime. So we should add all these vectors (or ‘arrows’ as Feynman calls them) – an infinite number of them really. In the meanwhile, you know it amounts to adding complex numbers, and that infinite sums are done by doing integrals, but let’s take a step back: how are vectors added?

Well…That’s easy, you’ll say… It’s the parallelogram rule… Well… Yes. And no. Let me take a step back here to show how adding a whole range of similar amplitudes works.

The illustration below shows a bunch of photons – real or imagined – from a source above a water surface (the sun for example), all taking different paths to arrive at a detector under the water (let’s say some fish looking at the sky from under the water). In this case, we make abstraction of all the photons leaving at different timesÂ and so we only look at a bunch that’s leaving at the same point in time. In other words, theirÂ stopwatches will be synchronized (i.e. there is no phase shift termÂ in the phase of their wave function) – let’s say at 12 o’clock when they leave the source. [If you think this simplification is not acceptable, well… Think again.]

When these stopwatches hit the retina of our poor fish’s eye (I feel we should put a detector there, instead of a fish), they will stop, and the hand of each stopwatch represents an amplitude: it has a modulus (its length) – which is assumed to be the same because all paths are equally likely (this is one of the first principles of QED) – but their direction is very different. However, by now we are quite familiar with these operations: we add all the ‘arrows’ indeed (or vectors or amplitudes or complex numbers or whatever you want to call them) and get one big final arrow, shown at the bottom – just above the caption. Look at itÂ very carefully.

If you look at the so-calledÂ contribution made by each of the individual arrows, you can see that it’s the arrows associated with the path of least time and the paths immediately left and right of it that make the biggest contributionÂ to the final arrow. Why? Because these stopwatches arrive around the same time and, hence, their hands point more or less in the same direction. It doesn’t matter what direction – as long as it’s more or less the same.

[As for the calculation of the path of least time, that has to do with the fact that light is slowed down in water. Feynman shows why in his 1985 Lectures on QED, but I cannot possibly copy the whole book here ! The principle is illustrated below.] Â

So, where are we? This digressions go on and on, don’t they? Let’s go back to the main story: we want to calculate P(A to B), remember?

As mentioned above, one of the first principlesÂ in QED is that all paths – in spacetime – are equally likely. So we need to add amplitudes for every possible path in spacetime using that Feynman propagator function. You can imagine that will be some kind of integral which you’ll never want to solve. Fortunately, Feynman’s disciples have done that for you already. The results is quite predictable: the grand result is that light has a tendency to travel in straight lines and at the speed of light.

WHAT!? Did Feynman get a Nobel prize for trivial stuff like that?

Yes. The math involved in adding amplitudes over all possible paths not only in space but also in time uses the so-called path integral formulation of quantum mechanics and so that’s got Feynman’s signature on it, and that’s the main reason why he got this award – together with Julian Schwinger and Sin-Itiro Tomonaga: both much less well known than Feynman, but so they shared the burden. Don’t complain about it. Just take a look at the ‘mechanics’ of it.

We already mentioned that the propagator has the spacetime interval IÂ in its denominator. Now, the way it works is that, for values of IÂ equal or close to zero, so the paths that are associated with light-like intervals, our propagator function will yield large contributions in the ‘same’ direction (wherever that direction is), but for the spacetime intervals that are very much time- or space-like, the magnitude of our amplitude will be smaller and – worse – our arrow will point in the ‘wrong’ direction. In short, the arrows associated with the time- and space-like intervals don’t add up to much, especially over longer distances. [When distances are short, there are (relatively) few arrows to add, and so the probability distribution will be flatter: in short, the likelihood of having the actualÂ photon travel faster or slower than speed is higher.]

Conclusion

Does this make sense? I am not sure, but I did what I promised to do. I told you how P(A to B) gets calculated; and from the formula for E(A to B), it is obvious that we can then also calculate E(A to B)Â provided we have a value for n. However, that value n is determined experimentally, just like the value of j, in order to ensure this amplitude theory yields probabilities that match the probabilities we observe in all kinds of crazy experiments that try to prove or disprove the theory; and then we can use these three amplitude formulas “to make the whole world”, as Feynman calls it, except the stuff that goes on inside of nucleiÂ (because that’s the domain of the weak and strong nuclear force) and gravitation, for which we have aÂ law (Newton’s Law)Â but no real ‘explanation’. [Now, you may wonder if this QED explanation of light is really all that good, but Mr Feynman thinks it is, and so I have no reason to doubt that – especially because there’s surely not anything more convincing lying around as far as I know.]

So what remains to be told? Lots of things, even within the realm of expertise of quantum electrodynamics. Indeed, Feynman applies the basics as described above to a number ofÂ real-lifeÂ phenomena – quite interesting, all of it ! – but, once again, it’s not my goal to copy all of his LecturesÂ here. [I am only hoping to offer some good summaries of key points in some attempt to convince myself that I am getting some of it at least.] And then there is the strong force, and the weak force, and the Higgs field, and so and so on. But that’s all very strange and new territory which I haven’t even started to explore. I’ll keep you posted as I am making my way towards it.

Post scriptum: On the values of j and n

In this post, I promised I would write something about how we can find j and nÂ because I realize it would just amount to copy three of four pages out of that book I mentioned above, and which inspired most of this post. Let me just say something more about that remarkable book, and then quote a few lines on what the author of that book – the great Mr Feynman ! – thinks of the math behind calculating these two constants (the coupling constantÂ j, and the ‘rest mass’Â of an ‘ideal’ electron). Now, before I do that, I should repeat that he actually invented that math (it makes use of a mathematical approximation method called perturbation theory) and that he gotÂ a Nobel Prize for it.

First, about the book. Feynman’s 1985 Lectures on Quantum ElectrodynamicsÂ are not like his 1965 Lectures on Physics.Â The Lectures on PhysicsÂ are proper courses for undergraduate and even graduate students in physics. This little 1985 book on QED is justÂ a series of four lectures for a lay audience, conceived in honor of Alix G. Mautner. She was a friend of Mr Feynman’s who died a few years before he gave and wrote these ‘lectures’ on QED. She had a degree in English literature and would ask Mr Feynman regularly to explain quantum mechanics and quantum electrodynamics in a way she would understand. While they had known each other for about 22 years, he had apparently never taken enough time to do so, as he writes in his Introduction to these Alix G. Mautner MemorialÂ Lectures: “So here are the lectures I really [should have] prepared for Alix, but unfortunately I can’t tell them to her directly, now.”

The great Richard Phillips Feynman himself died only three years later, in February 1988 – not of one but twoÂ rare forms ofÂ cancer. He was only 69 years old when he died. I don’t know if he was aware of the cancer(s) that would kill him, but I find his fourth and last lecture in the book, Loose Ends, just fascinating. Here we have aÂ brilliant mindÂ deprecating the math that earned him a Nobel Prize and without which the Standard Model would be unintelligible. I won’t try to paraphrase him. Let me just quote him. [If you want to check the quotes, the relevant pages are page 125 to 131):

The math behind calculating these constants] is a âdippy processâ and âhaving to resort to such hocus-pocus has prevented us from proving that the theory of quantum electrodynamics is mathematically self-consistentâ. He adds: âItâs surprising that the theory still hasnât been proved self-consistent one way or the other by now; I suspect that renormalization [“the shell game that we play to find n and j” as he calls it] Â is not mathematically legitimate.â […] Now,Â Mr Feynman writes this about quantum electrodynamics, not aboutÂ âthe rest of physicsâÂ (and so thatâs quantum chromodynamics (QCD) â the theory of the strong interactions â and quantum flavordynamics (QFD) â the theory of weak interactions) which, he adds,Â âhas not been checked anywhere near as well as electrodynamics.âÂ

Thatâs a pretty damning statement, isnât it? In one of my other posts (see:Â The End of the Road to Reality?), I explore these comments a bit. However, I have to admit I feel I really need to get back to math in order to appreciate these remarks. I’ve written way too much about physics anyway now (as opposed to the my first dozen of posts – which were much more math-oriented). So I’ll just have a look at some more stuff indeed (such as perturbation theory), and then I’ll get back blogging.Â Indeed, I’ve written like 20 posts or so in a few months only – so I guess I should shut up for while now !

In the meanwhile, you’re more than welcome to comment of course !Â