An introduction to virtual particles

We are going to¬†venture beyond quantum mechanics as it is usually understood – covering electromagnetic interactions only. Indeed, all of my posts so far – a bit less than 200, I think ūüôā – were all centered around electromagnetic interactions – with the model of the hydrogen atom as our most precious gem, so to speak.

In this post, we’ll be talking the strong force – perhaps not for the first time but surely for the first time at this level of detail. It’s an entirely different world – as I mentioned in one of my very first posts in this blog. Let me quote what I wrote there:

“The math describing the ‘reality’ of electrons and photons (i.e. quantum mechanics and quantum electrodynamics), as complicated as it is, becomes even more complicated ‚Äď and, important to note, also much less accurate¬†‚Äď when it is used to try to describe the behavior of ¬†quarks. Quantum chromodynamics (QCD) is a different world. […]¬†Of course, that should not surprise us, because we’re talking very different order of magnitudes here: femtometers (10‚Äď15 m), in the case of electrons, as opposed to attometers (10‚Äď18 m)¬†or even zeptometers (10‚Äď21¬†m) when we’re talking quarks.”

In fact, the femtometer scale is used to measure the radius¬†of both protons as well as electrons and, hence, is much smaller than the atomic scale, which is measured in nanometer (1 nm = 10‚ąí9¬†m). The so-called Bohr radius for example, which is a measure for the size of an atom, is measured in nanometer indeed, so that’s a scale that is a¬†million¬†times larger than the femtometer scale. This¬†gap¬†in the scale effectively separates entirely different worlds. In fact, the gap is probably as large a gap as the gap between our macroscopic world and the strange reality of quantum mechanics. What happens at the femtometer scale,¬†really?

The honest answer is: we don’t know, but we do have models¬†to describe what happens. Moreover, for want of better models, physicists sort of believe these models are credible. To be precise, we assume there’s a force down there which we refer to as the¬†strong¬†force. In addition, there’s also a weak force. Now, you probably know these forces are modeled as¬†interactions¬†involving an¬†exchange¬†of¬†virtual¬†particles. This may be related to what Aitchison and Hey refer to as the physicist’s “distaste for action-at-a-distance.” To put it simply: if one particle – through some force – influences some other particle, then something must be going on between the two of them.

Of course, now you’ll say that something is¬†effectively going on: there’s the electromagnetic field, right? Yes. But what’s the field? You’ll say: waves. But then you know electromagnetic waves also have a particle aspect. So we’re stuck with this weird theoretical framework: the conceptual distinction between particles and forces, or between particle and field, are not so clear. So that’s what the more advanced theories we’ll be looking at – like quantum field theory – try to bring together.

Note that we’ve been using a lot of confusing and/or ambiguous terms here: according to at least one leading physicist, for example, virtual particles should not be thought of as particles! But we’re putting the cart before the horse here. Let’s go step by step. To better understand the ‘mechanics’ of how the strong and weak interactions are being modeled in physics, most textbooks – including Aitchison and Hey, which we’ll follow here – start by explaining the original ideas as developed by the Japanese physicist Hideki Yukawa, who received a Nobel Prize for his work in 1949.

So what is it all about? As said, the ideas¬†– or the¬†model¬†as such, so to speak – are more important than Yukawa’s original application, which was to model the force between a proton and a neutron. Indeed, we now explain such force as a force between quarks, and the force carrier is the gluon, which carries the so-called¬†color¬†charge. To be precise, the force between protons and neutrons – i.e. the so-called nuclear¬†force – is¬†now¬†considered to be a rather minor¬†residual force: it’s just what’s left of the actual¬†strong force that binds quarks together. The Wikipedia article on this¬†has some¬†good text and¬†a really nice animation on this. But… Well… Again, note that we are only interested in the¬†model right now. So how does that look like?

First, we’ve got the equivalent of the electric charge: the nucleon is supposed to have some ‘strong’ charge, which we’ll write as gs. Now you know the formulas for the¬†potential¬†energy – because of the gravitational force – between two masses, or the¬†potential¬†energy between two charges – because of the electrostatic force. Let me jot them down once again:

  1. U(r) =¬†‚ÄďG¬∑M¬∑m/r
  2. U(r) = (1/4ŌÄőĶ0)¬∑q1¬∑q2/r

The two formulas are exactly the same. They both assume U = 0 for¬†r¬†‚Üí ‚ąě. Therefore, U(r) is always negative. [Just think of q1¬†and q2¬†as opposite charges, so the minus sign is not explicit – but it is also there!] We know that¬†U(r)¬†curve will look like the one below: some work (force times distance) is needed to move the two charges some distance¬†away from each other – from point 1 to point 2, for example. [The distance r is x here – but you got that, right?]potential energy

Now, physics textbooks – or other articles you might find, like on Wikipedia – will sometimes mention that the strong force is non-linear, but that’s very confusing because… Well… The electromagnetic force – or the gravitational force – aren’t linear either: their strength is inversely proportional to the square¬†of the distance and – as you can see from the formulas for the potential energy – that 1/r factor isn’t linear¬†either. So that isn’t very helpful. In order to further the discussion, I should now write down Yukawa’s¬†hypothetical¬†formula for the potential energy between a neutron and a proton, which we’ll refer to, logically, as the n-p potential:n-p potentialThe ‚ąígs2¬†factor is, obviously, the equivalent of the q1¬∑q2¬†product: think of the proton and the neutron having equal but opposite ‘strong’ charges. The 1/4ŌÄ factor reminds us of the Coulomb constant:¬†ke¬†= 1/4ŌÄőĶ0. Note this constant ensures the physical dimensions of both sides of the equation make sense: the dimension of őĶ0¬†is N¬∑m2/C2, so U(r) is – as we’d expect – expressed in newton¬∑meter, or¬†joule. We’ll leave the question of the units for gs¬†open – for the time being, that is. [As for the 1/4ŌÄ factor, I am not sure why Yukawa put it there. My best guess is that he wanted to remind us some constant should be there to ensure the units come out alright.]

So, when everything is said and done, the big new thing is the e‚ąír/a/r¬†factor, which replaces the usual 1/r dependency on distance. Needless to say, e is Euler’s number here –¬†not¬†the electric charge. The two green curves below show what the e‚ąír/a¬†factor does to the classical 1/r function for a¬†= 1 and¬†a¬†= 0.1 respectively: smaller values for¬†a¬†ensure the curve approaches zero more rapidly. In fact, for¬†a¬†= 1,¬†e‚ąír/a/r¬†is equal to 0.368 for¬†r¬†= 1, and remains significant for values r¬†that are greater than 1 too.¬†In contrast, for¬†a¬†= 0.1, e‚ąír/a/r¬†is equal to 0.004579 (more or less, that is) for r¬†= 4 and rapidly goes to zero for all values greater than that.

graph 1graph 2Aitchison and Hey call¬†a, therefore, a¬†range parameter: it effectively defines the¬†range¬†in which the n-p potential has a significant value: outside of the range, its value is, for all practical purposes, (close to) zero. Experimentally, this range was established as being more or less equal to r¬†‚ȧ 2 fm.¬†Needless to say, while this range factor may do its job, it’s obvious Yukawa’s formula for the n-p potential comes across as being somewhat random: what’s the theory behind? There’s none, really. It makes one think of the logistic function: the logistic function fits many statistical patterns, but it is (usually) not obvious why.

Next in Yukawa’s argument is the establishment of an equivalent, for the nuclear force, of the Poisson equation in electrostatics: using the¬†E = ‚Äď‚ąáő¶ formula, we can re-write Maxwell’s ‚ąá‚ÄĘE¬†= ŌĀ/őĶ0¬†equation (aka Gauss’ Law) as¬†‚ąá‚ÄĘE =¬†‚Äď‚ąá‚ÄĘ‚ąáő¶¬†= ‚Äď‚ąá2ő¶¬†‚áĒ¬†‚ąá2ő¶=¬†‚ÄďŌĀ/őĶ0¬†indeed. The divergence¬†operator¬†the¬†‚ąá‚ÄĘ operator gives us the¬†volume¬†density of the flux of E out of an infinitesimal volume around a given point. [You may want to check one of my post on this. The formula becomes somewhat more obvious if we re-write it as ‚ąá‚ÄĘE¬∑dV = ‚Äď(ŌĀ¬∑dV)/őĶ0: ‚ąá‚ÄĘE¬∑dV is then, quite simply, the flux of E out of the infinitesimally small volume dV, and the right-hand side of the equation says this is given by the product of the charge inside (ŌĀ¬∑dV) and 1/őĶ0, which accounts for the permittivity of the medium (which is the vacuum in this case).] Of course, you will also remember the ‚ąáő¶ notation: ‚ąá is just the gradient (or vector derivative) of the (scalar) potential ő¶, i.e. the electric (or electrostatic) potential in a space around that infinitesimally small volume with charge density ŌĀ. So… Well… The Poisson equation is probably not¬†so¬†obvious as it seems at first (again, check¬†my post on it¬†on it for more detail) and, yes, that ‚ąá‚ÄĘ operator – the divergence¬†operator – is a pretty impressive mathematical beast. However, I must assume you master this topic and move on. So… Well… I must now give you the equivalent of Poisson’s equation for the nuclear force. It’s written like this:Poisson nuclearWhat the heck? Relax. To derive this equation, we’d need to take a pretty complicated d√©tour, which we won’t do. [See Appendix G of Aitchison and Grey if you’d want the details.] Let me just point out the basics:

1. The Laplace operator (‚ąá2) is replaced by one that’s nearly the same: ‚ąá2¬†‚ąí 1/a2. And it operates on the same concept: a potential, which is a (scalar) function of the position r. Hence, U(r) is just the equivalent of¬†ő¶.

2. The right-hand side of the equation involves Dirac’s delta function. Now that’s a weird mathematical beast. Its definition seems to defy what I refer to as the ‘continuum assumption’ in math. ¬†I wrote a few things about it in one of my posts on Schr√∂dinger’s equation¬†– and I could give you its formula – but that won’t help you very much. It’s just a weird thing. As Aitchison and Grey¬†write, you should just think of the whole expression as a finite range analogue¬†of Poisson’s equation in electrostatics. So it’s only for extremely small¬†r¬†that the whole equation makes sense. Outside of the range defined by our range parameter¬†a, the whole equation just reduces to 0 = 0 – for all practical purposes, at least.

Now, of course, you know that the neutron and the proton are not supposed to just sit there. They’re also in these sort of intricate dance which – for the electron case – is described by some wavefunction, which we derive as a solution from Schr√∂dinger’s equation. So U(r) is going to vary not only in space but also in time and we should, therefore, write it as U(r, t). Now, we will, of course, assume it’s going to vary in space and time as some¬†wave¬†and we may, therefore, suggest some wave¬†equation¬†for it. To appreciate this point, you should review some of the posts I did on waves. More in particular, you may want to review the post I did on traveling fields, in which I showed you the following:¬†if we see an equation like:f8then the function¬†Ōą(x, t) must have the following general functional form:solutionAny¬†function Ōą¬†like that will work – so it will be a solution to the differential equation – and we’ll refer to it as a wavefunction. Now, the equation (and the function) is for a wave traveling in¬†one dimension only (x) but the same post shows we can easily generalize to waves traveling in three dimensions. In addition, we may generalize the analyse to include¬†complex-valued¬†functions as well. Now, you will still be shocked by Yukawa’s field equation for U(r, t) but, hopefully, somewhat less so after the above reminder on how wave equations generally look like:Yukawa wave equationAs said, you can look up the nitty-gritty in Aitchison and Grey¬†(or in its appendices) but, up to this point, you should be able to sort of appreciate what’s going on without getting lost in it all. Yukawa’s next step – and all that follows – is much more baffling. We’d think U, the nuclear potential, is just some scalar-valued wave, right? It varies in space and in time, but… Well… That’s what classical waves, like water or sound waves, for example do too. So far, so good. However, Yukawa’s next step is to associate a¬†de Broglie-type wavefunction with it. Hence, Yukawa imposes¬†solutions of the type:potential as particleWhat?¬†Yes. It’s a big thing to swallow, and it doesn’t help most physicists refer to U as a¬†force field. A force and the potential that results from it are two different things. To put it simply: the¬†force¬†on an object is¬†not¬†the same as the¬†work¬†you need to move it from here to there. Force and potential are¬†related¬†but¬†different¬†concepts. Having said that, it sort of make sense now, doesn’t it? If potential is energy, and if it behaves like some wave, then we must be able to associate it with a¬†de Broglie-type particle. This U-quantum, as it is referred to, comes in two varieties, which are associated with the ongoing¬†absorption-emission process that is supposed to take place inside of the nucleus (depicted below):

p + U‚ąí¬†‚Üí n and¬†n + U+¬†‚Üí p

absorption emission

It’s easy to see that the¬†U‚ąí¬†and¬†U+¬†particles are just each other’s anti-particle. When thinking about this, I can’t help remembering Feynman, when he enigmatically wrote – somewhere in his Strange Theory of Light and Matter¬†– that¬†an anti-particle might just be the same particle traveling back in time.¬†In fact, the¬†exchange¬†here is supposed to happen within a¬†time window¬†that is so short it allows for the brief¬†violation¬†of the energy conservation principle.

Let’s be more precise and try to find the properties of that mysterious U-quantum. You’ll need to refresh what you know about operators to understand how substituting Yukawa’s¬†de Broglie¬†wavefunction in the complicated-looking differential equation (the wave¬†equation) gives us the following relation between the energy and the momentum of our new particle:mass 1Now, it doesn’t take too many gimmicks to compare this against the relativistically correct energy-momentum relation:energy-momentum relationCombining both gives us the associated (rest) mass of the U-quantum:rest massFor¬†a¬†‚Čą 2 fm,¬†mU¬†is about 100 MeV. Of course, it’s always to check the dimensions and calculate stuff yourself. Note the physical dimension of¬†ńß/(a¬∑c) is N¬∑s2/m = kg (just think of the F = m¬∑a formula). Also note that N¬∑s2/m = kg = (N¬∑m)¬∑s2/m2¬†= J/(m2/s2), so that’s the [E]/[c2] dimension.¬†The calculation – and interpretation – is somewhat tricky though: if you do it, you’ll find that:

ńß/(a¬∑c) ‚Čą (1.0545718√ó10‚ąí34¬†N¬∑m¬∑s)/[(2√ó10‚ąí15¬†m)¬∑(2.997924583√ó108¬†m/s)] ‚Čą 0.176√ó10‚ąí27¬†kg

Now, most physics handbooks continue that terrible habit of writing particle weights in eV, rather than using the correct eV/c2¬†unit. So when they write: mU¬†is about 100 MeV, they actually mean to say that it’s 100 MeV/c2. In addition, the eV is not¬†an SI unit. Hence, to get that number, we should first write 0.176√ó10‚ąí27¬†kg as some value expressed in J/c2, and then convert the joule¬†(J) into electronvolt (eV). Let’s do that. First, note that c2¬†‚Čą 9√ó1016¬†m2/s2, so 0.176√ó10‚ąí27¬†kg¬†‚Čą¬†1.584√ó10‚ąí11¬†J/c2. Now we do the conversion from joule¬†to electronvolt. We¬†get: (1.584√ó10‚ąí11¬†J/c2)¬∑(6.24215√ó1018¬†eV/J)¬†‚Čą 9.9√ó107¬†eV/c2¬†= 99 MeV/c2.¬†Bingo!¬†So that was Yukawa’s prediction for the¬†nuclear force quantum.

Of course, Yukawa was wrong but, as mentioned above, his ideas are now generally accepted. First note the mass of the U-quantum is quite considerable:¬†100 MeV/c2¬†is a bit more than 10% of the individual proton or neutron mass (about 938-939 MeV/c2). While the¬†binding energy¬†causes the mass of an atom to be less than the mass of their constituent parts (protons, neutrons and electrons), it’s quite remarkably that the deuterium atom – a hydrogen atom with an extra neutron – has an excess mass of about 13.1 MeV/c2, and a binding energy with an equivalent mass of only 2.2 MeV/c2. So… Well… There’s something there.

As said, this post only wanted to introduce some basic ideas. The current model of nuclear physics is represented by the animation below, which I took from the Wikipedia article on it. The U-quantum appears as the pion here – and it does¬†not¬†really turn the proton into a neutron and vice versa. Those particles are assumed to be stable. In contrast, it is the¬†quarks¬†that change¬†color¬†by exchanging gluons between each other. And we know look at the exchange particle – which we refer to as the pion¬†–¬†between the proton and the neutron as consisting of two quarks in its own right: a quark and a anti-quark. So… Yes… All weird. QCD is just a different world. We’ll explore it more in the coming days and/or weeks. ūüôāNuclear_Force_anim_smallerAn alternative – and simpler – way of representing this exchange of a virtual particle (a neutral¬†pion¬†in this case) is obtained by drawing a so-called Feynman diagram:Pn_scatter_pi0OK. That’s it for today. More tomorrow. ūüôā

Light and matter

In my previous post, I discussed the¬†de Broglie¬†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 ő®(x ,t) – there is also a¬†phi¬†– i.e. the momentum-space wave function ő¶(p, t).

In that post, I also¬†compared it¬†– without much formalism – to the¬†de Broglie¬†wave of ‘matter particles’. Indeed, in physics, we look at ‘stuff’ as being made of particles and, while the taxonomy of the¬†particle zoo of the Standard Model¬†of physics is rather complicated, one ‘taxonomic’ principle stands out: particles are either¬†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 the¬†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 Model¬†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 1974¬†confirmed the existence of the (second-generation and, hence, heavy and unstable)¬†‘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 the¬†Large Hadron Collider¬†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 Committee¬†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 deny¬†its existence.]

Also note that the¬†particle zoo¬†is actually less complicated than it looks at first sight: the (composite) particles that are¬†stable¬†in our world – this world –¬†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,¬†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 my¬†my body,¬†right now and here,¬†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 largest¬†neutrino detector¬†on Earth is called¬†IceCube. It sits on the South Pole ‚Äď or¬†under¬†it, as it‚Äôs suspended¬†under¬†the Antarctic ice, and it regularly captures¬†high-energy neutrinos in the range of 1 to 10 TeV.¬†

Let me Рto conclude this introduction Рjust quickly list and explain the bosons (i.e the force carriers) in the table above:

1. Of all of the bosons, the photon (i.e. the topic of this post), is the most straightforward: there is only type 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.¬†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-called¬†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 ¬¶0‚Ć™ and ¬¶1‚Ć™ are the basis states, then any linear combination őĪ¬∑¬¶0‚Ć™ + √ü¬∑¬¶1‚Ć™ is also a valid state provided‚ĒāőĪ‚Ēā2 +¬†‚Ēāő≤‚Ēā2¬†= 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 kets¬†are used in conjunction with¬†bras. So we have a bra-ket notation: the ket gives the starting condition, and the bra – denoted as ‚Ć©¬†¬¶ – gives the final condition. They are combined in statements such as ‚Ć© particle arrives at x¬¶particle leaves from s‚Ć™ or – in short – ‚Ć© x¬¶s‚Ć™ and, while x and s would have some real-number value, ‚Ć©¬†x¬¶s‚Ć™ 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-called¬†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 four¬†directions of space (x, y and z) and time (t). ūüôā

2. Gluons, the exchange particles for the strong force, are more complicated: they come in eight so-called¬†colors.¬†In practice, one should think of these colors as different charges, but so we have more elementary charges¬†in this case¬†than just plus or minus one (¬Ī1) – as we have for the electric charge. So it’s just another type of¬†qubit¬†in quantum mechanics.

[Note that the so-called elementary ¬Ī1 values for electric charge are¬†not¬†really elementary: it’s ‚Äď1/3 (for the down¬†quark,¬†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 ¬Ī1 value is easily found from a linear combination of the ‚Äď1/3 and +2/3 values.]

3. Z and W bosons carry the so-called weak force, aka as Fermi‚Äôs interaction: they explain how one type of quark can change into another, thereby explaining phenomena such as¬†beta¬†decay. Beta decay explains why carbon-14 will, after a very long time (as compared to the ‘unstable’ particles mentioned above),¬†spontaneously decay into nitrogen-14. Indeed, carbon-12 is the (very) stable isotope, while¬†carbon-14 has a life-time of 5,730 ¬Ī 40 years ‘only’¬† (so one can’t call carbon-12 ‘unstable’: perhaps ‘less stable’ will do)¬†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, a¬†beta¬†particle can refer to an electron¬†or¬†a positron, so we can have¬†ő≤‚Ästdecay (e.g. the above-mentioned carbon-14 decay) as well as ő≤+¬†decay (e.g. magnesium-23 into sodium-23). There’s also alpha and gamma decay but that involves different things.¬†

As you can see from the table, W¬Ī¬†and Z0¬†bosons are very¬†heavy (157,000 and 178,000 times heavier than a electron!), and W¬Ī 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,¬†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, the¬†Higgs particle¬†– and, hence, of the associated Higgs field ‚Äď had been predicted since 1964 already but its existence was only (tentatively)¬†experimentally¬†confirmed last year. The Higgs field gives fermions, and also the W¬†and Z¬†bosons, mass (but not photons and gluons), and – as mentioned above – that‚Äôs why the weak force has such short range as compared to the electromagnetic and strong forces. Note, however, that the¬†Higgs particle does actually¬†not¬†explain the gravitational force, so it‚Äôs¬†not¬†the (theoretical) graviton and there is no quantum field theory for the gravitational force as yet. Just¬†Google¬†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 scalar¬†field (as opposed to a vector field). However, I have no idea how this so-called Higgs mechanism¬†(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 a¬†very¬†long way to go on this¬†Road to Reality.


In any case…¬†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 or¬†a fermion¬†makes a big difference. That being said, both fermions and bosons are¬†wavicles¬†described by a complex-valued¬†psi¬†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 some¬†observable property having a value in some relevant range, usually denoted by őĒ. [I also explained (in my post on Bose and Fermi) how the rules for¬†combining¬†amplitudes differ for bosons versus fermions, and how that explains why they are what they are: matter particles occupy space, while photons not only¬†can¬†but also¬†like¬†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 (őĒx,¬†őĒt). [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 őĒx is likely to be a cube or a sphere with an edge or a radius of a few picometer only (a picometer is a¬†thousandth¬†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 let¬†you¬†calculate a relevant őĒt. :-)]

If we want to do that, then we will need to square the modulus of the corresponding¬†wave function ő®(x, t). To be precise, we will have to do a summation¬†of all the values ‚Ēāő®(x, t)‚Ēā2¬†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’s¬†exact position but only in its momentum or energy. Well, we have another wave function for that: the momentum wave function ő¶(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 we¬†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 my¬†Bose and Fermi¬†post, so I won’t repeat that here. Just note that, when it comes to working with those probability¬†amplitudes¬†(that’s just another word for these psi and phi¬†functions), it makes a¬†huge¬†difference: fermions and bosons interact¬†very¬†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 principle¬†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-matter¬†interaction. Indeed, in my previous post (on Light), I¬†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 be¬†at point x at time t), and so that’s what I will do now.

In his 1985 Lectures on Quantum Electrodynamics¬†(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 third¬†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: “‚ÄúOut of these three amplitudes, we can make the¬†whole world, aside from what goes on in nuclei, and gravitation, as always!‚Ä̬†

Well… Thank you, Mr Feynman:¬†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 absorb¬†or¬†emit a photon. Indeed, absorbing or emitting makes no difference: we have the same complex number for both. It’s a constant – denoted by j¬†(for junction¬†number)¬†– equal to ‚Äď0.1 (a bit less actually but it’s good enough as an approximation in the context of this blog).

Huh?¬†Minus 0.1?¬†That’s not a complex number, is it? It is. Real numbers are complex numbers too: ‚Äď0.1 is 0.1eiŌĬ†in polar coordinates. As Feynman puts it: it’s “a shrink to about one-tenth, and half a turn.”¬†The ‘shrink’ is the 0.1 magnitude of this vector (or arrow), and the ‘half-turn’ is the angle of ŌÄ (i.e. 180 degrees). He obviously refers to multiplying (no adding here)¬†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)·j·E(5 to 3)·E(2 to 6)·j·E(6 to 3)

¬†As for how that¬†j¬†factor works,¬†please do read the caption of the illustration below: the same j¬†describes both emission as well as absorption. It’s just that we have¬†both¬†an emission¬†as well as¬†an as absorption here, so we have a¬†j2 factor here, which is less than 0.1¬∑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 way¬†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 predict¬†anything in the classical sense of the word.]

Feynman diagram of photon-electron coupling

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 usually¬†not¬†¬†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 a¬†photon (!)¬†to go from F to G – even if we are talking an electron here. But there’s a difference: we also¬†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)¬∑n2¬∑P(C to B)¬†+ P(A to D)¬∑n2¬∑P(D to E)¬∑n2¬∑P(E to B) + …¬†for all possible intermediate points C, D, E, and so on, as per the illustration below.

E(A to B)

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,¬†n4¬†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. ūüôā Hence, it’s not a probability amplitude like j: a proper¬†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 not¬†have such vanishing terms. I¬†will not dwell on this right¬†here, but I promise to discuss it in the Post Scriptum¬†of this post. The frightening possibility is that n might be¬†a number larger than one.

[As we’re freewheeling a bit anyway here, just a quick note on conventions: I should not be writing¬†j¬†in bold-face, because it’s a (complex- or real-valued) number and symbols representing numbers are usually not¬†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. j¬∑j) is an actual dot project¬†(aka as a scalar product or an inner¬†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)¬†+ 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,¬†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 function¬†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, I¬†could¬†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 Schr√∂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 must¬†be –¬†but I have not seen anything that’s equally ‘simple’ as the Schr√∂dinger equation for photons. [I have¬†Googled¬†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.]¬†
  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: that spacetime interval which I mentioned in my previous post already:

I = őĒr2¬†‚Äď őĒt2¬†= ¬†(z2– z1)2¬†+¬†(y2– y1)2¬†+¬†(x2– x1)2¬†‚Äst(t2–¬†t1)2

In this equation, we need to measure the time and spatial distance between two points in spacetime in equivalent units¬†(these ‘points’ are usually referred to as four-vectors), so we‚Äôd use light-seconds for the unit of distance or, for the unit of time, the time it takes for light to travel one meter.¬†[If we don’t want to transform time or distance scales, then we have to write¬†I¬†as I =¬†c2őĒt2¬†‚ÄstőĒr2.] Now, there are three types of intervals:

  1. For time-like intervals, we have a negative¬†value for I, so őĒt2¬†> őĒr2. For two events separated by a time-like interval, enough time passes between them so there could be a cause‚Äďeffect 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 őĒt2¬†< őĒr2. 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 photons¬†do have amplitudes to travel faster than light.
  3. Finally, for light-like intervals, I = 0, or¬†őĒt2 = őĒr2. 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 (+++‚Äď) here for I. There’s also a time-like convention, i.e. with¬†+‚Äď‚Äď‚Äď as signs: I =¬†őĒt2¬†‚ÄstőĒr2¬†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.]

Spacetime intervalsNow, what’s the relevance of this? To calculate P(A to B), we have to¬†add¬†the amplitudes¬†for all possible paths¬†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 times¬†and so we only look at a bunch that’s leaving at the same point in time. In other words, their¬†stopwatches will be synchronized (i.e. there is no phase shift term¬†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 it¬†very carefully.

adding arrows

If you look at the so-called¬†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 contribution¬†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.]  Least time principle

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 principles¬†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 I¬†in its denominator. Now, the way it works is that, for values of I¬†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 actual¬†photon travel faster or slower than speed is higher.]

Contribution interval


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)¬†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 nuclei¬†(because that’s the domain of the weak and strong nuclear force) and gravitation, for which we have a¬†law (Newton’s Law)¬†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 of¬†real-life¬†phenomena – quite interesting, all of it ! – but, once again, it’s not my goal to copy all of his Lectures¬†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 n¬†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 constant¬†j, and the ‘rest mass’¬†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 got¬†a Nobel Prize for it.

First, about the book. Feynman’s 1985 Lectures on Quantum Electrodynamics¬†are not like his 1965 Lectures on Physics.¬†The Lectures on Physics¬†are proper courses for undergraduate and even graduate students in physics. This little 1985 book on QED is just¬†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 Memorial¬†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 two¬†rare forms of¬†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 a¬†brilliant mind¬†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 ‚Äúdippy process‚ÄĚ and ‚Äúhaving to resort to such hocus-pocus has prevented us from proving that the theory of quantum electrodynamics is mathematically self-consistent‚Äú. He adds: ‚ÄúIt‚Äôs surprising that the theory still hasn‚Äôt 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] ¬†is not mathematically legitimate.‚ÄĚ […] Now,¬†Mr Feynman writes this about quantum electrodynamics, not about¬†‚Äúthe rest of physics‚Ä̬†(and so that‚Äôs quantum chromodynamics (QCD) ‚Äď the theory of the strong interactions ‚Äď and quantum flavordynamics (QFD) ‚Äď the theory of weak interactions) which, he adds,¬†‚Äúhas not been checked anywhere near as well as electrodynamics.‚Ä̬†

That‚Äôs a pretty damning statement, isn‚Äôt it? In one of my other posts (see:¬†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.¬†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 !¬†