The Emperor has no clothes

Hi guys (and ladies) – I should copy the paper into this post but… Well… That’s rather tedious. :-/ The topic is one that is of interest of you. You’re looking for a classical explanation of the anomalous magnetic moment, right? Well… We don’t have one – but we’re pretty sure this paper has all the right ingredients for one. We also designed a test for it. Also check out my other paper on the fine-structure constant. It explains everything.

Everything? Well…聽Almost聽everything. 馃檪聽The Revolution has started. The (quantum-mechanical) Emperor seems to have no clothes. 馃檪

I am damn serious. This is what I wrote on my FB page today:

The only thing I can be proud of this year is a series of papers on quantum math.聽I will probably turn them into a popular book on physics. Its working title is “The Emperor Has No Clothes !” Indeed – if anything – these papers show that a lot of the聽highbrow stuff is just unnecessary complexity or deliberate hyping up of models that can be simplified significantly.

Worse, through my interactions with some physicists, I found some serious research into the nature of matter and energy is being neglected or ignored just because it challenges the Copenhagen interpretation of quantum physics. Most papers of Alexander Burinskii, for example, a brilliant physicist who developed a very plausible model of an electron, have been re-classified from ‘quantum physics’ to ‘general physics’ – which means no one will read them. Worse, he has had trouble just getting stuff published over the last four years! It’s plain censorship!聽

I now summarize the Copenhagen interpretation as: “Calculate, don’t think !”聽It’s a Diktatur, really! And I now also understand why the founding fathers of quantum mechanics (Dirac, Heisenberg, Pauli, Schroedinger,…) thought the theory they helped to create didn’t quite make the cut. It’s going to be a sad story to tell. In fact, I think Burinskii is in trouble because his model may show that a lot of the research on the anomalous magnetic moment is plain humbug – but so that got some people a Nobel Prize in 1955 and it’s popularly referred to as the ‘high-precision test’ of QED, so… Well… I looked at it too, and for quite a long time, and I’ve come to the conclusion that it’s plain nonsense – but so that cannot be said.

Hmm… If the state of physics is so poor, then we should not be surprised that we are constantly being misled in other fields as well. Let us remember Boltzmann:

“Bring forth what is true. Write it so it it鈥檚 clear. Defend it to your last breath.”

Oh – and I have a sort of classical explanation for what happens in the one-photon Mach-Zehnder experiment too. Check it out here. Quantum mechanics is聽not聽a mystery. Mr. Bell has got it all wrong. 馃檪

Kind regards – JL

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Taking the magic out of God’s number

Note: I have published a paper that is very coherent and fully explains this so-called God-given number. There is nothing magical about it. It is just a scaling constant. Check it out: The Meaning of the Fine-Structure Constant. No ambiguity. No hocus-pocus.

Jean Louis Van Belle, 23 December 2018

Original post:

I think the post scriptum to my previous post is interesting enough to separate it out as a piece of its own, so let me do that here. You’ll remember that we were trying to find some kind of a model for the electron, picturing it like a tiny little ball of charge, and then we just applied the classical energy formulas to it to see what comes out of it. The聽key formula is the integral that gives us the energy that goes into assembling a charge. It was the following thing:

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.

graph

In fact, the need for a ‘cut-off factor’ so as to ensure our energy function doesn’t ‘blow up’ is not because of the exponent in the聽1/r4 expression: the need is also there for any 1/r relation, as illustrated below. All聽1/rn聽function have the same聽pivot聽point, as you can see from the simple illustration below. So, yes, we cannot go all the way to zero from there when integrating: we have to stop聽somewhere.

graph 2So what’s the ‘cut-off point’? What’s ‘too small’ a radius? Let’s look at the formula we got for our electron as a shell聽of charge (so the assumption here is that the charge is聽uniformly distributed on the surface聽of a sphere with radius a):

energy electron

So we’ve got an even simpler formula here: it’s just a 1/r聽relation (a is r聽in this formula), not 1/r4. Why is that? Well… It’s just the way the math turns out: we’re integrating over volumes and so that involves an r3聽factor and so it all simplifies to 1/r, and so that gives us this simple inversely聽proportional relationship between U and r, i.e聽a, in this case. 馃檪聽I copied the detail of Feynman’s calculation in my previous post, so you can double-check it. It’s quite wonderful, really. Look at it again: 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, with聽伪 = e2/2

Still… We need the cut-off point’. Also note that, as I pointed out, we don’t necessarily need to assume that the charge in our little ball of charge (i.e. our electron) sits on the surface only: if we’d assume it’s a uniformly charged sphere of charge, we’d just get another constant of proportionality: our 1/2 factor would become a 3/5 factor, so we’d write: Uelect聽聽= (3/5)路e2/a. But we’re not interested in finding the right聽model here. We know the聽Uelect聽聽= (3/5)路e2/a聽gives us a value for聽a聽that differs with a 2/5 factor as the classical electron radius. That’s not so bad and so let’s go along with it. 馃檪

We’re going to look at the simple聽structure聽of this relation, and all of its implications. The simple equation above says that the energy of our electron is (a) proportional to the square of its charge and (b) inversely proportional to its radius. Now,聽that聽is a very remarkable result. In fact,聽we’ve seen something like this before, and we were astonished.聽We saw it when we were discussing the wonderful properties of that magical number, the聽fine-structure constant, which we also denoted by聽伪. However, because we used聽伪 already, I’ll denote the fine-structure constant as 伪e here, so you don’t get confused. You’ll remember that聽the fine-structure constant is a God-like number indeed: it links all聽of the fundamental properties of the electron, i.e. its charge, its radius, its distance to the nucleus (i.e. the Bohr radius), its velocity, its mass (and, hence, its energy), its de Broglie聽wavelength. Whatever: all these physical constants are all related through the fine-structure constant.聽

In my various posts on this topic, I’ve repeatedly said that, but I never showed why it’s true, and so it was a very magical number indeed. I am going to take some of the magic out now. Not too much but… Well… You can judge for yourself how much of the magic remains after I am done here. 馃檪

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,聽or didn’t understand, I should say. Past tense. Indeed, we’re going to get some understanding here because we know that one of 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.] In fact, the U =聽(3/5)路e2/a聽and聽me聽= 伪e/re聽relations looks exactly聽the same, because one of the other equations involving the fine-structure constant was:聽伪e聽=聽eP2. So we’ve got the square of the charge here as well! Indeed, as I’ll explain in a moment, the difference between the two formulas is just a matter of units.

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. As I mentioned already, we also know that 伪e聽is the square of the electron charge expressed in Planck units, so we have:

聽伪e聽=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 Eelect聽聽= (2/3)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.

So we found God’s ‘cut-off factor’ 馃檪 It’s equal to 伪e聽鈮 0.0073 = 7.3脳10鈭3. So 7.3 thousands of… What? Well… Nothing. It’s just a pure ratio between the energy and the radius of an electron (if both are expressed in Planck units, of course). And so it determines the electron charge (again, expressed in Planck units). Indeed, we write:

eP聽= 鈭毼e

Really? Yes. Just聽do聽all these formulas:

eP聽= 鈭毼e聽鈮埪犫垰0.0073路1.9脳10鈭18聽coulomb聽鈮 1.6聽脳10鈭19 C

Just re-check it聽with聽all the known decimals: you’ll see it’s bang on. Let’s look at the Ee聽= me聽= 伪e/re聽ratio once again. What’s the聽meaning聽of it? Let’s first calculate the value of re聽and me,聽i.e. the electron radius and electron mass expressed in Planck units. It’s equal to the classical electron radius divided by the Planck length, and then the same for the mass, so we get the following thing:

re聽鈮 (2.81794脳10鈭15聽m)/(1.6162脳10鈭35聽m) = 1.7435脳1020聽

me聽鈮 (9.1脳10鈭31聽kg)/(2.17651脳10鈭8 kg) = 4.18脳10鈭23

e聽= (4.18脳10鈭23)路(1.7435脳1020) 鈮 0.0073

It works like a charm, but what does it mean? Well… It’s just a ratio between two physical quantities, and the scale聽you use to measure those quantities matters very much. We’ve explained that the Planck mass is a rather large unit at the atomic scale and, therefore, it’s perhaps not quite appropriate to use it here. In fact, out of the many interesting expressions for 伪e, I should highlight the following one:

e聽= e2/(魔路c) 鈮 (1.60217662脳10鈭19 C)2/(4蟺蔚0路[(1.054572脳10鈭34 N路m路s)路(2.998脳108 m/s)])聽鈮 0.0073 once more 馃檪

Note that the elementary charge e is actually equal to qe/4蟺蔚0, which is what I am using in the formula. I know that’s confusing, but it what it is. As for the units, it’s a bit tedious to write it all out, but you’ll get there. Note that 蔚0聽鈮 8.8542脳10鈭12聽C2/(N路m2) so… Well… All the units do cancel out, and we get a dimensionless number indeed, which is what 伪e聽is.

The point is: this expression links 伪e聽to the the de Broglie聽relation (p =聽h/位), with 位 the wavelength聽that’s associated with the electron. Of course, because of the Uncertainty Principle, we know we’re talking some wavelength聽range聽really, so we should write the de Broglie relation as聽螖p = h路螖(1/位). Now, that, in turn, allows us to try to work out the Bohr radius, which is the other ‘dimension’ we associate with an electron. Of course, now you’ll say: why would you do that. Why would you bring in the de Broglie聽relation here?

Well… We’re talking energy, and so we have the Planck-Einstein聽relation first: the energy of some particle can always be written as the product of h聽and some frequency聽f: E = h路f. The only thing that聽de Broglie relation adds is the Uncertainty Principle indeed: the frequency聽f聽will be some frequency range, associated with some聽momentum聽range, and so that’s what the Uncertainty Principle really says. I can’t dwell too much on that here, because otherwise this post would become a book. 馃檪 For more detail, you can check out one of my many posts on the Uncertainty Principle. In fact, the one I am referring to here has Feynman’s calculation of the Bohr radius, so I warmly recommend you check it out. The thrust of the argument is as follows:

  1. If we assume that (a) an electron takes some space 鈥 which I鈥檒l denote by r聽馃檪 鈥 and (b) that it has some momentum p because of its mass m and its velocity v, then the 螖x螖p =聽魔聽relation (i.e. the Uncertainty Principle in its roughest form) suggests that the order of magnitude of r and p should be related in the very same way. Hence, let鈥檚 just boldly write r 鈮 魔/p and see what we can do with that.
  2. We know that the kinetic energy of our electron equals mv2/2, which we can write as p2/2m so we get rid of the velocity factor.Well鈥 Substituting our p 鈮 魔/r conjecture, we get K.E. = 魔2/2mr2. So that鈥檚 a formula for the kinetic energy. Next is potential.
  3. The聽formula for the potential energy is U = q1q2/4蟺蔚0r12. Now, we鈥檙e actually talking about the size of an atom聽here, so one charge is the proton (+e) and the other is the electron (鈥揺), so the potential energy is U = P.E. = 鈥揺2/4蟺蔚0r, with r聽the 鈥榙istance鈥 between the proton and the electron鈥攕o that鈥檚 the Bohr radius we鈥檙e looking for!
  4. We can now write the total energy (which I鈥檒l denote by E, but don鈥檛 confuse it with the electric field vector!) as聽E = K.E. + P.E. =聽聽魔2/2mr2聽鈥撀爀2/4蟺蔚0r. Now,聽the electron (whatever it is) is, obviously, in some kind of equilibrium state. Why is that obvious? Well鈥 Otherwise our hydrogen atom wouldn鈥檛 or couldn鈥檛 exist. 馃檪 Hence, it鈥檚 in some kind of energy 鈥榳ell鈥 indeed, at the bottom. Such equilibrium point 鈥榓t the bottom鈥 is characterized by its derivative (in respect to whatever variable) being equal to zero. Now, the only 鈥榲ariable鈥 here is聽r聽(all the other symbols are physical constants), so we have to solve for dE/dr = 0. Writing it all out yields:聽dE/dr = 鈥撃2/mr3聽+ e2/4蟺蔚0r2聽= 0聽鈬 r =聽4蟺蔚02/me2
  5. We can now put the values in:聽r =聽4蟺蔚0h2/me2聽= [(1/(9脳109) C2/N路m2)路(1.055脳10鈥34聽J路s)2]/[(9.1脳10鈥31聽kg)路(1.6脳10鈥19聽C)2]聽= 53脳10鈥12聽m = 53 pico-meter (pm)

Done. We’re right on the spot.聽The Bohr radius is, effectively, about 53 trillionths聽of a meter indeed!

Phew!

Yes… I know… Relax. We’re almost done. You should now be able to figure out why the classical electron radius and the Bohr radius can also be related to each other through the fine-structure constant. We write:

me聽= 伪/re聽= 伪/伪2r聽= 1/伪r

So we get that 伪/re聽=聽1/伪r and, therefore, we get re/r = 伪2, which explains why 伪 is also equal to the so-called junction number, or the coupling constant, for an electron-photon coupling (see my post on the quantum-mechanical aspects of the photon-electron interaction). It gives a physical meaning to the probability (which, as you know, is the absolute square of the probability amplitude) in terms of the chance of a photon actually ‘hitting’ the electron as it goes through the atom. Indeed, the ratio of the Thomson scattering cross-section and the Bohr size of the atom should be of the same order as聽re/r, and so that’s 伪2.

[Note: To be fully correct and complete, I should add that the coupling constant itself is not聽伪2聽but 鈭毼 = eP. Why do we have this square root? You’re right: the fact that the probability is the absolute square聽of the amplitude explains one square root (鈭毼2聽= 伪), but not two. The thing is: the photon-electron interaction consists of two聽things. First, the electron sort of ‘absorbs’ the photon, and then it emits another one, that has the same or a different frequency depending on whether or not the ‘collision’ was elastic or not. So if we denote the coupling constant as j, then the whole interaction will have a probability amplitude equal to j2. In fact, the聽value which Feynman uses in his wonderful popular presentation of quantum mechanics (The Strange Theory of Light and Matter), is 鈭捨 鈮 鈭0.0073. I am not quite sure why the minus sign is there. It must be something with the angles involved (the emitted photon will not be following the trajectory of the incoming photon) or, else, with the special arithmetic involved in boson-fermion interactions (we add amplitudes when bosons are involved, but聽subtract聽amplitudes when it’s fermions interacting.聽I’ll probably find out once I am true through Feynman’s third volume of聽Lectures, which focus on quantum mechanics only.]

Finally, the last bit of unexplained ‘magic’ in the fine-structure constant is that the fine-structure constant (which I’ve started to write as 伪 again, instead of 伪e) also gives us the (classical) relative speed of an electron, so that’s its speed as it orbits around the nucleus (according to the classical theory, that is), so we write

伪 = v/c聽=聽尾

I should go through the motions here聽鈥 I’ll probably do so in the coming days聽鈥 but you can see we must be able to get it out somehow from all what we wrote above. See how powerful聽our聽Uelect聽聽鈭 e2/a relation really is? It links the electron, charge, its radius and its energy, and it’s all we need to all the rest out of it: its mass, its momentum, its speed and 鈥 through the Uncertainty Principle 鈥 the Bohr radius, which is the size of the atom.

We’ve come a long way. This is truly a milestone. We’ve taken the magic out of God’s number鈥攖o some extent at least. 馃檪

You’ll have one last question, of course: if proportionality constants are all about the聽scale聽in which we聽measure聽the physical quantities on either side of an equation, is there some way the fine-structure constant would come out differently? That’s the same as asking: what if we’d measure energy in units that are equivalent to the energy of an electron, and the radius of our electron just as… Well… What if we’d equate our unit of distance with the radius of the electron, so we’d write re = 1? What would happen to聽伪? Well…聽I’ll let you figure that one out yourself. I am tired and so I should go to bed now. 馃檪

[…] OK. OK. Let me tell you. It’s not that simple here. All those relationships involving 伪, in one form or the other, are very deep. They relate a lot of stuff to a lot of stuff, and we can appreciate that only when doing a dimensional analysis. A dimensional analysis of the Ee聽= 伪e/re聽= eP2/ryields [eP2/r] = C2/m on the right-hand side and [Ee] = J = N路mon the left-hand side. How can we reconcile both? The coulomb is an SI base unit聽, so we can’t ‘translate’ it into something with N and m. [To be fully correct, for some reason, the聽amp猫re聽(i.e. coulomb聽per second) was chosen as an SI base unit, but they’re interchangeable in regard to their place in the international system of units: they can’t be reduced.] So we’ve got a problem. Yes. That’s where we sort of ‘smuggled’ the 4蟺蔚0 factor in when doing our calculations above. That聽蔚0 constant is, obviously, not ‘as fundamental’ as c聽or聽伪 (just think of the c鈭2聽=聽蔚00 relationship to understand what I mean here) but, still, it was necessary to make the dimensions come out alright: we need the reciprocal聽dimension聽of 蔚0, i.e. (N路m2)/C2, to make the dimensional analysis work.聽We get: (C2/m)路(N路m2)/C2聽= N路m = J, i.e.聽joule, so that’s the unit in which we measure energy or 鈥 using the E = mc2聽equivalence聽鈥 mass, which is the aspect of energy emphasizing its聽inertia.

So the answer is: no. Changing units won’t change alpha. So all that’s left is to play with it now. Let’s try to do that. Let me first plot that Ee聽= me聽= 伪e/re = 0.00729735256/re:

graph 3Unsurprisingly, we find the聽pivot聽point of this curve is at the intersection of the diagonal and the curve itself, so that’s at the (0.00729735256,聽0.00729735256) point, where slopes are 卤 1, i.e. plus or minus unity.聽What does this show? Nothing much. What?聽I can hear you: I should be excited because… Well… Yes! Think of it. If you聽would have to chose a cut-off point, you’d chose this one, wouldn’t you? 馃檪 Sure, you’re right. How exciting! Let me show you. Look at it! It proves that God thinks in terms of logarithms. He has chosen聽伪 such that ln(E) = ln(伪/r) = ln伪 鈥 lnr聽= ln伪 鈥 lnr聽= 0, so ln 伪 = lnr and, therefore, 伪 = r. 馃檪

Huh? Excuse me?

I am sorry. […] Well… I am not, of course… 馃檪 I just wanted to illustrate the kind of exercise some people are tempted to do. It’s no use. The fine-structure constant is what it is: it sort of聽summarizes聽an awful lot of formulas. It basically shows what Maxwell’s equation imply in terms of the聽structure聽of an atom defined as a negativecharge orbiting around some positive charge. It shows we can get calculate everything as a function of something else, and that’s what the fine-structure constant tells us: it relates everything to everything. However, when everything is said and done, the fine-structure constant shows us two things:

  1. Maxwell’s equations are complete: we can聽construct a complete model of the electron and the atom, which includes: the electron’s energy and mass, its velocity, its own radius, and the radius of the atom. [I might have forgotten one of the dimensions here, but you’ll add it. :-)]
  2. God doesn’t want our equations to blow up. Our equations are all correct but, in reality, there’s a cut-off factor that ensures we don’t go to the limit with them.

So the fine-structure constant anchors our world, so to speak. In other words: of all the worlds that are possible, we live in this one.

[…] It’s pretty good as far as I am concerned. Isn’t it amazing that our mind is able to just聽grasp聽things like that? I know my approach here is pretty intuitive, and with ‘intuitive’, I mean ‘not scientific’ here. 馃檪 Frankly, I don’t like the talk about physicists “looking into God’s mind.” I don’t think that’s what they’re trying to do. I think they’re just trying to understand the fundamental聽unity聽behind it all. And that’s religion enough for me. 馃檪

So… What’s the conclusion? Nothing much. We’ve sort of concluded our description of the classical world… Well… Of its ‘electromagnetic sector’ at least. 馃檪 That sector can be summarized in Maxwell’s equations, which describe an infinite world of possible worlds. However, God fixed three constants:聽h,聽c聽and聽伪. So we live in a world that’s defined by this Trinity of fundamental physical constants. Why is it not two, or four?

My guts instinct tells me it’s because we live in three dimensions, and so there’s three degrees of freedom really. But what about time? Time is the fourth dimension, isn’t it? Yes. But time is symmetric in the ‘electromagnetic’ sector: we can reverse the arrow of time in our equations and everything still works. The聽arrow of time聽involves other theories: statistics (physicists refer to it as ‘statistical mechanics‘) and the ‘weak force’ sector, which I discussed when talking about symmetries in physics. So… Well… We’re not done. God gave us plenty of other stuff to try to understand. 馃檪