Some thoughts on the nature of reality

Some other comment on an article on my other blog, inspired me to structure some thoughts that are spread over various blog posts. What follows below, is probably the first draft of an article or a paper I plan to write. Or, who knows, I might re-write my two books on quantum physics and publish a new edition soon. 馃檪

Physical dimensions and Uncertainty

The physical dimension of the quantum of action (h or聽魔 = h/2蟺) is force (expressed in newton)聽times distance (expressed in meter)聽times time (expressed in seconds): N路m路s. This is also the unit in which angular momentum is expressed. Of course, a force of one newton will give a mass of 1 kg an acceleration of 1 m/s per second. Therefore, 1 N = 1 kg路m/s2聽and the physical dimension of h, or the unit of angular momentum, may also be written as 1 N路m路s = 1 (kg路m/s2)路m路s = 1 kg路m2/s.

The newton is a聽derived聽unit in the metric system, as opposed to the units of mass, length and time (kg, m, s). Nevertheless, I like to think of the quantum of action as representing the three fundamental physical dimensions: (1)聽force, (2)聽time and (3) distance – or space. We may then look at energy and (linear) momentum as physical quantities combining (1) force and distance and (2) force and time respectively.

  1. Force times length (think of force that is聽acting on some object over some distance) is energy: 1 joule聽(J) =聽1 newtonmeter (N). Hence, we may think of the concept of energy as a projection聽of action in space only: we make abstraction of time. The physical dimension of the quantum of action should then be written as [h] = [E]路[t]
  2. Conversely, the magnitude of linear momentum (p = m路v) is expressed in newtonseconds: 1 kg路m/s = 1 (kg路m/s2)路s = 1 N路s. Hence, we may think of (linear) momentum as a projection of action in time only: we make abstraction of its spatial dimension. Think of a force that is acting on some object聽during some time.聽The physical dimension of the quantum of action should then be written as [h] = [p]路[x]

Of course, a force that is acting on some object during some time, will usually also act on the same object over some distance but… Well… Just try to make abstraction of one of the two dimensions here: time聽or聽distance. It is a difficult thing to do because, when everything is said and done, we don’t live in space or in time alone, but in spacetime and, hence, such abstractions are not easy. Also, the聽principle of least action聽in physics tells us it’s action that matters:

  1. In classical physics, the path of some object in a force field will minimize聽the total action (which is usually written as S) along that path.
  2. In quantum mechanics, the same action integral will give us various values S – each corresponding to a particular path – and each path (and, therefore, each value of S, really) will be associated with a probability amplitude that will be proportional to some constant times e鈭抜路胃聽=聽ei路(S/魔). Because is so tiny, even a small change in S will give a completely different phase angle 胃. Therefore, most amplitudes will cancel each other out as we take the sum of the amplitudes over all possible paths: only the paths that nearly聽give the same phase matter. In practice, these are the paths that are associated with a variation in S of an order of magnitude that is equal to .

The paragraph above summarizes, in essence, Feynman’s path integral formulation of quantum mechanics. We may, therefore, think of the quantum of action聽expressing聽itself (1) in time only, (2) in space only, or – much more likely – (3) expressing itself in both dimensions at the same time. Hence, if the quantum of action gives us the order of magnitude聽of the uncertainty, we may re-write our dimensional [] = [E]路[t] and [] = [p]路[x] equations as the uncertainty equations:

  • 螖E路螖t =
  • 螖p路螖x =

It is best to think of the uncertainty relations as a聽pair聽of equations, if only because you should also think of the concept of energy and momentum as representing different aspects聽of the same reality, as evidenced by the (relativistic) energy-momentum relation (E2聽= p2c2聽鈥 m02c4). Also, as illustrated below, the actual path – or, to be more precise, what we might associate with the concept of the actual path – is likely to be some mix of 螖x and 螖t. If 螖t is very small, then 螖x will be very large. In order to move over such distance, our particle will require a larger energy, so 螖E will be large. Likewise, if 螖t is very large, then 螖x will be very small and, therefore, 螖E will be very small. You can also reason in terms of 螖x, and talk about momentum rather than energy. You will arrive at the same conclusions: the 螖E路螖t = h and 螖p路螖x = h聽relations represent two aspects of the same reality – or, at the very least, what we might聽think聽of as reality.


We will not further dwell on this here. We want to do some more thinking about those physical dimensions. The idea of a force implies the idea of some object – of some mass on which the force is acting. Hence, let’s think about the concept of mass now.

Note: The actual uncertainty relations have a factor 1/2 in them. This may be explained by thinking of both negative as well as positive variations in space and in time.

Action, energy and mass

Let’s look at the concept of energy once more. What is聽energy, really? In聽real life, we are usually not interested in the energy of a system as such, but by the energy it can deliver, or absorb, per second. This is referred to as the聽power聽of a system, and it’s expressed in J/s. However, in physics, we always talk energy, so what is the energy of a system?

We should – and will – obviously think of the kinetic energy of its parts, their potential energy, their rest聽energy, and – for an atomic system – we may add some internal energy, which may be binding energy, or excitation energy (think of a hydrogen atom in an excited state, for example). Einstein’s mass-equivalence formula comes to mind here: E = m路c2. [The m here refers to mass – not to meter, obviously.] But then… Well… What is it, really?

As I explained in several posts, it is very tempting to think of energy as some kind of two-dimensional oscillation of mass. A force over some distance will cause a mass to accelerate. This is reflected in the聽dimensional analysis:

[E] = [m]路[c2] = 1 kg路m2/s2聽= 1 kg路m/s2路m = 1 N路m

The kg and m/s2聽factors make this abundantly clear: m/s2聽is the physical dimension of acceleration: (the change in) velocity per time unit.

Other formulas now come to mind, such as the Planck-Einstein relation: E = h路f = 蠅路魔. We could also write: E = h/T. Needless to say, T = 1/f聽is the聽period聽of the oscillation. So we could say, for example, that the energy of some particle times the period of the oscillation gives us Planck’s constant again. What does that mean? Perhaps it’s easier to think of it the other way around: E/f = h = 6.626070040(81)脳10鈭34聽J路s. Now, f聽is the number of oscillations聽per second. Let’s write it as聽f聽= n/s, so we get:

E/f聽= E/(n/s) = E路s/n聽= 6.626070040(81)脳10鈭34聽J路s 鈬 E/n聽= 6.626070040(81)脳10鈭34聽J

What an amazing result! Our wavicle – be it a photon or a matter-particle – will always聽pack聽6.626070040(81)脳10鈭34joule聽in聽one聽oscillation, so that’s the numerical聽value of Planck’s constant which, of course, depends on our fundamental聽units (i.e. kg, meter, second, etcetera in the SI system).

Of course, the obvious question is: what’s one聽oscillation? If it’s a wave packet, the oscillations may not have the same amplitude, and we may also not be able to define an exact period. In fact, we should expect the amplitude and duration of each oscillation to be slightly different, shouldn’t we? And then…

Well… What’s an oscillation? We’re used to聽counting聽them:聽n聽oscillations per second, so that’s聽per time unit. How many do we have in total? We wrote about that in our posts on the shape and size of a photon. We know photons are emitted by atomic oscillators – or, to put it simply, just atoms going from one energy level to another. Feynman calculated the Q of these atomic oscillators: it鈥檚 of the order of 108聽(see his聽Lectures,聽I-33-3: it鈥檚 a wonderfully simple exercise, and one that really shows his greatness as a physics teacher), so… Well… This wave train will last about 10鈥8聽seconds (that鈥檚 the time it takes for the radiation to die out by a factor 1/e). To give a somewhat more precise example,聽for sodium light, which has a frequency of 500 THz (500脳1012聽oscillations per second) and a wavelength of 600 nm (600脳10鈥9聽meter), the radiation will lasts about 3.2脳10鈥8聽seconds. [In fact, that鈥檚 the time it takes for the radiation鈥檚 energy to die out by a factor 1/e, so(i.e. the so-called decay time 蟿), so the wavetrain will actually last聽longer, but so the amplitude becomes quite small after that time.]聽So… Well… That鈥檚 a very short time but… Still, taking into account the rather spectacular frequency (500 THz) of sodium light, that makes for some 16 million oscillations and, taking into the account the rather spectacular speed of light (3脳108聽m/s), that makes for a wave train with a length of, roughly,聽9.6 meter. Huh? 9.6 meter!? But a photon is supposed to be pointlike, isn’it it? It has no length, does it?

That’s where relativity helps us out: as I wrote in one of my posts, relativistic length contraction may explain the apparent paradox. Using the reference frame of the photon聽– so if we’d be traveling at speed c,鈥 riding鈥 with the photon, so to say, as it鈥檚 being emitted – then we’d 鈥榮ee鈥 the electromagnetic transient as it鈥檚 being radiated into space.

However, while we can associate some mass聽with the energy of the photon, none of what I wrote above explains what the (rest) mass of a matter-particle could possibly be.There is no real answer to that, I guess. You’ll think of the Higgs field now but… Then… Well. The Higgs field is a scalar field. Very simple: some number that’s associated with some position in spacetime. That doesn’t explain very much, does it? 馃槮 When everything is said and done, the scientists who, in 2013 only, got the Nobel Price for their theory on the Higgs mechanism, simply tell us mass is some number. That’s something we knew already, right? 馃檪

The reality of the wavefunction

The wavefunction is, obviously, a mathematical construct: a聽description聽of reality using a very specific language. What language? Mathematics, of course! Math may not be universal (aliens might not be able to decipher our mathematical models) but it’s pretty good as a global聽tool of communication, at least.

The real聽question is: is the description聽accurate? Does it match reality and, if it does, how聽good聽is the match? For example, the wavefunction for an electron in a hydrogen atom looks as follows:

蠄(r, t) = ei路(E/魔)路tf(r)

As I explained in previous posts (see, for example, my recent post聽on reality and perception), the聽f(r) function basically provides some envelope for the two-dimensional ei路胃聽=聽ei路(E/魔)路t聽= cos胃 + isin胃聽oscillation, with r= (x, y, z),聽胃 = (E/魔)路t聽= 蠅路t聽and 蠅 = E/魔. So it presumes the聽duration of each oscillation is some constant. Why? Well… Look at the formula: this thing has a constant frequency in time. It’s only the amplitude that is varying as a function of the r= (x, y, z) coordinates. 馃檪 So… Well… If each oscillation is to always聽pack聽6.626070040(81)脳10鈭34joule, but the amplitude of the oscillation varies from point to point, then… Well… We’ve got a problem. The wavefunction above is likely to be an approximation of reality only. 馃檪 The associated energy is the same, but… Well… Reality is probably聽not聽the nice geometrical shape we associate with those wavefunctions.

In addition, we should think of the聽Uncertainty Principle: there聽must聽be some uncertainty in the energy of the photons when our hydrogen atom makes a transition from one energy level to another. But then… Well… If our photon packs something like 16 million oscillations, and the order of magnitude of the uncertainty is only of the order of聽h聽(or 魔 = h/2蟺) which, as mentioned above, is the (average) energy of one聽oscillation only, then we don’t have much of a problem here, do we? 馃檪

Post scriptum: In previous posts, we offered some analogies – or metaphors – to a two-dimensional oscillation (remember the V-2 engine?). Perhaps it’s all relatively simple. If we have some tiny little ball of mass – and its center of mass has to stay where it is – then any rotation – around any axis – will be some combination of a rotation around our聽x- and z-axis – as shown below. Two axes only. So we may want to think of a two-dimensional聽oscillation as an oscillation of the polar and azimuthal angle. 馃檪

oscillation of a ball

The classical explanation for the electron’s mass and radius

Feynman’s 28th聽Lecturein his series on electromagnetism is one of the more interesting but, at the same time, it’s one of the few聽Lectures聽that is clearly (out)dated. In essence, it talks about the difficulties involved in applying Maxwell’s equations to the聽elementary charges聽themselves, i.e. the electron and the proton.聽We already signaled some of these problems in previous posts. For example, in our post on the energy in electrostatic fields, we showed how our formulas for the field energy and/or the potential of a charge blow up when we use it to calculate the energy we’d need to聽assemble聽a point charge. What comes out is infinity: 鈭. So our formulas tell us we’d need an infinite amount of energy to assemble a point charge.

Well… That’s no surprise, is it? The idea itself is impossible: how can one have a finite amount of charge in something that’s infinitely small? Something that has no size whatsoever? It’s pretty obvious we get some division by zero there. 馃檪 The mathematical聽approach is often inconsistent. Indeed, a lot of blah-blah聽in physics is obviously just about applying formulas to situations that are clearly not聽within the relevant area of application of the formula.聽So that’s why I went through the trouble (in my previous post, that is) of explaining you how we get聽these energy and potential formulas, and that’s by bringing charges聽(note the plural) together. Now, we may assume these charges are point charges, but that assumption is not so essential. What I tried to say when being so explicit聽was the following: yes, a聽charge causes a field, but the聽idea聽of a potential makes sense only when we’re thinking of placing some聽other charge in that field. So point charges with ‘infinite energy’ should not be a problem. Feynman admits as much when he writes:

“If the energy can鈥檛 get out, but must stay there forever, is there any real difficulty with an infinite energy? Of course, a quantity that comes out infinite may be annoying, but what really matters is only whether there are any observable physical effects.”

So… Well… Let’s see. There’s another, more interesting, way to look at an electron: let’s have a look at the field it creates. A electron 鈥 stationary or moving 鈥撀爓ill create a field in Maxwell’s world, which we know inside out now. So let’s just calculate it. In fact, Feynman calculates it for the unit charge (+1), so that’s a positron. It eases the analysis because we don’t have to drag any minus sign along. So how does it work? Well…

We’ll have an聽energy flux density vector聽鈥 i.e. the Poynting vector S聽鈥 as well as a momentum density聽vector g all over space. Both are related through the g = S/c2聽equation which, as I explained in my previous post, is probably best written as cg = S/c, because we’ve got units then, on both sides, that we can readily understand, like N/m2 (so that’s force per unit area) or J/m3聽(so that’s energy per unit volume). On the other hand, we’ll need something that’s written as a function of the velocity of our positron, so that’s v, and so it’s probably best to just calculate g, the momentum,聽which is measured in N路s or kg路(m/s2)路s (both are equivalent units for the聽momentum p = mv, indeed) per unit volume聽(so we need to add a 1/聽m3聽to the unit).聽So we’ll have some integral all over space, but I won’t bother you with it. Why not? Well… Feynman uses a rather particular volume element to solve the integral, and so I want you to focus on the solution. The geometry of the situation, and the solution for g, i.e. the momentum of the field per unit volume,聽is what matters here.

So let’s look at that geometry. It’s depicted below. We’ve got a radial electric field鈥攁 Coulomb field really, because our charge is moving at a non-relativistic speed, so v << c and we can approximate with a Coulomb field indeed. Maxwell’s equations imply that B = vE/c2, so g = 蔚0EB聽is what it is in the illustration below. Note that we’d have to reverse the direction of both E and B for an electron (because it’s negative), but g would be the same. It is directed obliquely toward the line of motion and its magnitude is g = (蔚0v/c2)路E2路sin胃. Don’t worry about it: Feynman integrates this thing for you. 馃檪 It’s not聽that聽difficult, but still… To solve it, he uses the fact that the fields are symmetric about the line of motion, which is indicated by the little聽arrow around the v-axis, with the 桅 symbol next to it (it symbolizes the potential). [The ‘rather particular volume element’ is a ring around the v-axis, and it’s because of this symmetry that Feynman picks the ring. Feynman’s Lectures are not only聽great to learn physics: they’re a treasure drove of mathematical tricks too. :-)]

momentum field

As said, I don’t want to bother you with the technicalities of the integral here. This is the result:


What does this say? It says that the momentum of the field 鈥 i.e. the electromagnetic momentum, integrated over all of space聽鈥 is proportional to the velocity v of our charge. That makes sense: when v = 0, we’ll have an electrostatic field all over space and, hence, some inertia, but it’s only when we try to聽move聽our charge that Newton’s Law comes into play: then we’ll need聽some聽force聽to overcome that inertia. It all works through the Poynting formula: S = EB/渭0. If nothing’s moving, then B = 0, and so we’ll have some E聽and, therefore, we’ll have field energy alright, but the energy聽flow聽will be zero. But when we move the charge, we’re moving the field, and so then B 鈮 0 and so it’s through B that the E in our S equation start kicking in. Does that make sense? Think about it: it’s good to try to visualize things in your mind. 馃檪

The constants in the聽proportionality constant (2e2)/(3ac2) of our pv formula above are:

  • e2=聽qe2/(4蟺蔚0), with聽qe聽the electron charge (without the minus sign) and 蔚0聽our ubiquitous electric constant. [Note that, unlike Feynman, I prefer to聽not聽write e in italics, so as to聽not聽confuse it with Euler’s number聽e聽鈮埪2.71828 etc. However, I know I am not always consistent in my notation. :-/ We don’t need Euler’s number in this post, so e or e聽is always an expression for the electron charge,聽not聽Euler’s number. Stupid remark, perhaps, but I don’t want you to be confused.]
  • a is the radius of our charge鈥攕ee we got away from the idea of a point charge? 馃檪
  • c2聽is just c2, i.e. our weird constant (the square of the speed of light) which seems to connect everything to everything. Indeed, think about stuff like this: S/g = c2聽= 1/(蔚00).

Now, p = mv, so that formula for p basically says that our elementary charge (as mentioned, g is the same for a positron or an electron: E and B聽will be reversed, but g is not) has an聽electromagnetic mass melec聽equal to:

emm - 2

That’s an amazing result. We don’t need to give our electron any聽rest mass: just its charge and its movement will do!聽Super!聽So we don’t need any Higgs fields here! 馃檪 The electromagnetic field will do!

Well… Maybe. Let’s explore what we’ve got here.

First, let’s compare that radius a in our formula to what’s found in experiments.聽Huh? Did someone ever try to measure the electron radius? Of course. There are all these聽scattering experiments in which electrons get fired at atoms. They can fly through or, else, hit something. Therefore, one can some statistical analysis and determine what is referred to as a聽cross-section. A cross-section is denoted by the same symbol as the standard deviation: 蟽 (sigma). In any case… So there’s something that’s referred to as the classical electron radius, and it’s equal to the so-called聽Thomsom scattering length.聽Thomson scattering, as opposed to Compton聽scattering, is elastic scattering, so it preserves kinetic energy (unlike Compton scattering, where energy gets absorbed and changes frequencies). So… Well… I won’t go into too much detail but, yes, this is the聽electron聽radius we need. [I am saying this rather explicitly because there are two other numbers around: the so-called Bohr radius聽and, as you might imagine, the Compton聽scattering cross-section.]

The Thomson scattering length is聽2.82 femtometer (so that’s聽2.82脳10鈭15聽m), more or less that is :-), and it’s usually related to the聽observed聽electron mass聽me聽through the fine-structure constant聽伪. In fact, using Planck units, we can write:聽 re路me=聽伪, which is an amazing formula but, unfortunately, I can’t dwell on it here. Using ordinary m, s, C and what have you units, we can write re聽as:

classical electron radius

That’s good, because if we equate聽me聽and melec聽and switch melec聽and a in our formula for melec, we get:


So, frankly, we’re聽spot on!聽Well… Almost. The two numbers differ by 1/3. But who cares about a 1/3 factor indeed? We’re talking rather fuzzy stuff here 鈥 scattering cross-sections and standard deviations and all that 鈥 so… Yes. Well done! Our theory works!

Well… Maybe. Physicists don’t think so. They think the 1/3 factor is聽an issue. It’s聽sad because it really makes a lot of sense. In fact, the Dutch physicist Hendrik Lorentz聽鈥 whom we know so well by now 馃檪聽鈥 had also worked out that, because of the length contraction effect, our spherical charge would contract into an ellipsoid and… Well… He worked it all out, and it was not a problem: he found that the momentum was altered by the factor聽(1鈭抳2/c2)鈭1/2, so that’s the ubiquitous Lorentz factor聽纬! He got this formula in the 1890s already, so that’s聽long before the theory of relativity had been developed. So, many years before Planck and Einstein would come up with their聽stuff, Hendrik Antoon聽Lorentz had the correct formulas already: the mass, or everything really, all should vary with that 纬-factor. 馃檪

Why bother about the 1/3 factor? [I should note it’s actually referred to as the 4/3 problem in physics.] Well… The critics do have a point: if we assume that (a) an electron is聽not聽a point charge聽鈥 so if we allow it to have some radius a聽鈥 and (b) that Maxwell’s Laws apply, then we should go all the way. The聽energy聽that’s needed to聽assemble聽an electron should then, effectively, be the same as the value we’d get out of those field energy formulas. So what do聽we get when we apply those formulas? Well… Let me quickly copy Feynman as he does the calculation for an electron, not聽looking at it as a point particle, but as a tiny聽shell聽of charge, i.e. a sphere with all charge sitting on the surface:

Feynman energy

聽Let me enlarge the formula:

energy electron

Now, if we combine that with our formula for聽melec聽above, then we get:

4-3 problem

So that formula does not聽respect Einstein’s universal mass-energy equivalence formula E = mc2. Now, you聽will聽agree that we really want Einstein’s mass-energy equivalence relation to be respected by all, so our electron should respect it too. 馃檪 So, yes, we’ve got a problem here, and it’s referred to as the聽4/3 problem (yes, the ratio got turned around).

Now, you may think it got solved in the meanwhile. Well… No. It’s still a bit of a puzzle today, and the current-day explanation is not really different from what the French scientist Henri Poincar茅聽proposed as a ‘solution’ to the problem back in the 1890s. He basically told Lorentz the following: “If the electron is some聽little聽ball of charge, then it should explode because of the repulsive forces inside. So there should be some binding forces there, and so that energy explains the ‘missing mass’ of the electron.” So these forces are effectively being referred to as Poincar茅 stresses, and the non-electromagnetic energy that’s associated with them聽鈥 which, of course,聽has to be equal to 1/3 of the electromagnetic energy (I am sure you see why) 馃檪聽鈥 adds to the聽total聽energy and all is alright now. We get:

U= mc2聽= (melec聽+聽mPoincar茅)c2

So… Yes… Pretty ad hoc.聽Worse, according to the Wikipedia article on electromagnetic mass, that’s still where we are. And, no, don’t read Feynman’s overview of all of the theories that were around then (so that’s in the 1960s, or earlier). As I said, it’s the聽one聽Lecture聽you don’t want to waste time on. So I won’t do that either.

In fact, let me try to do something else here, and that’s to de-construct the whole argument really. 馃檪聽Before I do so, let me highlight the聽essence聽of what was written above. It’s quite amazing really.聽Think of it: we say that the mass of an electron 鈥 i.e.聽its inertia, or the proportionality factor in Newton’s聽F = m路a聽law of motion聽鈥 is the energy in the electric and magnetic field it causes. So the electron itself is just a聽hook for the force law, so to say. There’s nothing there, except for the charge causing the field. But so its mass is everywhere and, hence, nowhere really. Well… I should correct that: the field strength falls of as 1/r2聽and, hence, the energy flow and momentum density that’s associated with it, falls of as 1/r4, so it falls of very聽rapidly and so the bulk of the energy is pretty near the charge. 馃檪

[Note: You’ll remember that the field that’s associated with electromagnetic聽radiation聽falls of as 1/r, not as 1/r2, which is why there is an energy flux there which is never lost, which can travel independently through space. It’s not the same here, so don’t get confused.]

So that’s something to note: the聽melec聽= (2c鈭2/3)路(e2/a) has the radius聽a聽in it, but that radius is only the聽hook, so to say. That’s fine, because it is not inconsistent with the聽idea聽of the Thomson scattering cross-section, which is the area that one聽can聽hit. Now, you’ll wonder how one can hit an electron: you can readily imagine an electron beam aimed at nuclei, but how would one hit electrons? Well… You can shoot photons at them, and see if they bounce back elastically or non-elastically. The cross-section area that bounces them off elastically must be pretty ‘hard’, and the cross-section that deflects them non-elastically somewhat less so. 馃檪

OK… But… Yes?Hey! How did we get that electron radius in that formula?聽

Good question! Brilliant, in fact! You’re right: it’s here that the whole argument falls apart really. We did a substitution. That radius a is the radius of a spherical聽shell of charge聽with an energy that’s equal to Uelec聽= (1/2)路(e2/a), so there’s another way of stating the inconsistency: the equivalent energy of melec聽= (2c鈭2/3)路(e2)/a) is equal to E =聽melecc2聽= (2/3)路(e2/a) and that’s not聽the same as Uelec聽= (1/2)路(e2/a). If we take the ratio of Uelec聽and melecc2聽=, we get the same factor: (1/2)/(2/3) = 3/4. But… Your question is superb! Look at it: putting it the way we put it reveals the inconsistency in the whole argument. We’re mixing two things here:

  1. We first calculate the momentum density, and the momentum, that’s caused聽by the unit charge,聽so we get some energy which I’ll denote as Eelec聽=聽melecc2
  2. Now, we then assume this energy must be聽equal to the energy that’s needed to assemble聽the unit charge from an infinite number of infinitesimally small charges, thereby also assuming聽the unit charge is a uniformly charged sphere of charge with radius a.
  3. We then use this radius a to simplify our formula for Eelec聽=聽melecc2

Now that聽is not kosher, really! First, it’s (a) a lot of assumptions, both implicit as well as explicit, and then (b) it’s, quite simply, not a legit mathematical procedure: calculating the energy in the field, or calculating the energy we need to assemble a uniformly charged sphere of radius a are two聽very聽different things.

Well… Let me put it differently. We’re using the same laws 鈥 it’s all Maxwell’s equations, really 鈥 but we should be clear about what we’re doing with them, and those two things are very聽different.聽The legitimate conclusion must be that our a is wrong. In other words, we should not assume that our electron is spherical shell of charge. So then what? Well… We could easily imagine something else, like a uniform or even a non-uniformly charged sphere. Indeed, if we’re just filling empty space with infinitesimally small charge ‘elements’, then we may want to think the density at the ‘center’ will be much higher, like what’s going on when planets form: the density of the inner core of our own planet Earth is more than four times the density of its surface material. [OK. Perhaps not very relevant here, but you get the idea.] Or, conversely, taking into account Poincar茅’s objection, we may want to think all of the charge will be on the surface, just like on a perfect conductor, where all charge is surface charge!

Note that the field outside of a uniformly charged sphere and the field of a spherical聽shell聽of charge is exactly the same, so we would聽not聽find a different number for聽Eelec聽=聽melecc2, but we surely would find a different number for Uelec. You may want to look up some formulas here: you’ll find that the energy of a uniformly distributed sphere聽of charge (so we do not聽assume that all of the charge sits on the surface here)聽is equal to (3/5)路(e2/a). So we’d already have much less of a problem, because the 3/4 factor in the Uelec聽= (3/4)路melecc2聽becomes a (5/3)路(2/3) = 10/9 factor. So now we have a discrepancy of some 10% only. 馃檪

You’ll say: 10% is 10%. It’s huge in physics, as it’s supposed to be an聽exact science.聽Well… It is and it isn’t. Do you realize we haven’t even started to talk about stuff like聽spin? Indeed, in modern physics, we think of聽electrons as something that also spins聽around one or the other axis, so there’s energy there too, and we didn’t include that in our analysis.

In short, Feynman’s approach here is disappointing. Naive聽even, but then… Well… Who knows? Perhaps he didn’t do this聽Lecture聽himself. Perhaps it’s just an assistant or so.聽In fact, I should wonder why there’s still physicists wasting time on this!聽I should also note that naively comparing that a radius with the classical electron radius also makes little or no sense. Unlike what you’d expect, the classical electron radius re聽and the Thomson scattering cross-section 蟽e聽are not聽related like you might think they are, i.e. like 蟽e聽= 蟺路re2聽or 蟽e聽= 蟺路(re/2)2聽or聽蟽e聽=聽re2聽or 蟽e聽= 蟺路(2路re)2聽or whatever circular surface calculation rule that might make sense here.聽No. The Thomson scattering cross-section is equal to:

e聽=聽(8蟺/3)路re2聽= (2蟺/3)路(2路re)2聽=聽(2/3)路蟺路(2路re)2聽鈮 66.5脳10鈭30聽m2聽= 66.5 (fm)2

Why? I am not sure. I must assume it’s got to do with the standard deviation and all that. The point is, we’ve got a 2/3 factor here too, so do we have a problem really? I mean… The a we got was equal to a = (2/3)路re, wasn’t it? It was. But, unfortunately, it doesn’t mean anything. It’s just a coincidence. In fact, looking at the Thomson scattering cross-section, instead of the Thomson scattering radius, makes聽the ‘problem’ a little bit worse. Indeed, applying the 蟺路r2聽rule for a circular surface, we get that the radius聽would be equal to (8/3)1/2re聽鈮 1.633路re, so we get something that’s much聽larger聽rather than something that’s smaller here.

In any case, it doesn’t matter. The point is: this kind of comparisons should not be taken too seriously. Indeed, when everything is said and done, we’re comparing three very different things here:

  1. The radius that’s associated with the energy that’s needed to聽assemble聽our electron from infinitesimally small charges, and so that’s based on聽Coulomb’s聽law and the model we use for our electron: is it a shell or a sphere of charge? If it’s a sphere, do we want to think of it as something that’s of uniform of non-uniform density.
  2. The second radius is associated with the field of an electron, which we calculate using Poynting’s formula for the energy flow and/or the momentum density. So that’s not聽about the internal structure of the electron but, of course, it would be nice if we could find some model of an electron that matches聽this聽radius.
  3. Finally, there’s the radius that’s associated with elastic聽scattering, which is also referred to as聽hard聽scattering because it’s like the collision of two hard spheres indeed. But so that’s some value that has to be established experimentally聽and so it involves judicious choices because there’s probabilities and standard deviations involved.

So should we worry about the gaps between these three different concepts? In my humble opinion: no. Why? Because they’re all damn close and so we’re actually talking about the same thing. I mean: isn’t terrific that we’ve got a model that brings the first and the second radius together with a difference of 10% only? As far as I am concerned, that shows the theory works. So what Feynman’s doing in that (in)famous chapter is some kind of ‘dimensional analysis’ which confirms rather than invalidates聽classical electromagnetic theory. So it shows classical theory’s strength, rather than its weakness. It actually shows our formula聽do聽work where we wouldn’t expect them to work. 馃檪

The thing is: when looking at the behavior of electrons themselves, we’ll need a different conceptual framework altogether. I am talking quantum mechanics here. Indeed, we’ll encounter other anomalies than the ones we presented above. There’s the issue of the anomalous magnetic moment of electrons, for example. Indeed, as I mentioned above, we’ll also want to think as electrons as spinning around their own axis, and so that implies some circulation聽of E聽that will generate a permanent magnetic dipole moment… […]聽OK, just think of some magnetic field if you don’t have a clue what I am saying here (but then you should check out my post on it). […] The point is: here too, the so-called ‘classical result’, so that’s its theoretical聽value, will differ from the experimentally measured value. Now, the difference here will be 0.0011614, so that’s about 0.1%, i.e. 100 times聽smaller聽than my 10%. 馃檪

Personally, I think that’s not so bad. 馃檪 But then physicists need to stay in business, of course. So, yes, it聽is聽a problem. 馃檪

Post scriptum on the math versus the physics

The key to the calculation of the energy that goes into assembling a charge was the following integral:

U 4

This is a double integral which we simplified in two stages, so we’re looking at an integral within an integral really, but聽we can substitute the integral over the 蟻(2)路dV2聽product by the formula we got for the potential, so we write that as聽桅(1), and so the integral above becomes:

U 5Now, this聽integral integrates the 蟻(1)路桅(1)路dV1聽product over all of space, so that鈥檚 over all points in space, and so we just dropped the index and wrote the whole thing as the integral of 蟻路桅路dV聽over all of space:

U 6

We thenestablished that this integral was mathematically equivalent to the following equation:

U 7

So this integral is actually quite simple: it just integrates EE = E2聽over all of space. The illustration below shows E as a function of the distance聽r聽for a sphere of radius R filled聽uniformly聽with charge.

uniform density

So the field (E) goes as聽r聽for r 鈮 R and as 1/r2聽for r 鈮 R. So, for r 鈮 R, the integral will have (1/r2)2聽= 1/r4聽in it. Now, you know that the integral of some function is the surface under the graph of that function. Look at the 1/r4 function below: it blows up between 1 and 0. That’s where the problem is: there needs to be some kind of cut-off, because that integral will effectively blow up when the radius of our little sphere of charge gets ‘too small’.聽So that makes it clear why it doesn’t make sense to use this formula to try to calculate the energy of a point charge. It just doesn’t make sense to do that.


What’s ‘too small’? Let’s look at the formula we got for our electron as a spherical shell聽of charge:

energy electron

So we’ve got an even simpler formula here: it’s just a 1/r聽relation. Why is that? Well… It’s just the way the math turns it out. I copied the detail of Feynman’s calculation above, so you can double-check it. It’s quite wonderful, really. We have a very simple聽inversely聽proportional relationship between the聽radius聽of our electron and聽its聽energy as a sphere of charge. We could write it as:

Uelect聽聽= 伪/a聽, with聽伪 = e2/2

But聽鈥撀Hey!Wait a minute! We’ve seen something like this before, haven’t we?聽We did. We did when we were discussing the wonderful properties of that magical number, the聽fine-structure constant, which we also denoted by聽伪. 馃檪 However, because we used聽伪 already, I’ll denote the fine-structure constant as 伪e here, so you don’t get confused. As you can see, the fine-structure constant links all聽of the fundamental properties of the electron: its charge, its radius, its distance to the nucleus (i.e. the Bohr radius), its velocity, and its mass (and, hence, its energy). So, at this stage of the argument,聽伪 can be anything, and聽伪e聽cannot, of course. It’s just that magical number out there, which relates everything to everything: it’s the God-given number we don’t understand. 馃檪 Having said that, it seems like we’re going to get some understanding here because we know that, one the聽many expressions involving 伪e was the following one:

me聽= 伪e/re

This says that the聽mass聽of the electron is equal to the ratio of the fine-structure constant and the electron radius. [Note that we express everything in natural units here, so that’s聽Planck units. For the detail of the conversion, please see the relevant section on that in my one of my posts on this and other stuff.] Now, mass is equivalent to energy, of course: it’s just a matter of units, so we can equate me聽with Ee聽(this amounts to expressing the energy of the electron in a聽kg unit鈥攂it weird, but OK) and so we get:

Ee聽= 伪e/re

So there we have: the fine-structure constant聽伪e聽is Nature’s ‘cut-off’ factor, so to speak. Why? Only God knows. 馃檪 But it’s now (fairly) easy to see why all the relations involving 伪e聽are what they are. For example, we also know that 伪e聽is the square of the electron charge expressed in Planck units, so we have:

聽伪 =聽eP2聽and, therefore,聽Ee聽= eP2/re

Now, you can check for yourself: it’s just a matter of re-expressing everything in standard SI units, and relating eP2聽to e2, and it should all work: you should get the聽Uelect聽聽= (1/2)路e2/a聽expression. So… Well… At least this takes some of the magic out the fine-structure constant. It’s still a wonderful thing, but so you see that the fundamental relationship between (a) the energy (and, hence, the mass), (b) the radius and (c) the charge of an electron is聽not聽something God-given. What’s God-given are Maxwell’s equations, and so the聽Ee聽= 伪e/re聽= eP2/re聽is just one of the many wonderful things that you can get out of 聽them.聽馃檪

Loose ends…

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

The three-forces model and coupling constants

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

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

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

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

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

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

1. There are three equivalent definitions of聽聽in terms of other fundamental physical constants:

\alpha = \frac{k_\mathrm{e} e^2}{\hbar c} = \frac{1}{(4 \pi \varepsilon_0)} \frac{e^2}{\hbar c} = \frac{e^2 c \mu_0}{2 h}
where聽e聽is the elementary charge (so that’s the electric charge of the proton);聽聽=聽h/2蟺 is the reduced Planck constant;聽c聽is the speed of light (in vacuum);聽0聽is the electric constant (i.e. the so-called聽permittivity of free space);聽碌0聽is the magnetic constant (i.e. the so-called permeability of free space); and聽ke聽is the Coulomb constant.

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

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

3. When using so-called聽natural units, we equate聽0聽,聽cand聽to 1. [That does not mean they are the same, but they just become the unit for measurement for whatever is measured in them. :-)] The value of the fine-structure constant then becomes:

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

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

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

Coupling constants, interactions, and Feynman diagrams

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

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

Compton scattering

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


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

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

The boson exchange model

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

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

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

Pandora’s Box

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

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

Indeed, terms with j2, j3,聽j4聽etcetera (i.e. the terms involved in adding amplitudes for all possible paths and all possible ways in which an event can happen) quickly become very聽small as the exponent increases, but terms with聽g2, g3,聽g4聽etcetera do not become negligibly small. In fact, they don’t become irrelevant at all. Indeed, if we wrote 伪 for the electromagnetic force as聽7.297脳10鈥3, then the 伪 for the strong force is one, and so none of these terms becomes vanishingly small. I won’t dwell on this, but just quote Wikipedia’s very succinct appraisal of the situation: “If 伪聽is much less than 1 [in a quantum field theory with a dimensionless coupling constant 伪], then the theory is said to be聽weakly coupled. In this case it is well described by an expansion in powers of 伪聽called perturbation theory. [However] If the coupling constant is of order one or larger, the theory is said to be聽strongly coupled. An example of the latter [the only聽example as far as I am aware: we don’t have like a dozen different forces out there !]聽is the hadronic theory of strong interactions,聽which is why it is called strong in the first place. [Hadrons is just a difficult word for particles composed of quarks – so don’t worry about it: you understand what is being said here.] In such a case non-perturbative methods have to be used to investigate the theory.”

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

The first-generation world

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

first-generation matter

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

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

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

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

Second- and third-generation matter

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

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

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

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

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


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

quark gluon exchange

So what聽are聽the loose ends then? The problem of ‘mathematical consistency’ associated with the techniques used to calculate (or estimate) these coupling constants – which Feynman identifies as a key defect聽in 1985 – is is a form of skepticism about the Standard Model that is聽not聽shared by others. It’s more about the other forces. So let’s now talk about these.

The weak force as the聽weird聽force: about symmetry breaking

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

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

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

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

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

2000px-Elementary_particle_interactions.svgBut so the Higgs field does聽not聽interact with photons and gluons. Why? Well… I am not sure. Let me copy the Wikipedia explanation: “The Higgs field consists of four components, two neutral ones and two charged component fields. Both of the charged components and one of the neutral fields are Goldstone bosons, which act as the longitudinal third-polarization components of the massive W+, W鈥 and Z bosons. The quantum of the remaining neutral component corresponds to (and is theoretically realized as) the massive Higgs boson.”

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

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

So… No聽real聽loose end here I’d say.


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

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

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

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

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

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

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

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

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

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

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

I don鈥檛 like that they鈥檙e not calculating anything. I don鈥檛 like that they don鈥檛 check their ideas. I don鈥檛 like that for anything that disagrees with an experiment, they cook up an explanation鈥揳 fix-up to say, 鈥淲ell, it might be true.鈥 For example, the theory requires ten dimensions. Well, maybe there鈥檚 a way of wrapping up six of the dimensions. Yes, that鈥檚 all possible mathematically, but why not seven? When they write their equation, the equation should decide how many of these things get wrapped up, not the desire to agree with experiment. In other words, there鈥檚 no reason whatsoever in superstring theory that it isn鈥檛 eight out of the ten dimensions that get wrapped up and that the result is only two dimensions, which would be completely in disagreement with experience. So the fact that it might disagree with experience is very tenuous, it doesn鈥檛 produce anything; it has to be excused most of the time. It doesn鈥檛 look right.鈥

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