The speed of light as an angular velocity

Over the weekend, I worked on a revised version of my paper on a physical interpretation of the wavefunction. However, I forgot to add the final remarks on the speed of light as an angular velocity. I know… This post is for my faithful followers only. It is dense, but let me add the missing bits here:

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Post scriptum (29 October): Einstein’s view on aether theories probably still holds true: “We may say that according to the general theory of relativity space is endowed with physical qualities; in this sense, therefore, there exists an aether. According to the general theory of relativity, space without aether is unthinkable – for in such space there not only would be no propagation of light, but also no possibility of existence for standards of space and time (measuring-rods and clocks), nor therefore any space-time intervals in the physical sense. But this aether may not be thought of as endowed with the quality characteristic of ponderable media, as consisting of parts which may be tracked through time. The idea of motion may not be applied to it.”

The above quote is taken from the Wikipedia article on aether theories. The same article also quotes Robert Laughlin, the 1998 Nobel Laureate in Physics, who said this about aether in 2005: “It is ironic that Einstein’s most creative work, the general theory of relativity, should boil down to conceptualizing space as a medium when his original premise [in special relativity] was that no such medium existed. […] The word ‘aether’ has extremely negative connotations in theoretical physics because of its past association with opposition to relativity. This is unfortunate because, stripped of these connotations, it rather nicely captures the way most physicists actually think about the vacuum. […]The modern concept of the vacuum of space, confirmed every day by experiment, is a relativistic aether. But we do not call it this because it is taboo.”

I really love this: a relativistic aether. My interpretation of the wavefunction is very consistent with that.

Re-visiting the speed of light, Planck’s constant, and the fine-structure constant

A brother of mine sent me a link to an article he liked. Now, because we share some interest in physics and math and other stuff, I looked at it and…

Well… I was disappointed. Despite the impressive credentials of its author – a retired physics professor – it was very poorly written. It made me realize how much badly written stuff is around, and I am glad I am no longer wasting my time on it. However, I do owe my brother some explanation of (a) why I think it was bad, and of (b) what, in my humble opinion, he should be wasting his time on. 🙂 So what it is all about?

The article talks about physicists deriving the speed of light from “the electromagnetic properties of the quantum vacuum.” Now, it’s the term ‘quantum‘, in ‘quantum vacuum’, that made me read the article.

Indeed, deriving the theoretical speed of light in empty space from the properties of the classical vacuum – aka empty space – is a piece of cake: it was done by Maxwell himself as he was figuring out his equations back in the 1850s (see my post on Maxwell’s equations and the speed of light). And then he compared it to the measured value, and he saw it was right on the mark. Therefore, saying that the speed of light is a property of the vacuum, or of empty space, is like a tautology: we may just as well put it the other way around, and say that it’s the speed of light that defines the (properties of the) vacuum!

Indeed, as I’ll explain in a moment: the speed of light determines both the electric as well as the magnetic constants μand ε0, which are the (magnetic) permeability and the (electric) permittivity of the vacuum respectively. Both constants depend on the units we are working with (i.e. the units for electric charge, for distance, for time and for force – or for inertia, if you want, because force is defined in terms of overcoming inertia), but so they are just proportionality coefficients in Maxwell’s equations. So once we decide what units to use in Maxwell’s equations, then μand ε0 are just proportionality coefficients which we get from c. So they are not separate constants really – I mean, they are not separate from c – and all of the ‘properties’ of the vacuum, including these constants, are in Maxwell’s equations.

In fact, when Maxwell compared the theoretical value of c with its presumed actual value, he didn’t compare c‘s theoretical value with the speed of light as measured by astronomers (like that 17th century Ole Roemer, to which our professor refers: he had a first go at it by suggesting some specific value for it based on his observations of the timing of the eclipses of one of Jupiter’s moons), but with c‘s value as calculated from the experimental values of μand ε0! So he knew very well what he was looking at. In fact, to drive home the point, it may also be useful to note that the Michelson-Morley experiment – which accurately measured the speed of light – was done some thirty years later. So Maxwell had already left this world by then—very much in peace, because he had solved the mystery all 19th century physicists wanted to solve through his great unification: his set of equations covers it all, indeed: electricity, magnetism, light, and even relativity!

I think the article my brother liked so much does a very lousy job in pointing all of that out, but that’s not why I wouldn’t recommend it. It got my attention because I wondered why one would try to derive the speed of light from the properties of the quantum vacuum. In fact, to be precise, I hoped the article would tell me what the quantum vacuum actually is. Indeed, as far as I know, there’s only one vacuum—one ’empty space’: empty is empty, isn’t it? 🙂 So I wondered: do we have a ‘quantum’ vacuum? And, if so, what is it, really?

Now, that is where the article is really disappointing, I think. The professor drops a few names (like the Max Planck Institute, the University of Paris-Sud, etcetera), and then, promisingly, mentions ‘fleeting excitations of the quantum vacuum’ and ‘virtual pairs of particles’, but then he basically stops talking about quantum physics. Instead, he wanders off to share some philosophical thoughts on the fundamental physical constants. What makes it all worse is that even those thoughts on the ‘essential’ constants are quite off the mark.

So… This post is just a ‘quick and dirty’ thing for my brother which, I hope, will be somewhat more thought-provoking than that article. More importantly, I hope that my thoughts will encourage him to try to grind through better stuff.

On Maxwell’s equations and the properties of empty space

Let me first say something about the speed of light indeed. Maxwell’s four equations may look fairly simple, but that’s only until one starts unpacking all those differential vector equations, and it’s only when going through all of their consequences that one starts appreciating their deep mathematical structure. Let me quickly copy how another blogger jotted them down: 🙂

god-said-maxwell-equation

As I showed in my above-mentioned post, the speed of light (i.e. the speed with which an electromagnetic pulse or wave travels through space) is just one of the many consequences of the mathematical structure of Maxwell’s set of equations. As such, the speed of light is a direct consequence of the ‘condition’, or the properties, of the vacuum indeed, as Maxwell suggested when he wrote that “we can scarcely avoid the inference that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena”.

Of course, while Maxwell still suggests light needs some ‘medium’ here – so that’s a reference to the infamous aether theory – we now know that’s because he was a 19th century scientist, and so we’ve done away with the aether concept (because it’s a redundant hypothesis), and so now we also know there’s absolutely no reason whatsoever to try to “avoid the inference.” 🙂 It’s all OK, indeed: light is some kind of “transverse undulation” of… Well… Of what?

We analyze light as traveling fields, represented by two vectors, E and B, whose direction and magnitude varies both in space as well as in time. E and B are field vectors, and represent the electric and magnetic field respectively. An equivalent formulation – more or less, that is (see my post on the Liénard-Wiechert potentials) – for Maxwell’s equations when only one (moving) charge is involved is:

E

B

This re-formulation, which is Feynman’s preferred formula for electromagnetic radiation, is interesting in a number of ways. It clearly shows that, while we analyze the electric and magnetic field as separate mathematical entities, they’re one and the same phenomenon really, as evidenced by the B = –er×E/c equation, which tells us the magnetic field from a single moving charge is always normal (i.e. perpendicular) to the electric field vector, and also that B‘s magnitude is 1/times the magnitude of E, so |B| = B = |E|/c = E/c. In short, B is fully determined by E, or vice versa: if we have one of the two fields, we have the other, so they’re ‘one and the same thing’ really—not in a mathematical sense, but in a real sense.

Also note that E and B‘s magnitude is just the same if we’re using natural units, so if we equate c with 1. Finally, as I pointed out in my post on the relativity of electromagnetic fields, if we would switch from one reference frame to another, we’ll have a different mix of E and B, but that different mix obviously describes the same physical reality. More in particular, if we’d be moving with the charges, the magnetic field sort of disappears to re-appear as an electric field. So the Lorentz force F = Felectric + Fmagnetic = qE + qv×B is one force really, and its ‘electric’ and ‘magnetic’ component appear the way they appear in our reference frame only. In some other reference frame, we’d have the same force, but its components would look different, even if they, obviously, would and should add up to the same. [Well… Yes and no… You know there’s relativistic corrections to be made to the forces to, but that’s a minor point, really. The force surely doesn’t disappear!]

All of this reinforces what you know already: electricity and magnetism are part and parcel of one and the same phenomenon, the electromagnetic force field, and Maxwell’s equations are the most elegant way of ‘cutting it up’. Why elegant? Well… Click the Occam tab. 🙂

Now, after having praised Maxwell once more, I must say that Feynman’s equations above have another advantage. In Maxwell’s equations, we see two constants, the electric and magnetic constant (denoted by μand ε0 respectively), and Maxwell’s equations imply that the product of the electric and magnetic constant is the reciprocal of c2: μ0·ε= 1/c2. So here we see εand only, so no μ0, so that makes it even more obvious that the magnetic and electric constant are related one to another through c.

[…] Let me digress briefly: why do we have c2 in μ0·ε= 1/c2, instead of just c? That’s related to the relativistic nature of the magnetic force: think about that B = E/c relation. Or, better still, think about the Lorentz equation F = Felectric + Fmagnetic = qE + qv×B = q[E + (v/c)×(E×er)]: the 1/c factor is there because the magnetic force involves some velocity, and any velocity is always relative—and here I don’t mean relative to the frame of reference but relative to the (absolute) speed of light! Indeed, it’s the v/c ratio (usually denoted by β = v/c) that enters all relativistic formulas. So the left-hand side of the μ0·ε= 1/c2 equation is best written as (1/c)·(1/c), with one of the two 1/c factors accounting for the fact that the ‘magnetic’ force is a relativistic effect of the ‘electric’ force, really, and the other 1/c factor giving us the proper relationship between the magnetic and the electric constant. To drive home the point, I invite you to think about the following:

  • μ0 is expressed in (V·s)/(A·m), while εis expressed in (A·s)/(V·m), so the dimension in which the μ0·εproduct is expressed is [(V·s)/(A·m)]·[(A·s)/(V·m)] = s2/m2, so that’s the dimension of 1/c2.
  • Now, this dimensional analysis makes it clear that we can sort of distribute 1/c2 over the two constants. All it takes is re-defining the fundamental units we use to calculate stuff, i.e. the units for electric charge, for distance, for time and for force – or for inertia, as explained above. But so we could, if we wanted, equate both μ0 as well as εwith 1/c.
  • Now, if we would then equate c with 1, we’d have μ0 = ε= c = 1. We’d have to define our units for electric charge, for distance, for time and for force accordingly, but it could be done, and then we could re-write Maxwell’s set of equations using these ‘natural’ units.

In any case, the nitty-gritty here is less important: the point is that μand εare also related through the speed of light and, hence, they are ‘properties’ of the vacuum as well. [I may add that this is quite obvious if you look at their definition, but we’re approaching the matter from another angle here.]

In any case, we’re done with this. On to the next!

On quantum oscillations, Planck’s constant, and Planck units 

The second thought I want to develop is about the mentioned quantum oscillation. What is it? Or what could it be? An electromagnetic wave is caused by a moving electric charge. What kind of movement? Whatever: the charge could move up or down, or it could just spin around some axis—whatever, really. For example, if it spins around some axis, it will have a magnetic moment and, hence, the field is essentially magnetic, but then, again, E and B are related and so it doesn’t really matter if the first cause is magnetic or electric: that’s just our way of looking at the world: in another reference frame, one that’s moving with the charges, the field would essential be electric. So the motion can be anything: linear, rotational, or non-linear in some irregular way. It doesn’t matter: any motion can always be analyzed as the sum of a number of ‘ideal’ motions. So let’s assume we have some elementary charge in space, and it moves and so it emits some electromagnetic radiation.

So now we need to think about that oscillation. The key question is: how small can it be? Indeed, in one of my previous posts, I tried to explain some of the thinking behind the idea of the ‘Great Desert’, as physicists call it. The whole idea is based on our thinking about the limit: what is the smallest wavelength that still makes sense? So let’s pick up that conversation once again.

The Great Desert lies between the 1032 and 1043 Hz scale. 1032 Hz corresponds to a photon energy of Eγ = h·f = (4×10−15 eV·s)·(1032 Hz) = 4×1017 eV = 400,000 tera-electronvolt (1 TeV = 1012 eV). I use the γ (gamma) subscript in my Eγ symbol for two reasons: (1) to make it clear that I am not talking the electric field E here but energy, and (2) to make it clear we are talking ultra-high-energy gamma-rays here.

In fact, γ-rays of this frequency and energy are theoretical only. Ultra-high-energy gamma-rays are defined as rays with photon energies higher than 100 TeV, which is the upper limit for very-high-energy gamma-rays, which have been observed as part of the radiation emitted by so-called gamma-ray bursts (GRBs): flashes associated with extremely energetic explosions in distant galaxies. Wikipedia refers to them as the ‘brightest’ electromagnetic events know to occur in the Universe. These rays are not to be confused with cosmic rays, which consist of high-energy protons and atomic nuclei stripped of their electron shells. Cosmic rays aren’t rays really and, because they consist of particles with a considerable rest mass, their energy is even higher. The so-called Oh-My-God particle, for example, which is the most energetic particle ever detected, had an energy of 3×1020 eV, i.e. 300 million TeV. But it’s not a photon: its energy is largely kinetic energy, with the rest mass m0 counting for a lot in the m in the E = m·c2 formula. To be precise: the mentioned particle was thought to be an iron nucleus, and it packed the equivalent energy of a baseball traveling at 100 km/h! 

But let me refer you to another source for a good discussion on these high-energy particles, so I can get get back to the energy of electromagnetic radiation. When I talked about the Great Desert in that post, I did so using the Planck-Einstein relation (E = h·f), which embodies the idea of the photon being valid always and everywhere and, importantly, at every scale. I also discussed the Great Desert using real-life light being emitted by real-life atomic oscillators. Hence, I may have given the (wrong) impression that the idea of a photon as a ‘wave train’ is inextricably linked with these real-life atomic oscillators, i.e. to electrons going from one energy level to the next in some atom. Let’s explore these assumptions somewhat more.

Let’s start with the second point. Electromagnetic radiation is emitted by any accelerating electric charge, so the atomic oscillator model is an assumption that should not be essential. And it isn’t. For example, whatever is left of the nucleus after alpha or beta decay (i.e. a nuclear decay process resulting in the emission of an α- or β-particle) it likely to be in an excited state, and likely to emit a gamma-ray for about 10−12 seconds, so that’s a burst that’s about 10,000 times shorter than the 10–8 seconds it takes for the energy of a radiating atom to die out. [As for the calculation of that 10–8 sec decay time – so that’s like 10 nanoseconds – I’ve talked about this before but it’s probably better to refer you to the source, i.e. one of Feynman’s Lectures.]

However, what we’re interested in is not the energy of the photon, but the energy of one cycle. In other words, we’re not thinking of the photon as some wave train here, but what we’re thinking about is the energy that’s packed into a space corresponding to one wavelength. What can we say about that?

As you know, that energy will depend both on the amplitude of the electromagnetic wave as well as its frequency. To be precise, the energy is (1) proportional to the square of the amplitude, and (2) proportional to the frequency. Let’s look at the first proportionality relation. It can be written in a number of ways, but one way of doing it is stating the following: if we know the electric field, then the amount of energy that passes per square meter per second through a surface that is normal to the direction in which the radiation is going (which we’ll denote by S – the s from surface – in the formula below), must be proportional to the average of the square of the field. So we write S ∝ 〈E2〉, and so we should think about the constant of proportionality now. Now, let’s not get into the nitty-gritty, and so I’ll just refer to Feynman for the derivation of the formula below:

S = ε0c·〈E2

So the constant of proportionality is ε0c. [Note that, in light of what we wrote above, we can also write this as S = (1/μ0·c)·〈(c·B)2〉 = (c0)·〈B2〉, so that underlines once again that we’re talking one electromagnetic phenomenon only really.] So that’s a nice and rather intuitive result in light of all of the other formulas we’ve been jotting down. However, it is a ‘wave’ perspective. The ‘photon’ perspective assumes that, somehow, the amplitude is given and, therefore, the Planck-Einstein relation only captures the frequency variable: Eγ = h·f.

Indeed, ‘more energy’ in the ‘wave’ perspective basically means ‘more photons’, but photons are photons: they have a definite frequency and a definite energy, and both are given by that Planck-Einstein relation. So let’s look at that relation by doing a bit of dimensional analysis:

  • Energy is measured in electronvolt or, using SI units, joule: 1 eV ≈ 1.6×10−19 J. Energy is force times distance: 1 joule = 1 newton·meter, which means that a larger force over a shorter distance yields the same energy as a smaller force over a longer distance. The oscillations we’re talking about here involve very tiny distances obviously. But the principle is the same: we’re talking some moving charge q, and the power – which is the time rate of change of the energy – that goes in or out at any point of time is equal to dW/dt = F·v, with W the work that’s being done by the charge as it emits radiation.
  • I would also like to add that, as you know, forces are related to the inertia of things. Newton’s Law basically defines a force as that what causes a mass to accelerate: F = m·a = m·(dv/dt) = d(m·v)/dt = dp/dt, with p the momentum of the object that’s involved. When charges are involved, we’ve got the same thing: a potential difference will cause some current to change, and one of the equivalents of Newton’s Law F = m·a = m·(dv/dt) in electromagnetism is V = L·(dI/dt). [I am just saying this so you get a better ‘feel’ for what’s going on.]
  • Planck’s constant is measured in electronvolt·seconds (eV·s) or in, using SI units, in joule·seconds (J·s), so its dimension is that of (physical) action, which is energy times time: [energy]·[time]. Again, a lot of energy during a short time yields the same energy as less energy over a longer time. [Again, I am just saying this so you get a better ‘feel’ for these dimensions.]
  • The frequency f is the number of cycles per time unit, so that’s expressed per second, i.e. in herz (Hz) = 1/second = s−1.

So… Well… It all makes sense: [x joule] = [6.626×10−34 joule]·[1 second]×[f cycles]/[1 second]. But let’s try to deepen our understanding even more: what’s the Planck-Einstein relation really about?

To answer that question, let’s think some more about the wave function. As you know, it’s customary to express the frequency as an angular frequency ω, as used in the wave function A(x, t) = A0·sin(kx − ωt). The angular frequency is the frequency expressed in radians per second. That’s because we need an angle in our wave function, and so we need to relate x and t to some angle. The way to think about this is as follows: one cycle takes a time T (i.e. the period of the wave) which is equal to T = 1/f. Yes: one second divided by the number of cycles per second gives you the time that’s needed for one cycle. One cycle is also equivalent to our argument ωt going around the full circle (i.e. 2π), so we write:  ω·T = 2π and, therefore:

ω = 2π/T = 2π·f

Now we’re ready to play with the Planck-Einstein relation. We know it gives us the energy of one photon really, but what if we re-write our equation Eγ = h·f as Eγ/f = h? The dimensions in this equation are:

[x joule]·[1 second]/[cyles] = [6.626×10−34 joule]·[1 second]

⇔ = 6.626×10−34 joule per cycle

So that means that the energy per cycle is equal to 6.626×10−34 joule, i.e. the value of Planck’s constant.

Let me rephrase truly amazing result, so you appreciate it—perhaps: regardless of the frequency of the light (or our electromagnetic wave, in general) involved, the energy per cycle, i.e. per wavelength or per period, is always equal to 6.626×10−34 joule or, using the electronvolt as the unit, 4.135667662×10−15 eV. So, in case you wondered, that is the true meaning of Planck’s constant!

Now, if we have the frequency f, we also have the wavelength λ, because the velocity of the wave is the frequency times the wavelength: = λ·f and, therefore, λ = c/f. So if we increase the frequency, the wavelength becomes smaller and smaller, and so we’re packing the same amount of energy – admittedly, 4.135667662×10−15 eV is a very tiny amount of energy – into a space that becomes smaller and smaller. Well… What’s tiny, and what’s small? All is relative, of course. 🙂 So that’s where the Planck scale comes in. If we pack that amount of energy into some tiny little space of the Planck dimension, i.e. a ‘length’ of 1.6162×10−35 m, then it becomes a tiny black hole, and it’s hard to think about how that would work.

[…] Let me make a small digression here. I said it’s hard to think about black holes but, of course, it’s not because it’s ‘hard’ that we shouldn’t try it. So let me just mention a few basic facts. For starters, black holes do emit radiation! So they swallow stuff, but they also spit stuff out. More in particular, there is the so-called Hawking radiation, as Roger Penrose and Stephen Hawking discovered.

Let me quickly make a few remarks on that: Hawking radiation is basically a form of blackbody radiation, so all frequencies are there, as shown below: the distribution of the various frequencies depends on the temperature of the black body, i.e. the black hole in this case. [The black curve is the curve that Lord Rayleigh and Sir James Jeans derived in the late 19th century, using classical theory only, so that’s the one that does not correspond to experimental fact, and which led Max Planck to become the ‘reluctant’ father of quantum mechanics. In any case, that’s history and so I shouldn’t dwell on this.]

600px-Black_body

The interesting thing about blackbody radiation, including Hawking radiation, is that it reduces energy and, hence, the equivalent mass of our blackbody. So Hawking radiation reduces the mass and energy of black holes and is therefore also known as black hole evaporation. So black holes that lose more mass than they gain through other means are expected to shrink and ultimately vanish. Therefore, there’s all kind of theories that say why micro black holes, like that Planck scale black hole we’re thinking of right now, should be much larger net emitters of radiation than large black holes and, hence, whey they should shrink and dissipate faster.

Hmm… Interesting… What do we do with all of this information? Well… Let’s think about it as we continue our trek on this long journey to reality over the next year or, more probably, years (plural). 🙂

The key lesson here is that space and time are intimately related because of the idea of movement, i.e. the idea of something having some velocity, and that it’s not so easy to separate the dimensions of time and distance in any hard and fast way. As energy scales become larger and, therefore, our natural time and distance units become smaller and smaller, it’s the energy concept that comes to the fore. It sort of ‘swallows’ all other dimensions, and it does lead to limiting situations which are hard to imagine. Of course, that just underscores the underlying unity of Nature, and the mysteries involved.

So… To relate all of this back to the story that our professor is trying to tell, it’s a simple story really. He’s talking about two fundamental constants basically, c and h, pointing out that c is a property of empty space, and h is related to something doing something. Well… OK. That’s really nothing new, and surely not ground-breaking research. 🙂

Now, let me finish my thoughts on all of the above by making one more remark. If you’ve read a thing or two about this – which you surely have – you’ll probably say: this is not how people usually explain it. That’s true, they don’t. Anything I’ve seen about this just associates the 1043 Hz scale with the 1028 eV energy scale, using the same Planck-Einstein relation. For example, the Wikipedia article on micro black holes writes that “the minimum energy of a microscopic black hole is 1019 GeV [i.e. 1028 eV], which would have to be condensed into a region on the order of the Planck length.” So that’s wrong. I want to emphasize this point because I’ve been led astray by it for years. It’s not the total photon energy, but the energy per cycle that counts. Having said that, it is correct, however, and easy to verify, that the 1043 Hz scale corresponds to a wavelength of the Planck scale: λ = c/= (3×10m/s)/(1043 s−1) = 3×10−35 m. The confusion between the photon energy and the energy per wavelength arises because of the idea of a photon: it travels at the speed of light and, hence, because of the relativistic length contraction effect, it is said to be point-like, to have no dimension whatsoever. So that’s why we think of packing all of its energy in some infinitesimally small place. But you shouldn’t think like that. The photon is dimensionless in our reference frame: in its own ‘world’, it is spread out, so it is a wave train. And it’s in its ‘own world’ that the contradictions start… 🙂

OK. Done!

My third and final point is about what our professor writes on the fundamental physical constants, and more in particular on what he writes on the fine-structure constant. In fact, I could just refer you to my own post on it, but that’s probably a bit too easy for me and a bit difficult for you 🙂 so let me summarize that post and tell you what you need to know about it.

The fine-structure constant

The fine-structure constant α is a dimensionless constant which also illustrates the underlying unity of Nature, but in a way that’s much more fascinating than the two or three things the professor mentions. Indeed, it’s quite incredible how this number (α = 0.00729735…, but you’ll usually see it written as its reciprocal, which is a number that’s close to 137.036…) links charge with the relative speeds, radii, and the mass of fundamental particles and, therefore, how this number also these concepts with each other. And, yes, the fact that it is, effectively, dimensionless, unlike h or c, makes it even more special. Let me quickly sum up what the very same number α all stands for:

(1) α is the square of the electron charge expressed in Planck units: α = eP2.

(2) α is the square root of the ratio of (a) the classical electron radius and (b) the Bohr radius: α = √(re /r). You’ll see this more often written as re = α2r. Also note that this is an equation that does not depend on the units, in contrast to equation 1 (above), and 4 and 5 (below), which require you to switch to Planck units. It’s the square of a ratio and, hence, the units don’t matter. They fall away.

(3) α is the (relative) speed of an electron: α = v/c. [The relative speed is the speed as measured against the speed of light. Note that the ‘natural’ unit of speed in the Planck system of units is equal to c. Indeed, if you divide one Planck length by one Planck time unit, you get (1.616×10−35 m)/(5.391×10−44 s) = m/s. However, this is another equation, just like (2), that does not depend on the units: we can express v and c in whatever unit we want, as long we’re consistent and express both in the same units.]

(4) α is also equal to the product of (a) the electron mass (which I’ll simply write as me here) and (b) the classical electron radius re (if both are expressed in Planck units): α = me·re. Now think that’s, perhaps, the most amazing of all of the expressions for α. [If you don’t think that’s amazing, I’d really suggest you stop trying to study physics. :-)]

Also note that, from (2) and (4), we find that:

(5) The electron mass (in Planck units) is equal me = α/r= α/α2r = 1/αr. So that gives us an expression, using α once again, for the electron mass as a function of the Bohr radius r expressed in Planck units.

Finally, we can also substitute (1) in (5) to get:

(6) The electron mass (in Planck units) is equal to me = α/r = eP2/re. Using the Bohr radius, we get me = 1/αr = 1/eP2r.

So… As you can see, this fine-structure constant really links all of the fundamental properties of the electron: its charge, its radius, its distance to the nucleus (i.e. the Bohr radius), its velocity, its mass (and, hence, its energy),…

So… Why is what it is?

Well… We all marvel at this, but what can we say about it, really? I struggle how to interpret this, just as much – or probably much more 🙂 – as the professor who wrote the article I don’t like (because it’s so imprecise, and that’s what made me write all what I am writing here).

Having said that, it’s obvious that it points to a unity beyond these numbers and constants that I am only beginning to appreciate for what it is: deep, mysterious, and very beautiful. But so I don’t think that professor does a good job at showing how deep, mysterious and beautiful it all is. But then that’s up to you, my brother and you, my imaginary reader, to judge, of course. 🙂

[…] I forgot to mention what I mean with ‘Planck units’. Well… Once again, I should refer you to one of my other posts. But, yes, that’s too easy for me and a bit difficult for you. 🙂 So let me just note we get those Planck units by equating not less than five fundamental physical constants to 1, notably (1) the speed of light, (2) Planck’s (reduced) constant, (3) Boltzmann’s constant, (4) Coulomb’s constant and (5) Newton’s constant (i.e. the gravitational constant). Hence, we have a set of five equations here (ħ = kB = ke = G = 1), and so we can solve that to get the five Planck units, i.e. the Planck length unit, the Planck time unit, the Planck mass unit, the Planck energy unit, the Planck charge unit and, finally (oft forgotten), the Planck temperature unit. Of course, you should note that all mass and energy units are directly related because of the mass-energy equivalence relation E = mc2, which simplifies to E = m if c is equated to 1. [I could also say something about the relation between temperature and (kinetic) energy, but I won’t, as it would only further confuse you.]

OK. Done! 🙂

Addendum: How to think about space and time?

If you read the argument on the Planck scale and constant carefully, then you’ll note that it does not depend on the idea of an indivisible photon. However, it does depend on that Planck-Einstein relation being valid always and everywhere. Now, the Planck-Einstein relation is, in its essence, a fairly basic result from classical electromagnetic theory: it incorporates quantum theory – remember: it’s the equation that allowed Planck to solve the black-body radiation problem, and so it’s why they call Planck the (reluctant) ‘Father of Quantum Theory’ – but it’s not quantum theory.

So the obvious question is: can we make this reflection somewhat more general, so we can think of the electromagnetic force as an example only. In other words: can we apply the thoughts above to any force and any movement really?

The truth is: I haven’t advanced enough in my little study to give the equations for the other forces. Of course, we could think of gravity, and I developed some thoughts on how gravity waves might look like, but nothing specific really. And then we have the shorter-range nuclear forces, of course: the strong force, and the weak force. The laws involved are very different. The strong force involves color charges, and the way distances work is entirely different. So it would surely be some different analysis. However, the results should be the same. Let me offer some thoughts though:

  • We know that the relative strength of the nuclear force is much larger, because it pulls like charges (protons) together, despite the strong electromagnetic force that wants to push them apart! So the mentioned problem of trying to ‘pack’ some oscillation in some tiny little space should be worse with the strong force. And the strong force is there, obviously, at tiny little distances!
  • Even gravity should become important, because if we’ve got a lot of energy packed into some tiny space, its equivalent mass will ensure the gravitational forces also become important. In fact, that’s what the whole argument was all about!
  • There’s also all this talk about the fundamental forces becoming one at the Planck scale. I must, again, admit my knowledge is not advanced enough to explain how that would be possible, but I must assume that, if physicists are making such statements, the argument must be fairly robust.

So… Whatever charge or whatever force we are talking about, we’ll be thinking of waves or oscillations—or simply movement, but it’s always a movement in a force field, and so there’s power and energy involved (energy is force times distance, and power is the time rate of change of energy). So, yes, we should expect the same issues in regard to scale. And so that’s what’s captured by h.

As we’re talking the smallest things possible, I should also mention that there are also other inconsistencies in the electromagnetic theory, which should (also) have their parallel for other forces. For example, the idea of a point charge is mathematically inconsistent, as I show in my post on fields and charges. Charge, any charge really, must occupy some space. It cannot all be squeezed into one dimensionless point. So the reasoning behind the Planck time and distance scale is surely valid.

In short, the whole argument about the Planck scale and those limits is very valid. However, does it imply our thinking about the Planck scale is actually relevant? I mean: it’s not because we can imagine how things might look like  – they may look like those tiny little black holes, for example – that these things actually exist. GUT or string theorists obviously think they are thinking about something real. But, frankly, Feynman had a point when he said what he said about string theory, shortly before his untimely death in 1988: “I don’t like that they’re not calculating anything. I don’t like that they don’t check their ideas. I don’t like that for anything that disagrees with an experiment, they cook up an explanation—a fix-up to say, ‘Well, it still might be true.'”

It’s true that the so-called Standard Model does not look very nice. It’s not like Maxwell’s equations. It’s complicated. It’s got various ‘sectors’: the electroweak sector, the QCD sector, the Higgs sector,… So ‘it looks like it’s got too much going on’, as a friend of mine said when he looked at a new design for mountainbike suspension. 🙂 But, unlike mountainbike designs, there’s no real alternative for the Standard Model. So perhaps we should just accept it is what it is and, hence, in a way, accept Nature as we can see it. So perhaps we should just continue to focus on what’s here, before we reach the Great Desert, rather than wasting time on trying to figure out how things might look like on the other side, especially because we’ll never be able to test our theories about ‘the other side.’

On the other hand, we can see where the Great Desert sort of starts (somewhere near the 1032 Hz scale), and so it’s only natural to think it should also stop somewhere. In fact, we know where it stops: it stops at the 1043 Hz scale, because everything beyond that doesn’t make sense. The question is: is there actually there? Like fundamental strings or whatever you want to call it. Perhaps we should just stop where the Great Desert begins. And what’s the Great Desert anyway? Perhaps it’s a desert indeed, and so then there is absolutely nothing there. 🙂

Hmm… There’s not all that much one can say about it. However, when looking at the history of physics, there’s one thing that’s really striking. Most of what physicists can think of, in the sense that it made physical sense, turned out to exist. Think of anti-matter, for instance. Paul Dirac thought it might exist, that it made sense to exist, and so everyone started looking for it, and Carl Anderson found in a few years later (in 1932). In fact, it had been observed before, but people just didn’t pay attention, so they didn’t want to see it, in a way. […] OK. I am exaggerating a bit, but you know what I mean. The 1930s are full of examples like that. There was a burst of scientific creativity, as the formalism of quantum physics was being developed, and the experimental confirmations of the theory just followed suit.

In the field of astronomy, or astrophysics I should say, it was the same with black holes. No one could really imagine the existence of black holes until the 1960s or so: they were thought of a mathematical curiosity only, a logical possibility. However, the circumstantial evidence now is quite large and so… Well… It seems a lot of what we can think of actually has some existence somewhere. 🙂

So… Who knows? […] I surely don’t. And so I need to get back to the grind and work my way through the rest of Feynman’s Lectures and the related math. However, this was a nice digression, and so I am grateful to my brother he initiated it. 🙂

Maxwell’s equations and the speed of light

We know how electromagnetic waves travel through space: they do so because of the mechanism described in Maxwell’s equation: a changing magnetic field causes a changing electric field, and a changing magnetic field causes a (changing) electric field, as illustrated below.

Maxwell interaction

So we need some First Cause to get it all started 🙂 i.e. some current, i.e. some moving charge, but then the electromagnetic wave travels, all by itself, through empty space, completely detached from the cause. You know that by now – indeed, you’ve heard this a thousand times before – but, if you’re reading this, you want to know how it works exactly. 🙂

In my post on the Lorentz gauge, I included a few links to Feynman’s Lectures that explain the nitty-gritty of this mechanism from various angles. However, they’re pretty horrendous to read, and so I just want to summarize them a bit—if only for myself, so as to remind myself what’s important and not. In this post, I’ll focus on the speed of light: why do electromagnetic waves – light – travel at the speed of light?

You’ll immediately say: that’s a nonsensical question. It’s light, so it travels at the speed of light. Sure, smart-arse! Let me be more precise: how can we relate the speed of light to Maxwell’s equations? That is the question here. Let’s go for it.

Feynman deals with the matter of the speed of an electromagnetic wave, and the speed of light, in a rather complicated exposé on the fields from some infinite sheet of charge that is suddenly set into motion, parallel to itself, as shown below. The situation looks – and actually is – very simple, but the math is rather messy because of the rather exotic assumptions: infinite sheets and infinite acceleration are not easy to deal with. 🙂 But so the whole point of the exposé is just to prove that the speed of propagation (v) of the electric and magnetic fields is equal to the speed of light (c), and it does a marvelous job at that. So let’s focus on that here only. So what I am saying is that I am going to leave out most of the nitty-gritty and just try to get to that v = result as fast as I possibly can. So, fasten your seat belt, please.

sheet of charge

Most of the nitty-gritty in Feynman’s exposé is about how to determine the direction and magnitude of the electric and magnetic fields, i.e. E and B. Now, when the nitty-gritty business is finished, the grand conclusion is that both E and B travel out in both the positive as well as the negative x-direction at some speed v and sort of ‘fill’ the entire space as they do. Now, the region they are filling extends infinitely far in both the y- and z-direction but, because they travel along the x-axis, there are no fields (yet) in the region beyond x = ± v·t (t = 0 is the moment when the sheet started moving, and it moves in the positive y-direction). As you can see, the sheet of charge fills the yz-plane, and the assumption is that its speed goes from zero to u instantaneously, or very very quickly at least. So the E and B fields move out like a tidal wave, as illustrated below, and thereby ‘fill’ the space indeed, as they move out.

tidal wave

The magnitude of E and B is constant, but it’s not the same constant, and part of the exercise here is to determine the relationship between the two constants. As for their direction, you can see it in the first illustration: B points in the negative z-direction for x > 0 and in the positive z-direction for x < 0, while E‘s direction is opposite to u‘s direction everywhere, so E points in the negative y-direction. As said, you should just take my word for it, because the nitty-gritty on this – which we do not want to deal with here – is all in Feynman and so I don’t want to copy that.

The crux of the argument revolves around what happens at the wavefront itself, as it travels out. Feynman relates flux and circulation there. It’s the typical thing to do: it’s at the wavefront itself that the fields change: before they were zero, and now they are equal to that constant. The fields do not change anywhere else, so there’s no changing flux or circulation business to be analyzed anywhere else. So we define two loops at the wavefront itself: Γ1 and Γ2. They are normal to each other (cf. the top and side view of the situation below), because the E and B fields are normal to each other. And so then we use Maxwell’s equations to check out what happens with the flux and circulation there and conclude what needs to be concluded. 🙂

top view side view

We start with rectangle Γ2. So one side is in the region where there are fields, and one side is in the region where the fields haven’t reached yet. There is some magnetic flux through this loop, and it is changing, so there is an emf around it, i.e. some circulation of E. The flux changes because the area in which B exists increases at speed v. Now, the time rate of change of the flux is, obviously, the width of the rectangle L times the rate of change of the area, so that’s (B·L·v·Δt)/Δt = B·L·v, with Δt some differential time interval co-defining how slow or how fast the field changes. Now, according to Faraday’s Law (see my previous post), this will be equal to minus the line integral of E around Γ2, which is E·L. So E·L = B·L·v and, hence, we find: E = v·B.

Interesting! To satisfy Faraday’s equation (which is just one of Maxwell’s equations in integral rather than in differential form), E must equal B times v, with v the speed of propagation of our ‘tidal’ wave. Now let’s look at Γ1. There we should apply:

IntegralNow the line integral is just B·L, and the right-hand side is E·L·v, so, not forgetting that c2 in front—i.e. the square of the speed of light, as you know!—we get: c2B = E·v, or E = (c2/v)·B. 

Now, the E = v·B and E = (c2/v)·B equations must both apply (we’re talking one wave and one and the same phenomenon) and, obviously, that’s only possible if v = c2/v, i.e. if v = c. So the wavefront must travel at the speed of light! Waw ! That’s fast. 🙂 Yes. […] Jokes aside, that’s the result we wanted here: we just proved that the speed of travel of an electromagnetic wave must be equal to the speed of light.

As an added bonus, we also showed the mechanism of travel. It’s obvious from the equations we used to prove the result: it works through the derivatives of the fields with respect to time, i.e. ∂E/∂t and ∂B/∂t.

Done! Great! Enjoy the view!

Well… Yes and no. If you’re smart, you’ll say: we got this result because of the c2 factor in that equation, so Maxwell had already put it in, so to speak. Waw! You really are a smart-arse, aren’t you? 🙂

The thing is… Well… The answer is: no. Maxwell did not put it in. Well… Yes and no. Let me explain. Maxwell’s first equation was the electric flux law ·E = σ/ε0: the flux of E through a closed surface is proportional to the charge inside. So that’s basically an other way of writing Coulomb’s Law, and ε0 was just some constant in it, the electric constant. So it’s a constant of proportionality that depends on the unit in which we measure electric charge. The only reason that it’s there is to make the units come out alright, so if we’d measure charge not in coulomb (C) in a unit equal to 1 C/ε0, it would disappear. If we’d do that, our new unit would be equivalent to the charge of some 700,000 protons. You can figure that magical number yourself by checking the values of the proton charge and ε0. 🙂

OK. And then Faraday came up with the exact laws for magnetism, and they involved current and some other constant of proportionality, and Maxwell formalized that by writing ×B = μ0j, with μ0 the magnetic constant. It’s not a flux law but a circulation law: currents cause circulation of B. We get the flux rule from it by integrating it. But currents are moving charges, and so Maxwell knew magnetism was related to the same thing: electric charge. So Maxwell knew the two constants had to be related. In fact, when putting the full set of equations together – there are four, as you know – Maxwell figured out that μtimes εwould have to be equal to the reciprocal of c2, with the speed of propagation of the wave. So Maxwell knew that, whatever the unit of charge, we’d get two constants of proportionality, and electric and a magnetic constant, and that μ0·εwould be equal to 1/c2. However, while he knew that, at the time, light and electromagnetism were considered to be separate phenomena, and so Maxwell did not say that c was the speed of light: the only thing his equations told him was that is the speed of propagation of that ‘electromagnetic’ wave that came out of his equations.

The rest is history. In 1856, the great Wilhelm Eduard Weber – you’ve seen his name before, didn’t you? – did a whole bunch of experiments which measured the electric constant rather precisely, and Maxwell jumped on it and calculated all the rest, i.e. μ0, and so then he took the reciprocal of the square root of μ0·εand – Bang! – he had c, the speed of propagation of the electromagnetic wave he was thinking of. Now, was some value of the order of 3×108 m/s, and so that happened to be the same as the speed of light, which suggested that Maxwell’s c and the speed of light were actually one and the same thing!

Now, I am a smart-arse too 🙂 and, hence, when I first heard this story, I actually wondered how Maxwell could possibly know the speed of light at the time: Maxwell died many years before the Michelson-Morley experiment unequivocally established the value of the speed of light. [In case, you wonder: the Michelson-Morley experiment was done in 1887. So I check it. The fact is that the Michelson-Morley experiment concluded that the speed of light was an absolute value and that, in the process of doing so, they got a rather precise value for it, but the value of itself has already been established, more or less, that is, by a Danish astronomer, Ole Römer, in 1676 ! He did so by carefully observing the timing of the repeating eclipses of Io, one of Jupiter’s moons. Newton mentioned his results in his Principia, which he wrote in 1687, duly noting that it takes about seven to eight minutes for light to travel from the Sun to the Earth. Done! The whole story is fascinating, really, so you should check it out yourself. 🙂

In any case, to make a long story short, Maxwell was puzzled by this mysterious coincidence, but he was bold enough to immediately point to the right conclusion, tentatively at least, and so he told the Cambridge Philosophical Society, in the very same year, i.e. 1856, that “we can scarcely avoid the inference that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena.”

So… Well… Maxwell still suggests light needs some medium here, so the ‘medium’ is a reference to the infamous aether theory, but that’s not the point: what he says here is what we all take for granted now: light is an electromagnetic wave. So now we know there’s absolute no reason whatsoever to avoid the ‘inference’, but… Well… 160 years ago, it was quite a big deal to suggest something like that. 🙂

So that’s the full story. I hoped you like it. Don’t underestimate what you just did: understanding an argument like this is like “climbing a great peak”, as Feynman puts it. So it is “a great moment” indeed. 🙂 The only thing left is, perhaps, to explain the ‘other’ flux rules I used above. Indeed, you know Faraday’s Law:

emf

But that other one? Well… As I explained in my previous post, Faraday’s Law is the integral form of Maxwell’s second equation: −∂B/∂t = ×E. The ‘other’ flux rule above – so that’s the one with the c2 in front and without a minus sign, is the integral form of Maxwell’s fourth equation: c2×= j+ ∂E/∂t, taking into account that we’re talking a wave traveling in free space, so there are no charges and currents (it’s just a wave in empty space—whatever that means) and, hence, the Maxwell equation reduces to c2×= ∂E/∂t. Now, I could take you through the same gymnastics as I did in my previous post but, if I were you, I’d just apply the general principle that ”the same equations must yield the same solutions” and so I’d just switch E for B and vice versa in Faraday’s equation. 🙂

So we’re done… Well… Perhaps one more thing. We’ve got these flux rules above telling us that the electromagnetic wave will travel all by itself, through empty space, completely detached from its First Cause. But… […] Well… Again you may think there’s some trick here. In other words, you may think the wavefront has to remain connected to the First Cause somehow, just like the whip below is connected to some person whipping it. 🙂

Bullwhip_effect

There’s no such connection. The whip is not needed. 🙂 If we’d switch off the First Cause after some time T, so our moving sheet stops moving, then we’d have the pulse below traveling through empty space. As Feynman puts it: “The fields have taken off: they are freely propagating through space, no longer connected in any way with the source. The caterpillar has turned into a butterfly! 

wavefront

Now, the last question is always the same: what are those fields? What’s their reality? Here, I should refer you to one of the most delightful sections in Feynman’s Lectures. It’s on the scientific imagination. I’ll just quote the introduction to it, but I warmly recommend you go and check it out for yourself: it has no formulas whatsoever, and so you should understand all of it without any problem at all. 🙂

“I have asked you to imagine these electric and magnetic fields. What do you do? Do you know how? How do I imagine the electric and magnetic field? What do I actually see? What are the demands of scientific imagination? Is it any different from trying to imagine that the room is full of invisible angels? No, it is not like imagining invisible angels. It requires a much higher degree of imagination to understand the electromagnetic field than to understand invisible angels. Why? Because to make invisible angels understandable, all I have to do is to alter their properties a little bit—I make them slightly visible, and then I can see the shapes of their wings, and bodies, and halos. Once I succeed in imagining a visible angel, the abstraction required—which is to take almost invisible angels and imagine them completely invisible—is relatively easy. So you say, “Professor, please give me an approximate description of the electromagnetic waves, even though it may be slightly inaccurate, so that I too can see them as well as I can see almost invisible angels. Then I will modify the picture to the necessary abstraction.”

I’m sorry I can’t do that for you. I don’t know how. I have no picture of this electromagnetic field that is in any sense accurate. I have known about the electromagnetic field a long time—I was in the same position 25 years ago that you are now, and I have had 25 years more of experience thinking about these wiggling waves. When I start describing the magnetic field moving through space, I speak of the and fields and wave my arms and you may imagine that I can see them. I’ll tell you what I see. I see some kind of vague shadowy, wiggling lines—here and there is an E and a B written on them somehow, and perhaps some of the lines have arrows on them—an arrow here or there which disappears when I look too closely at it. When I talk about the fields swishing through space, I have a terrible confusion between the symbols I use to describe the objects and the objects themselves. I cannot really make a picture that is even nearly like the true waves. So if you have some difficulty in making such a picture, you should not be worried that your difficulty is unusual.

Our science makes terrific demands on the imagination. The degree of imagination that is required is much more extreme than that required for some of the ancient ideas. The modern ideas are much harder to imagine. We use a lot of tools, though. We use mathematical equations and rules, and make a lot of pictures. What I realize now is that when I talk about the electromagnetic field in space, I see some kind of a superposition of all of the diagrams which I’ve ever seen drawn about them. I don’t see little bundles of field lines running about because it worries me that if I ran at a different speed the bundles would disappear, I don’t even always see the electric and magnetic fields because sometimes I think I should have made a picture with the vector potential and the scalar potential, for those were perhaps the more physically significant things that were wiggling.

Perhaps the only hope, you say, is to take a mathematical view. Now what is a mathematical view? From a mathematical view, there is an electric field vector and a magnetic field vector at every point in space; that is, there are six numbers associated with every point. Can you imagine six numbers associated with each point in space? That’s too hard. Can you imagine even one number associated with every point? I cannot! I can imagine such a thing as the temperature at every point in space. That seems to be understandable. There is a hotness and coldness that varies from place to place. But I honestly do not understand the idea of a number at every point.

So perhaps we should put the question: Can we represent the electric field by something more like a temperature, say like the displacement of a piece of jello? Suppose that we were to begin by imagining that the world was filled with thin jello and that the fields represented some distortion—say a stretching or twisting—of the jello. Then we could visualize the field. After we “see” what it is like we could abstract the jello away. For many years that’s what people tried to do. Maxwell, Ampère, Faraday, and others tried to understand electromagnetism this way. (Sometimes they called the abstract jello “ether.”) But it turned out that the attempt to imagine the electromagnetic field in that way was really standing in the way of progress. We are unfortunately limited to abstractions, to using instruments to detect the field, to using mathematical symbols to describe the field, etc. But nevertheless, in some sense the fields are real, because after we are all finished fiddling around with mathematical equations—with or without making pictures and drawings or trying to visualize the thing—we can still make the instruments detect the signals from Mariner II and find out about galaxies a billion miles away, and so on.

The whole question of imagination in science is often misunderstood by people in other disciplines. They try to test our imagination in the following way. They say, “Here is a picture of some people in a situation. What do you imagine will happen next?” When we say, “I can’t imagine,” they may think we have a weak imagination. They overlook the fact that whatever we are allowed to imagine in science must be consistent with everything else we know: that the electric fields and the waves we talk about are not just some happy thoughts which we are free to make as we wish, but ideas which must be consistent with all the laws of physics we know. We can’t allow ourselves to seriously imagine things which are obviously in contradiction to the known laws of nature. And so our kind of imagination is quite a difficult game. One has to have the imagination to think of something that has never been seen before, never been heard of before. At the same time the thoughts are restricted in a strait jacket, so to speak, limited by the conditions that come from our knowledge of the way nature really is. The problem of creating something which is new, but which is consistent with everything which has been seen before, is one of extreme difficulty.”

Isn’t that great? I mean: Feynman, one of the greatest physicists of all time, didn’t write what he wrote above when he was a undergrad student or so. No. He did so in 1964, when he was 45 years old, at the height of his scientific career! And it gets better, because Feynman then starts talking about beauty. What is beauty in science? Well… Just click and check what Feynman thinks about it. 🙂

Oh… Last thing. So what is the magnitude of the E and B field? Well… You can work it out yourself, but I’ll give you the answer. The geometry of the situation makes it clear that the electric field has a y-component only, and the magnetic field a z-component only. Their magnitudes are given in terms of J, i.e. the surface current density going in the positive y-direction:

equation

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 Ψ(x ,t) – there is also a phi – i.e. the momentum-space wave function Φ(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.

Standard_Model_of_Elementary_Particles

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 – or under it, as it’s suspended under the Antarctic ice, and it regularly captures high-energy neutrinos in the range of 1 to 10 TeV. 

Let me – to conclude this introduction – just 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〉 and ¦1〉 are the basis states, then any linear combination α·¦0〉 + ß·¦1〉 is also a valid state provided│α│2 + │β│= 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 〈 ¦ – gives the final condition. They are combined in statements such as 〈 particle arrives at x¦particle leaves from s〉 or – in short – 〈 x¦s〉 and, while x and s would have some real-number value, 〈 x¦s〉 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). 🙂

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 (±1) – as we have for the electric charge. So it’s just another type of qubit in quantum mechanics.

[Note that the so-called elementary ±1 values for electric charge are not really elementary: it’s –1/3 (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 ±1 value is easily found from a linear combination of the –1/3 and +2/3 values.]

3. Z and W bosons carry the so-called weak force, aka as Fermi’s 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 ± 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 β decay (e.g. the above-mentioned carbon-14 decay) as well as β+ 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± and Zbosons are very heavy (157,000 and 178,000 times heavier than a electron!), and W± 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 – 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’s 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’s 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.

2000px-Elementary_particle_interactions.svg

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 Δ. [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 (Δx, Δt). [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 Δx 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 Δt. :-)]

If we want to do that, then we will need to square the modulus of the corresponding wave function Ψ(x, t). To be precise, we will have to do a summation of all the values │Ψ(x, t)│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 Φ(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: ““Out of these three amplitudes, we can make the whole world, aside from what goes on in nuclei, and gravitation, as always!” 

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 –0.1 (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: –0.1 is 0.1eiπ 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 π (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 n2netcetera 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. 🙂 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 Schrödinger 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 Schrödinger 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 = Δr– Δt2 =  (z2– z1)+ (y2– y1)+ (x2– x1)– (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’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. [If we don’t want to transform time or distance scales, then we have to write I as I = c2Δt2 – Δr2.] Now, there are three types of intervals:

  1. For time-like intervals, we have a negative value for I, so Δt> Δr2. For two events separated by a time-like interval, enough time passes between them so there could be a cause–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 Δt< Δr2. 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 Δt2 = Δr2. 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 (+++–) here for I. There’s also a time-like convention, i.e. with +––– as signs: I = Δt2 – Δrso 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

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) 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 “dippy process” and “having to resort to such hocus-pocus has prevented us from proving that the theory of quantum electrodynamics is mathematically self-consistent“. He adds: “It’s surprising that the theory still hasn’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]  is not mathematically legitimate.” […] Now, Mr Feynman writes this about quantum electrodynamics, not about “the rest of physics” (and so that’s quantum chromodynamics (QCD) – the theory of the strong interactions – and quantum flavordynamics (QFD) – the theory of weak interactions) which, he adds, “has not been checked anywhere near as well as electrodynamics.” 

That’s a pretty damning statement, isn’t 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 !