I’ve did what I promised to do – and that is to start posting on my other blog. On quantum chromodynamics, that is. But I think this paper deserves wider distribution. 🙂
The paper below probably sort of sums up my views on quantum field theory. I am not sure if I am going to continue to blog. I moved my papers to an academia.edu site and… Well… I think that’s about it. 🙂
One of the readers of this blog asked me what I thought of the following site: Rational Science (https://www.youtube.com/channel/UC_I_L6pPCwxTgAH7yutyxqA). I watched it – for a brief while – and I must admit I am thoroughly disappointed by it. I think it’s important enough to re-post what I posted on this YouTube channel itself:
“I do believe there is an element of irrationality in modern physics: a realist interpretation of quantum electrodynamics is possible but may not gain acceptance because religion and other factors may make scientists somewhat hesitant to accept a common-sense explanation of things. The mystery needs to be there, and it needs to be protected – somehow. Quantum mechanics may well be the only place where God can hide – in science, that is.
But – in his attempt to do away with the notion of God – Bill Gaede takes things way too far – and so I think he errs on the other side of the spectrum. Mass, energy and spacetime are essential categories of the mind (or concepts if you want) to explain the world. Mass is a measure of inertia to a change in the state of motion of an object, kinetic energy is the energy of motion, potential energy is energy because of an object’s position in spacetime, etcetera. So, yes, these are concepts – and we need these concepts to explain what we human beings refer to as ‘the World’. Space and time are categories of the mind as well – philosophical or mathematical concepts, in other words – but they are related and well-defined.
In fact, space and time define each other also because the primordial idea of motion implies both: the idea of motion implies we imagine something moving in space and in time. So that’s space-time, and it’s a useful idea. That also explains why time goes in one direction only. If we’d allow time to reverse, then we’d also an object to be in two places at the same time (if an object can go back in time, then it can also go back to some other place – and so then it’s in two places at the same time). This is just one example where math makes sense of physical realities – or where our mind meets ‘the World’.
When Bill Gaede quotes Wheeler and other physicists in an attempt to make you feel he’s on the right side of history, he quotes him very selectively. John Wheeler, for example, believed in the idea of ‘mass’ – but it was ‘mass without mass’ for him: the mass of an object was the equivalent mass of the object’s energy. The ideas of Wheeler have been taken forward by a minority of physicists, such as David Hestenes and Alexander Burinskii. They’ve developed a fully-fledged electron model that combines wave and particle characteristics. It effectively does away with all of the hocus-pocus in QED – which Bill Gaede criticizes, and rightly so.
In short, while it’s useful to criticize mainstream physics as hocus-pocus, Bill Gaede is taking it much too far and, unfortunately, gives too much ammunition to critics to think of people like us – amateur physicists or scientists who try to make sense of it all – as wackos or crackpots. Math is, effectively, descriptive but, just like anything else, we need a language to describe stuff, and math is the language in which we describe actual physics. Trying to discredit the mathematical approach to science is at least as bad – much worse, actually – than attaching too much importance to it. Yes, we need to remind ourselves constantly that we are describing something physical, but we need concepts for that – and these concepts are mathematical.
PS: Bill Gaede also has very poor credentials, but you may want to judge these for yourself: https://en.wikipedia.org/wiki/Bill_Gaede. These poor credentials do not imply that his views are automatically wrong, but it does introduce an element of insincerity. In short, watch what you’re watching and always check sources and backgrounds when googling for answers to questions, especially when you’re googling for answers to fundamental questions ! 🙂
This is it, folks ! I am moving on ! It was nice camping out here. 🙂
This has been a very interesting journey for me. I wrote my first post in October 2013, so that’s almost five years ago. As mentioned in the ‘About‘ page, I started writing this blog because — with all those breakthroughs in science (some kind of experimental verification of what is referred to as the Higgs field in July 2012 and, more recently, the confirmation of the reality of gravitational waves in 2016 by Caltech’s LIGO Lab) — I felt I should make an honest effort to try to understand what it was all about.
Despite all of my efforts (including enrolling in MIT’s edX QM course, which I warmly recommend as an experience, especially because it’s for free), I haven’t moved much beyond quantum electrodynamics (QED). Hence, that Higgs field is a still a bit of a mystery to me. In any case, the summaries I’ve read about it say it’s just some scalar field. So that’s not very exciting: mass is some number associated with some position in spacetime. That’s nothing new, right?
In contrast, I am very enthusiastic about the LIGO Lab discovery. Why? Because it confirms Einstein was right all along.
If you have read any of my posts, you will know I actually disagree with Feynman. I have to thank him for his Lectures — and I would, once again, like to thank Michael Gottlieb and Rudolf Pfeiffer, who have worked for decades to get those Lectures online — but my explorations did confirm that guts feeling I had deep inside when starting this journey: the complexity in the quantum-mechanical framework does not match the intuition that, if the theory has a simple circle group structure, one should not be calculating a zillion integrals all over space over 891 4-loop Feynman diagrams to explain the magnetic moment of an electron in a Penning trap. And the interference of a photon with itself in the Mach-Zehnder interference experiment has a classical explanation too. The ‘zero state’ of a photon – or its zero states (plural), I should say – are the linear components of the circular polarization. In fact, I really wish someone would have gently told me that an actual beam splitter changes the polarization of light. I could have solved the Mach-Zehnder puzzle with that information like a year ago.]
This will probably sound like Chinese to you, so let me translate it: there is no mystery. Not in the QED sector of the Standard Model, at least. All can be explained by simple geometry and the idea of a naked charge: something that has no other property but its electric charge and – importantly – some tiny radius, which is given by the fine-structure constant (the ratio becomes a distance if we think of the electron’s Compton radius as a natural (distance) unit). So the meaning of God’s Number is clear now: there is nothing miraculous about it either. Maxwell’s equations combined with the Planck-Einstein Law (E = h·f) are all we need to explain the whole QED sector. No hocus-pocus needed. The elementary wavefunction exp(±i·θ) = exp(±ω·t) = exp[±(E/ħ)·t] represents an equally elementary oscillation. Physicists should just think some more about the sign convention and, more generally, think some more about Occam’s Razor Principle when modeling their problems. 🙂
Am I a crackpot? Maybe. I must be one, because I think the academics have a problem, not me. So… Well… That’s the definition of a crackpot, isn’t it? 🙂 It feels weird. Almost all physicists I got in touch with – spare two or three (I won’t mention their names because they too don’t quite know what to do with me) – are all stuck in their Copenhagen interpretation of quantum mechanics: reality is some kind of black box and we’ll never understand it the way we would want to understand it. Almost none of them is willing to think outside of the box. I blame vested interests (we’re talking Nobel Prize stuff, unfortunately) and Ivory Tower culture.
In any case, I found the answers to the questions I started out with, and I don’t think the academics I crossed (s)words with have found that peace of mind yet. So if I am a crackpot, then I am a happy one. 😊
The Grand Conclusion is that the Emperor is not wearing any clothes. Not in the QED sector, at least. In fact, I think the situation is a lot worse. The Copenhagen interpretation of quantum mechanics feels like a Bright Shining Lie. [Yes, I know that’s an ugly reference.] But… Yes. Just mathematical gimmicks to entertain students – and academics ! Of course, I can appreciate the fact that Nobel Prizes have been awarded and that academic reputations have to be upheld — posthumously or… I would want to write ‘humously’ here but that word doesn’t exist so I should replace it by ‘humorously’. 🙂 […] OK. Poor joke. 🙂
Frankly, it is a sad situation. Physics has become the domain of hype and canonical nonsense. To the few readers who have been faithful followers (this blog attracted about 154,034 visitors so far which is — of course — close to nothing), I’d say: think for yourself. Honor Boltzman’s spirit: “Bring forward what is true. Write it so that it is clear. Defend it to your last breath.” I actually like another quote of him too: “If you are out to describe the truth, leave elegance to the tailor.” But that’s too rough, isn’t it? And then I am also not sure he really said that. 🙂
Of course, QCD is another matter altogether — because of the non-linearity of the force(s) involved, and the multiplication of ‘colors’ — but my research over the past five years (longer than that, actually) have taught me that there is no ‘deep mystery’ in the QED sector. All is logical – including the meaning of the fine-structure constant: that’s just the radius of the naked charge expressed in natural units. All the rest can be derived. And 99% of what you’ll read or google about quantum mechanics is about QED: perturbation theory, propagators, the quantized field, etcetera to talk about photons and electrons, and their interactions. If you have a good idea about what an electron and a photon actually are, then you do not need anything of that to understand QED.
In short, quantum electrodynamics – as a theory, and in its current shape and form – is incomplete: it is all about electrons and photons – and the interactions between the two – but the theory lacks a good description of what electrons and photons actually are. All of the weirdness of Nature is, therefore, in this weird description of the fields: gauge theories, Feynman diagrams, quantum field theory, etcetera. And the common-sense is right there: right in front of us. It’s easy and elegant: a plain common-sense interpretation of quantum mechanics — which, I should remind the reader, is based on Erwin Schrödinger’s trivial solution for Dirac’s wave equation for an electron in free space.
So is no one picking this up? Let’s see. Truth cannot be hidden, right? Having said that, I must admit I have been very surprised by the rigidity of thought of academics (which I know all too well from my experience as a PhD student in economics) in this domain. If math is the queen of science, then physics is the king, right? Well… Maybe not. The brightest minds seem to have abandoned the field.
But I will stop my rant here. I want to examine the QCD sector now. What theories do we have for the non-linear force(s) that keep(s) protons together? What explains electron capture by a proton—turning it into a neutron in the process? What’s the nature of neutrinos? How should we think of all these intermediary particles—which are probably just temporary resonances rather than permanent fixtures?
My new readingeinstein.blog will be devoted to that. I think I’ll need some time to post my first posts (pun intended)—but… Well… We’ve started this adventure and so I want to get to the next destination. It’s a mind thing, right? 🙂
My posts on the fine-structure constant – God’s Number as it is often referred to – have always attracted a fair amount of views. I think that’s because I have always tried to clarify this or that relation by showing how and why exactly it pops us in this or that formula (e.g. Rydberg’s energy formula, the ratio of the various radii of an electron (Thomson, Compton and Bohr radius), the coupling constant, the anomalous magnetic moment, etcetera), as opposed to what most seem to try to do, and that is to further mystify it. You will probably not want to search through all of my writing so I will just refer you to my summary of these efforts on the viXra.org site: “Layered Motions: the Meaning of the Fine-Structure Constant.”
However, I must admit that – till now – I wasn’t quite able to answer this very simple question: what is that fine-structure constant? Why exactly does it appear as a scaling constant or a coupling constant in almost any equation you can think of but not in, say, Einstein’s mass-energy equivalence relation, or the de Broglie relations?
I finally have a final answer (pun intended) to the question, and it’s surprisingly easy: it is the radius of the naked charge in the electron expressed in terms of the natural distance unit that comes out of our realist interpretation of what an electron actually is. [For those who haven’t read me before, this realist interpretation is based on Schrödinger’s discovery of the Zitterbewegung of an electron.] That natural distance unit is the Compton radius of the electron: it is the effective radius of an electron as measured in inelastic collisions between high-energy photons and the electron. I like to think of it as a quantum of space in which interference happens but you will want to think that through for yourself.
The point is: that’s it. That’s all. All the other calculations follow from it. Why? It would take me a while to explain that but, if you carefully look at the logic in my classical calculations of the anomalous magnetic moment, then you should be able to understand why these calculations are somewhat more fundamental than the others and why we can, therefore, get everything else out of them. 🙂
Post scriptum: I quickly checked the downloads of my papers on Phil Gibbs’ site, and I am extremely surprised my very first paper (the quantum-mechanical wavefunction as a gravitational wave) of mine still gets downloads. To whomever is interested in this paper, I would say: the realist interpretation we have been pursuing – based on the Zitterbewegung model of an electron – is based on the idea of a naked charge (with zero rest mass) orbiting around some center. The energy in its motion – a perpetual current ring, really – gives the electron its (equivalent) mass. That’s just Wheeler’s idea of ‘mass without mass’. But the force is definitely not gravitational. It cannot be. The force has to grab onto something, and all it can grab onto here is that naked charge. The force is, therefore, electromagnetic. It must be. I now look at my very first paper as a first immature essay. It did help me to develop some basic intuitive ideas on what any realist interpretation of QM should look like, but the quantum-mechanical wavefunction has nothing to do with gravity. Quantum mechanics is electromagnetics: we just add the quantum. The idea of an elementary cycle. Gravity is dealt with by general relativity theory: energy – or its equivalent mass – bends spacetime. That’s very significant, but it doesn’t help you when analyzing the QED sector of physics. I should probably pull this paper of the site – but I won’t. Because I think it shows where I come from: very humble origins. 🙂
My book is moving forward. I just produced a very first promotional video. Have a look and let me know what you think of it ! 🙂
Note: Check the revised paper on this topic. The substance is the same, but it is a more coherent development.
Jean Louis Van Belle, 23 December 2018
I am going to expose a bright shining lie in (quantum) physics in this post: what is referred to as the electron’s anomalous magnetic moment is actually not a magnetic moment, and it is not anomalous. Let’s start with the first remark. The anomalous magnetic moment is not a magnetic moment. It is just some (real) number: it’s a ratio, to be precise. It does not have any physical dimension. If it would be an actual magnetic moment then we would measure it as we usually do in the context of quantum mechanics, and that is in terms of the Bohr magneton, which is equal to: μB = qeħ/2m ≈ 9.274×10−24 joule per tesla.
So what is the electron’s anomalous magnetic moment – denoted by ae – then? It is defined as the (half-)difference between (1) some supposedly real gyromagnetic ratio (ge) and (2) Dirac’s theoretical value for the gyromagnetic ratio of a spin-only electron (g = 2):This immediately triggers an obvious question: why would we use the g-factor of a spin-only electron. This is a very weird thing to do, because the electron in the cyclotron (a Penning trap) is actually not a spin-only electron: it follows an orbital motion – as we will explain shortly.
So… Well… It is also routinely said (and written) that its measured value is equal to 0.00115965218085(76). The 76 (between brackets) is the uncertainty – which looks pretty certain, because it is equal to 0.00000000000076. Hence, the precision here is equivalent to 76 parts per trillion (ppt). It is measured as a standard deviation. However, the problem is that these experiments actually do not directly measure ae. What is being measured in the so-called Penning traps that are used in these experiments (think of them as a sort of cyclotron) are two slightly different frequencies – an orbital frequency and a precession frequency, to be precise – and ae is then calculated as the fractional difference between the two:Let us go through the motions here – literally. The orbital frequency fc is the cyclotron frequency: a charged particle in a Penning trap will move in a circular orbit whose frequency depends on the charge, its mass and the strength of the magnetic field only. Let us give you the formula (we will derive it for you in an instant):The subscript c stands for cyclotron – or circular, if you want. We should not think of the speed of light here! In fact, the orbital velocity is a (relatively small) fraction of the speed of light and we can, therefore, use non-relativistic formulas. The derivation of the formula is quite straightforward – but we find it useful to recap it. It is based on a simple analysis of the Lorentz force, which is just the magnetic force here: F = v(q×B). Note that the frequency does not depend on the velocity or the radius of the circular motion. This is actually the whole idea of the trap: the electron can be inserted into the trap with a precise kinetic energy and will follow a circular trajectory if the frequency of the alternating voltage is kept constant. This is why we italicized only when writing that the orbital frequency depends on the charge, the mass and the strength of the magnetic field only. So what is the derivation? The Lorentz force is equal to the centripetal force here. We can therefore write:The v2/r factor is the centripetal acceleration. Hence, the F = m·v2/r does effectively represent Newton’s force law. The equation above yields the following formula for v and the v/r ratio:Now, the cyclotron frequency fc will respect the following equation:Re-arranging and substituting v for q·r·b/m yields:The associated current will be equal to:Hence, the magnetic moment is equal to:The angular momentum – which we will denote by – is equal to:Hence, we can write the g-factor as:It is what we would expect it to be: it is the gyromagnetic ratio for the orbital moment of the electron. It is one, not 2. Because gc is 1, we can write something very obvious:We should also note another equality here:Let us now look at the other frequency fs. It is the Larmor or precession frequency. It is (also) a classical thing: if we think of the electron as a tiny magnet with a magnetic moment that is proportional to its angular momentum, then it should, effectively, precess in a magnetic field. The analysis of precession is quite straightforward. The geometry of the situation is shown below and we may refer to (almost) any standard physics textbook for the derivation.
It is tempting to use the equality above and write this as:However, we should not do this. The precession causes the electron to wobble: its plane of rotation – and, hence, the axis of the angular momentum (and the magnetic moment) – is no longer fixed. This wobbling motion changes the orbital and, therefore, we can no longer trust the values we have used in our formulas for the angular momentum and the magnetic moment. There is, therefore, nothing anomalous about the anomalous magnetic moment. In fact, we should not wonder why it is not zero, but – as we will argue – we should wonder why it is so nearly zero.
Let us continue our analysis. It is, in fact, a bit weird to associate a gyromagnetic ratio with this motion, but that is what the physicists doing these experiments do. We will denote this g-factor by gp:Hence, we can write the following tautology:You can verify that this is nothing but a tautology by writing it all out:We can, of course, measure the frequency in cycles per second (as opposed to radians per second):Hence, we get the following expression for the so-called anomalous magnetic moment of an electron ae:Hence, the so-called anomalous magnetic moment of an electron is nothing but the ratio of two mathematical factors – definitions, basically – which we can express in terms of actual frequencies:Our formula for ae now becomes:Of course, if we use the μ/J = 2m/q equality, then the fp/fc ratio will be equal to 1/2, and ae will not be zero but −1/2:However, as mentioned above, we should not do that. The precession causes the magnetic moment and the angular momentum to wobble. Hence, there is nothing anomalous about the anomalous magnetic moment. We should not wonder why its value is not zero. We should wonder why it is so nearly zero.
 Needless to say, the tesla is the SI unit for the magnitude of a magnetic field. We can also write it as [B] = N/(m∙A), using the SI unit for current, i.e. the ampere (A). Now, 1 C = 1 A∙s and, hence, 1 N/(m∙A) = 1 (N/C)/(m/s). Hence, the physical dimension of the magnetic field is the physical dimension of the electric field (N/C) divided by m/s. We like the [E] = [B]·m/s expression because it reflects the geometry of the electric and magnetic field vectors.
 See: Physics Today, 1 August 2006, p. 15 (https://physicstoday.scitation.org/doi/10.1063/1.2349714). The article also explains the methodology of the experiment in terms of the frequency measurements, which we explain above.
 See: G. Gabrielse, D. Hanneke, T. Kinoshita, M. Nio, and B. Odom, New Determination of the Fine Structure Constant from the Electron g Value and QED, Phys. Rev. Lett. 97, 030802 (2006). More recent theory and experiments may have come up with an even more precise number.
 Our derivation is based on the following reference: https://www.didaktik.physik.uni-muenchen.de/elektronenbahnen/en/b-feld/anwendung/zyklotron2.php.
 J is the symbol which Feynman uses. In many articles and textbooks, one will read L instead of J. Note that the symbols may be confusing: I is a current, but I is the moment of inertia. It is equal to m·r2 for a rotating mass.
 We like the intuitive – but precise – explanation in Feynman’s Lectures (II-34-3), from which we also copied the illustration.
My dear readers – I haven’t published much lately, because I try to summarize my ideas now in short articles that might be suitable for publication in a journal. I think the latest one (on Einstein’s mass-energy relation) should be of interest. Let me just insert the summary here:
The radial velocity formula and the Planck-Einstein relation give us the Zitterbewegung (zbw) frequency (E = ħω = E/ħ) and zbw radius (a = c/ω = cħ/mc2 = ħ/mc) of the electron. We interpret this by noting that the c = aω identity gives us the E = mc2 = ma2ω2 equation, which suggests we should combine the total energy (kinetic and potential) of two harmonic oscillators to explain the electron mass. We do so by interpreting the elementary wavefunction as a two-dimensional (harmonic) electromagnetic oscillation in real space which drives the pointlike charge along the zbw current ring. This implies a dual view of the reality of the real and imaginary part of the wavefunction:
- The x = acos(ωt) and y = a·sin(ωt) equations describe the motion of the pointlike charge.
- As an electromagnetic oscillation, we write it as E0 = E0cos(ωt+π/2) + i·E0·sin(ωt+π/2).
The magnitudes of the oscillation a and E0 are expressed in distance (m) and force per unit charge (N/C) respectively and are related because the energy of both oscillations is one and the same. The model – which implies the energy of the oscillation and, therefore, the effective mass of the electron is spread over the zbw disk – offers an equally intuitive explanation for the angular momentum, magnetic moment and the g-factor of charged spin-1/2 particles. Most importantly, the model also offers us an intuitive interpretation of Einstein’s enigmatic mass-energy equivalence relation. Going from the stationary to the moving reference frame, we argue that the plane of the zbw oscillation should be parallel to the direction of motion so as to be consistent with the results of the Stern-Gerlach experiment.
So… Well… Have fun with it ! I think I am going to sign off. 🙂 Yours – JL
This post explores the limits of the physical interpretation of the wavefunction we have been building up in previous posts. It does so by examining if it can be used to provide a hidden-variable theory for explaining quantum-mechanical interference. The hidden variable is the polarization state of the photon.
The outcome is as expected: the theory does not work. Hence, this paper clearly shows the limits of any physical or geometric interpretation of the wavefunction.
This post sounds somewhat academic because it is, in fact, a draft of a paper I might try to turn into an article for a journal. There is a useful addendum to the post below: it offers a more sophisticated analysis of linear and circular polarization states (see: Linear and Circular Polarization States in the Mach-Zehnder Experiment). Have fun with it !
Duns Scotus wrote: pluralitas non est ponenda sine necessitate. Plurality is not to be posited without necessity. And William of Ockham gave us the intuitive lex parsimoniae: the simplest solution tends to be the correct one. But redundancy in the description does not seem to bother physicists. When explaining the basic axioms of quantum physics in his famous Lectures on quantum mechanics, Richard Feynman writes:
“We are not particularly interested in the mathematical problem of finding the minimum set of independent axioms that will give all the laws as consequences. Redundant truth does not bother us. We are satisfied if we have a set that is complete and not apparently inconsistent.”
Also, most introductory courses on quantum mechanics will show that both ψ = exp(iθ) = exp[i(kx-ωt)] and ψ* = exp(-iθ) = exp[-i(kx-ωt)] = exp[i(ωt-kx)] = -ψ are acceptable waveforms for a particle that is propagating in the x-direction. Both have the required mathematical properties (as opposed to, say, some real-valued sinusoid). We would then think some proof should follow of why one would be better than the other or, preferably, one would expect as a discussion on what these two mathematical possibilities might represent¾but, no. That does not happen. The physicists conclude that “the choice is a matter of convention and, happily, most physicists use the same convention.”
Instead of making a choice here, we could, perhaps, use the various mathematical possibilities to incorporate spin in the description, as real-life particles – think of electrons and photons here – have two spin states (up or down), as shown below.
Table 1: Matching mathematical possibilities with physical realities?
|Spin and direction||Spin up||Spin down|
|Positive x-direction||ψ = exp[i(kx-ωt)]||ψ* = exp[i(ωt-kx)]|
|Negative x-direction||χ = exp[i(ωt-kx)]||χ* = exp[i(kx+ωt)]|
That would make sense – for several reasons. First, theoretical spin-zero particles do not exist and we should therefore, perhaps, not use the wavefunction to describe them. More importantly, it is relatively easy to show that the weird 720-degree symmetry of spin-1/2 particles collapses into an ordinary 360-degree symmetry and that we, therefore, would have no need to describe them using spinors and other complicated mathematical objects. Indeed, the 720-degree symmetry of the wavefunction for spin-1/2 particles is based on an assumption that the amplitudes C’up = -Cup and C’down = -Cdown represent the same state—the same physical reality. As Feynman puts it: “Both amplitudes are just multiplied by −1 which gives back the original physical system. It is a case of a common phase change.”
In the physical interpretation given in Table 1, these amplitudes do not represent the same state: the minus sign effectively reverses the spin direction. Putting a minus sign in front of the wavefunction amounts to taking its complex conjugate: -ψ = ψ*. But what about the common phase change? There is no common phase change here: Feynman’s argument derives the C’up = -Cup and C’down = -Cdown identities from the following equations: C’up = eiπCup and C’down = e–iπCdown. The two phase factors are not the same: +π and -π are not the same. In a geometric interpretation of the wavefunction, +π is a counterclockwise rotation over 180 degrees, while -π is a clockwise rotation. We end up at the same point (-1), but it matters how we get there: -1 is a complex number with two different meanings.
We have written about this at length and, hence, we will not repeat ourselves here. However, this realization – that one of the key propositions in quantum mechanics is basically flawed – led us to try to question an axiom in quantum math that is much more fundamental: the loss of determinism in the description of interference.
The reader should feel reassured: the attempt is, ultimately, not successful—but it is an interesting exercise.
The standard MIT course on quantum physics vaguely introduces Bell’s Theorem – labeled as a proof of what is referred to as the inevitable loss of determinism in quantum mechanics – early on. The argument is as follows. If we have a polarizer whose optical axis is aligned with, say, the x-direction, and we have light coming in that is polarized along some other direction, forming an angle α with the x-direction, then we know – from experiment – that the intensity of the light (or the fraction of the beam’s energy, to be precise) that goes through the polarizer will be equal to cos2α.
But, in quantum mechanics, we need to analyze this in terms of photons: a fraction cos2α of the photons must go through (because photons carry energy and that’s the fraction of the energy that is transmitted) and a fraction 1-cos2α must be absorbed. The mentioned MIT course then writes the following:
“If all the photons are identical, why is it that what happens to one photon does not happen to all of them? The answer in quantum mechanics is that there is indeed a loss of determinism. No one can predict if a photon will go through or will get absorbed. The best anyone can do is to predict probabilities. Two escape routes suggest themselves. Perhaps the polarizer is not really a homogeneous object and depending exactly on where the photon is it either gets absorbed or goes through. Experiments show this is not the case.
A more intriguing possibility was suggested by Einstein and others. A possible way out, they claimed, was the existence of hidden variables. The photons, while apparently identical, would have other hidden properties, not currently understood, that would determine with certainty which photon goes through and which photon gets absorbed. Hidden variable theories would seem to be untestable, but surprisingly they can be tested. Through the work of John Bell and others, physicists have devised clever experiments that rule out most versions of hidden variable theories. No one has figured out how to restore determinism to quantum mechanics. It seems to be an impossible task.”
The student is left bewildered here. Are there only two escape routes? And is this the way how polarization works, really? Are all photons identical? The Uncertainty Principle tells us that their momentum, position, or energy will be somewhat random. Hence, we do not need to assume that the polarizer is nonhomogeneous, but we need to think of what might distinguish the individual photons.
Considering the nature of the problem – a photon goes through or it doesn’t – it would be nice if we could find a binary identifier. The most obvious candidate for a hidden variable would be the polarization direction. If we say that light is polarized along the x-direction, we should, perhaps, distinguish between a plus and a minus direction? Let us explore this idea.
The simple experiment above – linearly polarized light going through a polaroid – involves linearly polarized light. We can easily distinguish between left- and right-hand circular polarization, but if we have linearly polarized light, can we distinguish between a plus and a minus direction? Maybe. Maybe not. We can surely think about different relative phases and how that could potentially have an impact on the interaction with the molecules in the polarizer.
Suppose the light is polarized along the x-direction. We know the component of the electric field vector along the y-axis will then be equal to Ey = 0, and the magnitude of the x-component of E will be given by a sinusoid. However, here we have two distinct possibilities: Ex = cos(ω·t) or, alternatively, Ex = sin(ω·t). These are the same functions but – crucially important – with a phase difference of 90°: sin(ω·t) = cos(ω·t + π/2).
Figure 1: Two varieties of linearly polarized light?
Would this matter? Sure. We can easily come up with some classical explanations of why this would matter. Think, for example, of birefringent material being defined in terms of quarter-wave plates. In fact, the more obvious question is: why would this not make a difference?
Of course, this triggers another question: why would we have two possibilities only? What if we add an additional 90° shift to the phase? We know that cos(ω·t + π) = –cos(ω·t). We cannot reduce this to cos(ω·t) or sin(ω·t). Hence, if we think in terms of 90° phase differences, then –cos(ω·t) = cos(ω·t + π) and –sin(ω·t) = sin(ω·t + π) are different waveforms too. In fact, why should we think in terms of 90° phase shifts only? Why shouldn’t we think of a continuum of linear polarization states?
We have no sensible answer to that question. We can only say: this is quantum mechanics. We think of a photon as a spin-one particle and, for that matter, as a rather particular one, because it misses the zero state: it is either up, or down. We may now also assume two (linear) polarization states for the molecules in our polarizer and suggest a basic theory of interaction that may or may not explain this very basic fact: a photon gets absorbed, or it gets transmitted. The theory is that if the photon and the molecule are in the same (linear) polarization state, then we will have constructive interference and, somehow, a photon gets through. If the linear polarization states are opposite, then we will have destructive interference and, somehow, the photon is absorbed. Hence, our hidden variables theory for the simple situation that we discussed above (a photon does or does not go through a polarizer) can be summarized as follows:
|Linear polarization state||Incoming photon up (+)||Incoming photon down (-)|
|Polarizer molecule up (+)||Constructive interference: photon goes through||Destructive interference: photon is absorbed|
|Polarizer molecule down (-)||Destructive interference: photon is absorbed||Constructive interference: photon goes through|
Nice. No loss of determinism here. But does it work? The quantum-mechanical mathematical framework is not there to explain how a polarizer could possibly work. It is there to explain the interference of a particle with itself. In Feynman’s words, this is the phenomenon “which is impossible, absolutely impossible, to explain in any classical way, and which has in it the heart of quantum mechanics.”
So, let us try our new theory of polarization states as a hidden variable on one of those interference experiments. Let us choose the standard one: the Mach-Zehnder interferometer experiment.
The setup of the Mach-Zehnder interferometer is well known and should, therefore, probably not require any explanation. We have two beam splitters (BS1 and BS2) and two perfect mirrors (M1 and M2). An incident beam coming from the left is split at BS1 and recombines at BS2, which sends two outgoing beams to the photon detectors D0 and D1. More importantly, the interferometer can be set up to produce a precise interference effect which ensures all the light goes into D0, as shown below. Alternatively, the setup may be altered to ensure all the light goes into D1.
Figure 2: The Mach-Zehnder interferometer
The classical explanation is easy enough. It is only when we think of the beam as consisting of individual photons that we get in trouble. Each photon must then, somehow, interfere with itself which, in turn, requires the photon to, somehow, go through both branches of the interferometer at the same time. This is solved by the magical concept of the probability amplitude: we think of two contributions a and b (see the illustration above) which, just like a wave, interfere to produce the desired result¾except that we are told that we should not try to think of these contributions as actual waves.
So that is the quantum-mechanical explanation and it sounds crazy and so we do not want to believe it. Our hidden variable theory should now show the photon does travel along one path only. If the apparatus is set up to get all photons in the D0 detector, then we might, perhaps, have a sequence of events like this:
|Photon polarization||At BS1||At BS2||Final result|
|Up (+)||Photon is reflected||Photon is reflected||Photon goes to D0|
|Down (–)||Photon is transmitted||Photon is transmitted||Photon goes to D0|
Of course, we may also set up the apparatus to get all photons in the D1 detector, in which case the sequence of events might be this:
|Photon polarization||At BS1||At BS2||Final result|
|Up (+)||Photon is reflected||Photon is transmitted||Photon goes to D1|
|Down (–)||Photon is transmitted||Photon is reflected||Photon goes to D1|
This is a nice symmetrical explanation that does not involve any quantum-mechanical weirdness. The problem is: it cannot work. Why not? What happens if we block one of the two paths? For example, let us block the lower path in the setup where all photons went to D0. We know – from experiment – that the outcome will be the following:
|Photon is absorbed at the block||0.50|
|Photon goes to D0||0.25|
|Photon goes to D1||0.25|
How is this possible? Before blocking the lower path, no photon went to D1. They all went to D0. If our hidden variable theory was correct, the photons that do not get absorbed should also go to D0, as shown below.
|Photon polarization||At BS1||At BS2||Final result|
|Up (+)||Photon is reflected||Photon is reflected||Photon goes to D0|
|Down (–)||Photon is absorbed||Photon was absorbed||Photon was absorbed|
Our hidden variable theory does not work. Physical or geometric interpretations of the wavefunction are nice, but they do not explain quantum-mechanical interference. Their value is, therefore, didactic only.
Jean Louis Van Belle, 2 November 2018
This paper discusses general principles in physics only. Hence, references were limited to references to general textbooks and courses and physics textbooks only. The two key references here are the MIT introductory course on quantum physics and Feynman’s Lectures – both of which can be consulted online. Additional references to other material are given in the text itself (see footnotes).
 Duns Scotus, Commentaria.
 Feynman’s Lectures on Quantum Mechanics, Vol. III, Chapter 5, Section 5.
 See, for example, the MIT’s edX Course 8.04.1x (Quantum Physics), Lecture Notes, Chapter 4, Section 3.
 Photons are spin-one particles but they do not have a spin-zero state.
 Of course, the formulas only give the elementary wavefunction. The wave packet will be a Fourier sum of such functions.
 See, for example, https://warwick.ac.uk/fac/sci/physics/staff/academic/mhadley/explanation/spin/, accessed on 30 October 2018
 Feynman’s Lectures on Quantum Mechanics, Vol. III, Chapter 6, Section 3.
 See: MIT edX Course 8.04.1x (Quantum Physics), Lecture Notes, Chapter 1, Section 3 (Loss of determinism).
 The z-direction is the direction of wave propagation in this example. In quantum mechanics, we often define the direction of wave propagation as the x-direction. This will, hopefully, not confuse the reader. The choice of axes is usually clear from the context.
 Source of the illustration: https://upload.wikimedia.org/wikipedia/commons/7/71/Sine_cosine_one_period.svg..
 Classical theory assumes an atomic or molecular system will absorb a photon and, therefore, be in an excited state (with higher energy). The atomic or molecular system then goes back into its ground state by emitting another photon with the same energy. Hence, we should probably not think in terms of a specific photon getting through.
 Feynman’s Lectures on Quantum Mechanics, Vol. III, Chapter 1, Section 1.
 Source of the illustration: MIT edX Course 8.04.1x (Quantum Physics), Lecture Notes, Chapter 1, Section 4 (Quantum Superpositions).
I think I cracked the nut. Academics always throw two nasty arguments into the discussion on any geometric or physical interpretations of the wavefunction:
- The superposition of wavefunctions is done in the complex space and, hence, the assumption of a real-valued envelope for the wavefunction is, therefore, not acceptable.
- The wavefunction for spin-1/2 particles cannot represent any real object because of its 720-degree symmetry in space. Real objects have the same spatial symmetry as space itself, which is 360 degrees. Hence, physical interpretations of the wavefunction are nonsensical.
Well… I’ve finally managed to deconstruct those arguments – using, paradoxically, Feynman’s own arguments against him. Have a look: click the link to my latest paper ! Enjoy !
I realized that my last posts were just some crude and rude soundbites, so I thought it would be good to briefly summarize them into something more coherent. Please let me know what you think of it.
The Uncertainty Principle: epistemology versus physics
Anyone who has read anything about quantum physics will know that its concepts and principles are very non-intuitive. Several interpretations have therefore emerged. The mainstream interpretation of quantum mechanics is referred to as the Copenhagen interpretation. It mainly distinguishes itself from more frivolous interpretations (such as the many-worlds and the pilot-wave interpretations) because it is… Well… Less frivolous. Unfortunately, the Copenhagen interpretation itself seems to be subject to interpretation.
One such interpretation may be referred to as radical skepticism – or radical empiricism: we can only say something meaningful about Schrödinger’s cat if we open the box and observe its state. According to this rather particular viewpoint, we cannot be sure of its reality if we don’t make the observation. All we can do is describe its reality by a superposition of the two possible states: dead or alive. That’s Hilbert’s logic: the two states (dead or alive) are mutually exclusive but we add them anyway. If a tree falls in the wood and no one hears it, then it is both standing and not standing. Richard Feynman – who may well be the most eminent representative of mainstream physics – thinks this epistemological position is nonsensical, and I fully agree with him:
“A real tree falling in a real forest makes a sound, of course, even if nobody is there. Even if no one is present to hear it, there are other traces left. The sound will shake some leaves, and if we were careful enough we might find somewhere that some thorn had rubbed against a leaf and made a tiny scratch that could not be explained unless we assumed the leaf were vibrating.” (Feynman’s Lectures, III-2-6)
So what is the mainstream physicist’s interpretation of the Copenhagen interpretation of quantum mechanics then? To fully answer that question, I should encourage the reader to read all of Feynman’s Lectures on quantum mechanics. But then you are reading this because you don’t want to do that, so let me quote from his introductory Lecture on the Uncertainty Principle: “Making an observation affects the phenomenon. The point is that the effect cannot be disregarded or minimized or decreased arbitrarily by rearranging the apparatus. When we look for a certain phenomenon we cannot help but disturb it in a certain minimum way.” (ibidem)
It has nothing to do with consciousness. Reality and consciousness are two very different things. After having concluded the tree did make a noise, even if no one was there to hear it, he wraps up the philosophical discussion as follows: “We might ask: was there a sensation of sound? No, sensations have to do, presumably, with consciousness. And whether ants are conscious and whether there were ants in the forest, or whether the tree was conscious, we do not know. Let us leave the problem in that form.” In short, I think we can all agree that the cat is dead or alive, or that the tree is standing or not standing¾regardless of the observer. It’s a binary situation. Not something in-between. The box obscures our view. That’s all. There is nothing more to it.
Of course, in quantum physics, we don’t study cats but look at the behavior of photons and electrons (we limit our analysis to quantum electrodynamics – so we won’t discuss quarks or other sectors of the so-called Standard Model of particle physics). The question then becomes: what can we reasonably say about the electron – or the photon – before we observe it, or before we make any measurement. Think of the Stein-Gerlach experiment, which tells us that we’ll always measure the angular momentum of an electron – along any axis we choose – as either +ħ/2 or, else, as -ħ/2. So what’s its state before it enters the apparatus? Do we have to assume it has some definite angular momentum, and that its value is as binary as the state of our cat (dead or alive, up or down)?
We should probably explain what we mean by a definite angular momentum. It’s a concept from classical physics, and it assumes a precise value (or magnitude) along some precise direction. We may challenge these assumptions. The direction of the angular momentum may be changing all the time, for example. If we think of the electron as a pointlike charge – whizzing around in its own space – then the concept of a precise direction of its angular momentum becomes quite fuzzy, because it changes all the time. And if its direction is fuzzy, then its value will be fuzzy as well. In classical physics, such fuzziness is not allowed, because angular momentum is conserved: it takes an outside force – or torque – to change it. But in quantum physics, we have the Uncertainty Principle: some energy (force over a distance, remember) can be borrowed – so to speak – as long as it’s swiftly being returned – within the quantitative limits set by the Uncertainty Principle: ΔE·Δt = ħ/2.
Mainstream physicists – including Feynman – do not try to think about this. For them, the Stern-Gerlach apparatus is just like Schrödinger’s box: it obscures the view. The cat is dead or alive, and each of the two states has some probability – but they must add up to one – and so they will write the state of the electron before it enters the apparatus as the superposition of the up and down states. I must assume you’ve seen this before:
|ψ〉 = Cup|up〉 + Cdown|down〉
It’s the so-called Dirac or bra-ket notation. Cup is the amplitude for the electron spin to be equal to +ħ/2 along the chosen direction – which we refer to as the z-direction because we will choose our reference frame such that the z-axis coincides with this chosen direction – and, likewise, Cup is the amplitude for the electron spin to be equal to -ħ/2 (along the same direction, obviously). Cup and Cup will be functions, and the associated probabilities will vary sinusoidally – with a phase difference so as to make sure both add up to one.
The model is consistent, but it feels like a mathematical trick. This description of reality – if that’s what it is – does not feel like a model of a real electron. It’s like reducing the cat in our box to the mentioned fuzzy state of being alive and dead at the same time. Let’s try to come up with something more exciting. 😊
 Academics will immediately note that radical empiricism and radical skepticism are very different epistemological positions but we are discussing some basic principles in physics here rather than epistemological theories.
 The reference to Hilbert’s logic refers to Hilbert spaces: a Hilbert space is an abstract vector space. Its properties allow us to work with quantum-mechanical states, which become state vectors. You should not confuse them with the real or complex vectors you’re used to. The only thing state vectors have in common with real or complex vectors is that (1) we also need a base (aka as a representation in quantum mechanics) to define them and (2) that we can make linear combinations.
The ‘flywheel’ electron model
Physicists describe the reality of electrons by a wavefunction. If you are reading this article, you know how a wavefunction looks like: it is a superposition of elementary wavefunctions. These elementary wavefunctions are written as Ai·exp(-iθi), so they have an amplitude Ai and an argument θi = (Ei/ħ)·t – (pi/ħ)·x. Let’s forget about uncertainty, so we can drop the index (i) and think of a geometric interpretation of A·exp(-iθ) = A·e–iθ.
Here we have a weird thing: physicists think the minus sign in the exponent (-iθ) should always be there: the convention is that we get the imaginary unit (i) by a 90° rotation of the real unit (1) – but the rotation is counterclockwise rotation. I like to think a rotation in the clockwise direction must also describe something real. Hence, if we are seeking a geometric interpretation, then we should explore the two mathematical possibilities: A·e–iθ and A·e+iθ. I like to think these two wavefunctions describe the same electron but with opposite spin. How should we visualize this? I like to think of A·e–iθ and A·e+iθ as two-dimensional harmonic oscillators:
A·e–iθ = cos(-θ) + i·sin(-θ) = cosθ – i·sinθ
A·e+iθ = cosθ + i·sinθ
So we may want to imagine our electron as a pointlike electric charge (see the green dot in the illustration below) to spin around some center in either of the two possible directions. The cosine keeps track of the oscillation in one dimension, while the sine (plus or minus) keeps track of the oscillation in a direction that is perpendicular to the first one.
Figure 1: A pointlike charge in orbit
So we have a weird oscillator in two dimensions here, and we may calculate the energy in this oscillation. To calculate such energy, we need a mass concept. We only have a charge here, but a (moving) charge has an electromagnetic mass. Now, the electromagnetic mass of the electron’s charge may or may not explain all the mass of the electron (most physicists think it doesn’t) but let’s assume it does for the sake of the model that we’re trying to build up here. The point is: the theory of electromagnetic mass gives us a very simple explanation for the concept of mass here, and so we’ll use it for the time being. So we have some mass oscillating in two directions simultaneously: we basically assume space is, somehow, elastic. We have worked out the V-2 engine metaphor before, so we won’t repeat ourselves here.
Figure 2: A perpetuum mobile?
Previously unrelated but structurally similar formulas may be related here:
- The energy of an oscillator: E = (1/2)·m·a2ω2
- Kinetic energy: E = (1/2)·m·v2
- The rotational (kinetic) energy that’s stored in a flywheel: E = (1/2)·I·ω2 = (1/2)·m·r2·ω2
- Einstein’s energy-mass equivalence relation: E = m·c2
Of course, we are mixing relativistic and non-relativistic formulas here, and there’s the 1/2 factor – but these are minor issues. For example, we were talking not one but two oscillators, so we should add their energies: (1/2)·m·a2·ω2 + (1/2)·m·a2·ω2 = m·a2·ω2. Also, one can show that the classical formula for kinetic energy (i.e. E = (1/2)·m·v2) morphs into E = m·c2 when we use the relativistically correct force equation for an oscillator. So, yes, our metaphor – or our suggested physical interpretation of the wavefunction, I should say – makes sense.
If you know something about physics, then you know the concept of the electromagnetic mass – its mathematical derivation, that is – gives us the classical electron radius, aka as the Thomson radius. It’s the smallest of a trio of radii that are relevant when discussing electrons: the other two radii are the Bohr radius and the Compton scattering radius respectively. The Thomson radius is used in the context of elastic scattering: the frequency of the incident particle (usually a photon), and the energy of the electron itself, do not change. In contrast, Compton scattering does change the frequency of the photon that is being scattered, and also impacts the energy of our electron. [As for the Bohr radius, you know that’s the radius of an electron orbital, roughly speaking – or the size of a hydrogen atom, I should say.]
Now, if we combine the E = m·a2·ω2 and E = m·c2 equations, then a·ω must be equal to c, right? Can we show this? Maybe. It is easy to see that we get the desired equality by substituting the amplitude of the oscillation (a) for the Compton scattering radius r = ħ/(m·c), and ω (the (angular) frequency of the oscillation) by using the Planck relation (ω = E/ħ):
a·ω = [ħ/(m·c)]·[E/ħ] = E/(m·c) = m·c2/(m·c) = c
We get a wonderfully simple geometric model of an electron here: an electric charge that spins around in a plane. Its radius is the Compton electron radius – which makes sense – and the radial velocity of our spinning charge is the speed of light – which may or may not make sense. Of course, we need an explanation of why this spinning charge doesn’t radiate its energy away – but then we don’t have such explanation anyway. All we can say is that the electron charge seems to be spinning in its own space – that it’s racing along a geodesic. It’s just like mass creates its own space here: according to Einstein’s general relativity theory, gravity becomes a pseudo-force—literally: no real force. How? I am not sure: the model here assumes the medium – empty space – is, somehow, perfectly elastic: the electron constantly borrows energy from one direction and then returns it to the other – so to speak. A crazy model, yes – but is there anything better? We only want to present a metaphor here: a possible visualization of quantum-mechanical models.
However, if this model is to represent anything real, then many more questions need to be answered. For starters, let’s think about an interpretation of the results of the Stern-Gerlach experiment.
A spinning charge is a tiny magnet – and so it’s got a magnetic moment, which we need to explain the Stern-Gerlach experiment. But it doesn’t explain the discrete nature of the electron’s angular momentum: it’s either +ħ/2 or -ħ/2, nothing in-between, and that’s the case along any direction we choose. How can we explain this? Also, space is three-dimensional. Why would electrons spin in a perfect plane? The answer is: they don’t.
Indeed, the corollary of the above-mentioned binary value of the angular momentum is that the angular momentum – or the electron’s spin – is never completely along any direction. This may or may not be explained by the precession of a spinning charge in a field, which is illustrated below (illustration taken from Feynman’s Lectures, II-35-3).
Figure 3: Precession of an electron in a magnetic field
So we do have an oscillation in three dimensions here, really – even if our wavefunction is a two-dimensional mathematical object. Note that the measurement (or the Stein-Gerlach apparatus in this case) establishes a line of sight and, therefore, a reference frame, so ‘up’ and ‘down’, ‘left’ and ‘right’, and ‘in front’ and ‘behind’ get meaning. In other words, we establish a real space. The question then becomes: how and why does an electron sort of snap into place?
The geometry of the situation suggests the logical angle of the angular momentum vector should be 45°. Now, if the value of its z-component (i.e. its projection on the z-axis) is to be equal to ħ/2, then the magnitude of J itself should be larger. To be precise, it should be equal to ħ/√2 ≈ 0.7·ħ (just apply Pythagoras’ Theorem). Is that value compatible with our flywheel model?
Maybe. Let’s see. The classical formula for the magnetic moment is μ = I·A, with I the (effective) current and A the (surface) area. The notation is confusing because I is also used for the moment of inertia, or rotational mass, but… Well… Let’s do the calculation. The effective current is the electron charge (qe) divided by the period (T) of the orbital revolution: : I = qe/T. The period of the orbit is the time that is needed for the electron to complete one loop. That time (T) is equal to the circumference of the loop (2π·a) divided by the tangential velocity (vt). Now, we suggest vt = r·ω = a·ω = c, and the circumference of the loop is 2π·a. For a, we still use the Compton radius a = ħ/(m·c). Now, the formula for the area is A = π·a2, so we get:
μ = I·A = [qe/T]·π·a2 = [qe·c/(2π·a)]·[π·a2] = [(qe·c)/2]·a = [(qe·c)/2]·[ħ/(m·c)] = [qe/(2m)]·ħ
In a classical analysis, we have the following relation between angular momentum and magnetic moment:
μ = (qe/2m)·J
Hence, we find that the angular momentum J is equal to ħ, so that’s twice the measured value. We’ve got a problem. We would have hoped to find ħ/2 or ħ/√2. Perhaps it’s because a = ħ/(m·c) is the so-called reduced Compton scattering radius…
Maybe we’ll find the solution one day. I think it’s already quite nice we have a model that’s accurate up to a factor of 1/2 or 1/√2. 😊
Post scriptum: I’ve turned this into a small article which may or may not be more readable. You can link to it here. Comments are more than welcome.
A lot of the Uncertainty in quantum mechanics is suspiciously certain. For example, we know an electron will always have its spin up or down, in any direction along which we choose to measure it, and the value of the angular momentum will, accordingly, be measured as plus or minus ħ/2. That doesn’t sound uncertain to me. In fact, it sounds remarkably certain, doesn’t it? We know – we are sure, in fact, because of countless experiments – that the electron will be in either of those two states, and we also know that these two states are separated by ħ, Planck’s quantum of action, exactly.
Of course, the corollary of this is that the idea of the direction of the angular momentum is a rather fuzzy concept. As Feynman convincingly demonstrates, it is ‘never completely along any direction’. Why? Well… Perhaps it can be explained by the idea of precession?
In fact, the idea of precession might also explain the weird 720° degree symmetry of the wavefunction.
Hmm… Now that is an idea to look into ! 🙂
When modeling electromagnetic waves, the notion of left versus right circular polarization is quite clear and fully integrated in the mathematical treatment. In contrast, quantum math sticks to the very conventional idea that the imaginary unit (i) is – always! – a counter-clockwise rotation by 90 degrees. We all know that –i would do just as an imaginary unit as i, because the definition of the imaginary unit says the only requirement is that its square has to be equal to –1, and (–i)2 is also equal to –1.
So we actually have two imaginary units: i and –i. However, physicists stubbornly think there is only one direction for measuring angles, and that is counter-clockwise. That’s a mathematical convention, Professor: it’s something in your head only. It is not real. Nature doesn’t care about our conventions and, therefore, I feel the spin ‘up’ versus spin ‘down’ should correspond to the two mathematical possibilities: if the ‘up’ state is represented by some complex function, then the ‘down’ state should be represented by its complex conjugate.
This ‘additional’ rule wouldn’t change the basic quantum-mechanical rules – which are written in terms of state vectors in a Hilbert space (and, yes, a Hilbert space is an unreal as it sounds: its rules just say you should separate cats and dogs while adding them – which is very sensible advice, of course). However, they would, most probably (just my intuition – I need to prove it), avoid these crazy 720 degree symmetries which inspire the likes of Penrose to say there is no physical interpretation on the wavefunction.
Oh… As for the title of my post… I think it would be a great title for a book – because I’ll need some space to work it all out. 🙂
This post is basically a continuation of my previous one but – as you can see from its title – it is much more aggressive in its language, as I was inspired by a very thoughtful comment on my previous post. Another advantage is that it avoids all of the math. 🙂 It’s… Well… I admit it: it’s just a rant. 🙂 [Those who wouldn’t appreciate the casual style of what follows, can download my paper on it – but that’s much longer and also has a lot more math in it – so it’s a much harder read than this ‘rant’.]
My previous post was actually triggered by an attempt to re-read Feynman’s Lectures on Quantum Mechanics, but in reverse order this time: from the last chapter to the first. [In case you doubt, I did follow the correct logical order when working my way through them for the first time because… Well… There is no other way to get through them otherwise. 🙂 ] But then I was looking at Chapter 20. It’s a Lecture on quantum-mechanical operators – so that’s a topic which, in other textbooks, is usually tackled earlier on. When re-reading it, I realize why people quickly turn away from the topic of physics: it’s a lot of mathematical formulas which are supposed to reflect reality but, in practice, few – if any – of the mathematical concepts are actually being explained. Not in the first chapters of a textbook, not in its middle ones, and… Well… Nowhere, really. Why? Well… To be blunt: I think most physicists themselves don’t really understand what they’re talking about. In fact, as I have pointed out a couple of times already, Feynman himself admits so much:
“Atomic behavior appears peculiar and mysterious to everyone—both to the novice and to the experienced physicist. Even the experts do not understand it the way they would like to.”
So… Well… If you’d be in need of a rather spectacular acknowledgement of the shortcomings of physics as a science, here you have it: if you don’t understand what physicists are trying to tell you, don’t worry about it, because they don’t really understand it themselves. 🙂
Take the example of a physical state, which is represented by a state vector, which we can combine and re-combine using the properties of an abstract Hilbert space. Frankly, I think the word is very misleading, because it actually doesn’t describe an actual physical state. Why? Well… If we look at this so-called physical state from another angle, then we need to transform it using a complicated set of transformation matrices. You’ll say: that’s what we need to do when going from one reference frame to another in classical mechanics as well, isn’t it?
Well… No. In classical mechanics, we’ll describe the physics using geometric vectors in three dimensions and, therefore, the base of our reference frame doesn’t matter: because we’re using real vectors (such as the electric of magnetic field vectors E and B), our orientation vis-á-vis the object – the line of sight, so to speak – doesn’t matter.
In contrast, in quantum mechanics, it does: Schrödinger’s equation – and the wavefunction – has only two degrees of freedom, so to speak: its so-called real and its imaginary dimension. Worse, physicists refuse to give those two dimensions any geometric interpretation. Why? I don’t know. As I show in my previous posts, it would be easy enough, right? We know both dimensions must be perpendicular to each other, so we just need to decide if both of them are going to be perpendicular to our line of sight. That’s it. We’ve only got two possibilities here which – in my humble view – explain why the matter-wave is different from an electromagnetic wave.
I actually can’t quite believe the craziness when it comes to interpreting the wavefunction: we get everything we’d want to know about our particle through these operators (momentum, energy, position, and whatever else you’d need to know), but mainstream physicists still tell us that the wavefunction is, somehow, not representing anything real. It might be because of that weird 720° symmetry – which, as far as I am concerned, confirms that those state vectors are not the right approach: you can’t represent a complex, asymmetrical shape by a ‘flat’ mathematical object!
Huh? Yes. The wavefunction is a ‘flat’ concept: it has two dimensions only, unlike the real vectors physicists use to describe electromagnetic waves (which we may interpret as the wavefunction of the photon). Those have three dimensions, just like the mathematical space we project on events. Because the wavefunction is flat (think of a rotating disk), we have those cumbersome transformation matrices: each time we shift position vis-á-vis the object we’re looking at (das Ding an sich, as Kant would call it), we need to change our description of it. And our description of it – the wavefunction – is all we have, so that’s our reality. However, because that reality changes as per our line of sight, physicists keep saying the wavefunction (or das Ding an sich itself) is, somehow, not real.
Frankly, I do think physicists should take a basic philosophy course: you can’t describe what goes on in three-dimensional space if you’re going to use flat (two-dimensional) concepts, because the objects we’re trying to describe (e.g. non-symmetrical electron orbitals) aren’t flat. Let me quote one of Feynman’s famous lines on philosophers: “These philosophers are always with us, struggling in the periphery to try to tell us something, but they never really understand the subtleties and depth of the problem.” (Feynman’s Lectures, Vol. I, Chapter 16)
Now, I love Feynman’s Lectures but… Well… I’ve gone through them a couple of times now, so I do think I have an appreciation of the subtleties and depth of the problem now. And I tend to agree with some of the smarter philosophers: if you’re going to use ‘flat’ mathematical objects to describe three- or four-dimensional reality, then such approach will only get you where we are right now, and that’s a lot of mathematical mumbo-jumbo for the poor uninitiated. Consistent mumbo-jumbo, for sure, but mumbo-jumbo nevertheless. 🙂 So, yes, I do think we need to re-invent quantum math. 🙂 The description may look more complicated, but it would make more sense.
I mean… If physicists themselves have had continued discussions on the reality of the wavefunction for almost a hundred years now (Schrödinger published his equation in 1926), then… Well… Then the physicists have a problem. Not the philosophers. 🙂 As to how that new description might look like, see my papers on viXra.org. I firmly believe it can be done. This is just a hobby of mine, but… Well… That’s where my attention will go over the coming years. 🙂 Perhaps quaternions are the answer but… Well… I don’t think so either – for reasons I’ll explain later. 🙂
Post scriptum: There are many nice videos on Dirac’s belt trick or, more generally, on 720° symmetries, but this links to one I particularly like. It clearly shows that the 720° symmetry requires, in effect, a special relation between the observer and the object that is being observed. It is, effectively, like there is a leather belt between them or, in this case, we have an arm between the glass and the person who is holding the glass. So it’s not like we are walking around the object (think of the glass of water) and making a full turn around it, so as to get back to where we were. No. We are turning it around by 360°! That’s a very different thing than just looking at it, walking around it, and then looking at it again. That explains the 720° symmetry: we need to turn it around twice to get it back to its original state. So… Well… The description is more about us and what we do with the object than about the object itself. That’s why I think the quantum-mechanical description is defective.