Paul Ehrenfest and the search for truth

On 25 September 1933, Paul Ehrenfest took his son Wassily, who was suffering from Down syndrome, for a walk in the park. He shot him, and then killed himself. He was only 53. That’s my age bracket. From the letters he left (here is a summary in Dutch), we know his frustration of not being able to arrive at some kind of common-sense interpretation of the new quantum physics played a major role in the anxiety that had brought him to this point. He had taken courses from Ludwig Boltzmann as an aspiring young man. We, therefore, think Boltzmann’s suicide – for similar reasons – might have troubled him too.

His suicide did not come unexpectedly: he had announced it. In one of his letters to Einstein, he complains about ‘indigestion’ from the ‘unendlicher Heisenberg-Born-Dirac-Schrödinger Wurstmachinen-Physik-Betrieb.’ I’ll let you google-translate that. :-/ He also seems to have gone through the trouble of summarizing all his questions on the new approach in an article in what was then one of the top journals for physics: Einige die Quantenmechanik betreffende Erkundigungsfrage, Zeitschrift für Physik 78 (1932) 555-559 (quoted in the above-mentioned review article). This I’ll translate: Some Questions about Quantum Mechanics.

Ehrenfest

Paul Ehrenfest in happier times (painting by Harm Kamerlingh Onnes in 1920)

A diplomat-friend of mine once remarked this: “It is good you are studying physics only as a pastime. Professional physicists are often troubled people—miserable.” It is an interesting observation from a highly intelligent outsider. To be frank, I understand this strange need to probe things at the deepest level—to be able to explain what might or might not be the case (I am using Wittgenstein’s definition of reality here). Even H.A. Lorentz, who – fortunately, perhaps – died before his successor did what he did, was becoming quite alarmist about the sorry state of academic physics near the end of his life—and he, Albert Einstein, and so many others were not alone. Not then, and not now. All of the founding fathers of quantum mechanics ended up becoming pretty skeptical about the theory they had created. We have documented that elsewhere so we won’t talk too much about it here. Even John Stewart Bell himself – one of the third generation of quantum physicists, we may say – did not like his own ‘No Go Theorem’ and thought that some “radical conceptual renewal”[1] might disprove his conclusions.

The Born-Heisenberg revolution has failed: most – if not all – of contemporary high-brow physicist are pursuing alternative theories—in spite, or because, of the academic straitjackets they have to wear. If a genius like Ehrenfest didn’t buy it, then I won’t buy it either. Furthermore, the masses surely don’t buy it and, yes, truth – in this domain too – is, fortunately, being defined more democratically nowadays. The Nobel Prize Committee will have to do some serious soul-searching—if not five years from now, then ten.

We feel sad for the physicists who died unhappily—and surely for those who took their life out of depression—because the common-sense interpretation they were seeking is so self-evident: de Broglie’s intuition in regard to matter being wavelike was correct. He just misinterpreted its nature: it is not a linear but a circular wave. We quickly insert the quintessential illustration (courtesy of Celani, Vassallo and Di Tommaso) but we refer the reader for more detail to our articles or – more accessible, perhaps – our manuscript for the general public.

aa 2

The equations are easy. The mass of an electron – any matter-particle, really – is the equivalent mass of the oscillation of the charge it carries. This oscillation is, most probably, statistically regular only. So we think it’s chaotic, actually, but we also think the words spoken by Lord Pollonius in Shakespeare’s Hamlet apply to it: “Though this be madness, yet there is method in ‘t.” This means we can meaningfully speak of a cycle time and, therefore, of a frequency. Erwin Schrödinger stumbled upon this motion while exploring solutions to Dirac’s wave equation for free electrons, and Dirac immediately grasped the significance of Schrödinger’s discovery, because he mentions Schrödinger’s discovery rather prominently in his Nobel Prize Lecture:

“It is found that an electron which seems to us to be moving slowly, must actually have a very high frequency oscillatory motion of small amplitude superposed on the regular motion which appears to us. As a result of this oscillatory motion, the velocity of the electron at any time equals the velocity of light. This is a prediction which cannot be directly verified by experiment, since the frequency of the oscillatory motion is so high and its amplitude is so small. But one must believe in this consequence of the theory, since other consequences of the theory which are inseparably bound up with this one, such as the law of scattering of light by an electron, are confirmed by experiment.” (Paul A.M. Dirac, Theory of Electrons and Positrons, Nobel Lecture, December 12, 1933)

Unfortunately, Dirac confuses the concept of the electron as a particle with the concept of the (naked) charge inside. Indeed, the idea of an elementary (matter-)particle must combine the idea of a charge and its motion to account for both the particle- as well as the wave-like character of matter-particles. We do not want to dwell on all of this because we’ve written too many papers on this already. We just thought it would be good to sum up the core of our common-sense interpretation of physics. Why? To honor Boltzmann and Ehrenfest: I think of their demise as a sacrifice in search for truth.

[…]

OK. That sounds rather tragic—sorry for that! For the sake of brevity, we will just describe the electron here.

I. Planck’s quantum of action (h) and the speed of light (c) are Nature’s most fundamental constants. Planck’s quantum of action relates the energy of a particle to its cycle time and, therefore, to its frequency:

(1) h = E·T = E/f ⇔ ħ = E/ω

The charge that is whizzing around inside of the electron has zero rest mass, and so it whizzes around at the speed of light: the slightest force on it gives it an infinite acceleration. It, therefore, acquires a relativistic mass which is equal to mγ = me/2 (we refer to our paper(s) for a relativistically correct geometric argument). The momentum of the pointlike charge, in its circular or orbital motion, is, therefore, equal to p = mγ·c = me·c/2.

The (angular) frequency of the oscillation is also given by the formula for the (angular) velocity:

(2) c = a·ω ⇔ ω = c/a

While Eq. (1) is a fundamental law of Nature, Eq. (2) is a simple geometric or mathematical relation only.

II. From (1) and (2), we can now calculate the radius of this tiny circular motion as:

(3a) ħ = E/ω = E·a/c a = (ħ·c)/E

Because we know the mass of the electron is the inertial mass of the state of motion of the pointlike charge, we may use Einstein’s mass-energy equivalence relation to rewrite this as the Compton radius of the electron:

(3b) a = (ħ·c)/E = (ħ·c)/(me·c2) = ħ/(me·c)

Note that we only used two fundamental laws of Nature so far: the Planck-Einstein relation and Einstein’s mass-energy equivalence relation.

III. We must also be able to express the Planck-Einstein quantum as the product of the momentum (p) of the pointlike charge and some length λ:

(4) h = p·λ

The question here is: what length? The circumference of the loop, or its radius? The same geometric argument we used to derive the effective mass of the pointlike charge as it whizzes around at lightspeed around its center, tells us the centripetal force acts over a distance that is equal to two times the radius. Indeed, the relevant formula for the centripetal force is this:

(5) F = (mγ/me)·(E/a) = E/2a

We can therefore reduce Eq. (4) by dividing it by 2π. We then get reduced, angular or circular (as opposed to linear) concepts:

(6) ħ = (p·λ)/(2π) = (me·c/2)·(λ/π) = (me·c/2)·(2a) = me·c·a ⇔ ħ/a = me·c

We can verify the logic of our reasoning by substituting for the Compton radius:

ħ = p·λ = me·c·= me·c·a = me·c·ħ/(me·c) = ħ

IV. We can, finally, re-confirm the logic of our reason by re-deriving Einstein’s mass-energy equivalence relation as well as the Planck-Einstein relation using the ω = c/a and the ħ/a = me·c relations:

(7) ħ·ω = ħ·c/a = (ħ/ac = (me·cc = me·c2 = E

Of course, we note all of the formulas we have derived are interdependent. We, therefore, have no clear separation between axioms and derivations here. If anything, we are only explaining what Nature’s most fundamental laws (the Planck-Einstein relation and Einstein’s mass-energy equivalence relation) actually mean or represent. As such, all we have is a simple description of reality itself—at the smallest scale, of course! Everything that happens at larger scales involves Maxwell’s equations: that’s all electromagnetic in nature. No need for strong or weak forces, or for quarks—who invented that? Ehrenfest, Lorentz and all who suffered with truly understanding the de Broglie’s concept of the matter-wave might have been happier physicists if they would have seen these simple equations!

The gist of the matter is this: the intuition of Einstein and de Broglie in regard to the wave-nature of matter was, essentially, correct. However, de Broglie’s modeling of it as a wave packet was not: modeling matter-particles as some linear oscillation does not do the trick. It is extremely surprising no one thought of trying to think of some circular oscillation. Indeed, the interpretation of the elementary wavefunction as representing the mentioned Zitterbewegung of the electric charge solves all questions: it amounts to interpreting the real and imaginary part of the elementary wavefunction as the sine and cosine components of the orbital motion of a pointlike charge. We think that, in our 60-odd papers, we’ve shown such easy interpretation effectively does the trick of explaining all of the quantum-mechanical weirdness but, of course, it is up to our readers to judge that. 🙂

[1] See: John Stewart Bell, Speakable and unspeakable in quantum mechanics, pp. 169–172, Cambridge University Press, 1987 (quoted from Wikipedia). J.S. Bell died from a cerebral hemorrhage in 1990 – the year he was nominated for the Nobel Prize in Physics and which he, therefore, did not receive (Nobel Prizes are not awarded posthumously). He was just 62 years old then.

The wavefunction in a medium: amplitudes as signals

We finally did what we wanted to do for a while already: we produced a paper on the meaning of the wavefunction and wave equations in the context of an atomic lattice (think of a conductor or a semiconductor here). Unsurprisingly, we came to the following conclusions:

1. The concept of the matter-wave traveling through the vacuum, an atomic lattice or any medium can be equated to the concept of an electric or electromagnetic signal traveling through the same medium.

2. There is no need to model the matter-wave as a wave packet: a single wave – with a precise frequency and a precise wavelength – will do.

3. If we do want to model the matter-wave as a wave packet rather than a single wave with a precisely defined frequency and wavelength, then the uncertainty in such wave packet reflects our own limited knowledge about the momentum and/or the velocity of the particle that we think we are representing. The uncertainty is, therefore, not inherent to Nature, but to our limited knowledge about the initial conditions or, what amounts to the same, what happened to the particle(s) in the past.

4. The fact that such wave packets usually dissipate very rapidly, reflects that even our limited knowledge about initial conditions tends to become equally rapidly irrelevant. Indeed, as Feynman puts it, “the tiniest irregularities tend to get magnified very quickly” at the micro-scale.

In short, as Hendrik Antoon Lorentz noted a few months before his demise, there is, effectively, no reason whatsoever “to elevate indeterminism to a philosophical principle.” Quantum mechanics is just what it should be: common-sense physics.

The paper confirms intuitions we had highlighted in previous papers already, but uses the formalism of quantum mechanics itself to demonstrate this.

PS: We put the paper on academia.edu and ResearchGate as well, but Phil Gibbs’ site has easy access (no log-in or membership required). Long live Phil Gibbs!

The flywheel model of an electron

One of my readers sent me the following question on the geometric (or even physical) interpretation of the wavefunction that I’ve been offering in recent posts:

Does this mean that the wave function is merely describing excitations in a matter field; or is this unsupported?

My reply was very short: “Yes. In fact, we can think of a matter-particle as a tiny flywheel that stores energy.”

However, I realize this answer answers the question only partially. Moreover, I now feel I’ve been quite ambiguous in my description. When looking at the geometry of the elementary wavefunction (see the animation below, which shows us a left- and right-handed wave respectively), two obvious but somewhat conflicting interpretations readily come to mind:

(1) One is that the components of the elementary wavefunction represent an oscillation (in two dimensions) of a field. We may call it a matter field (yes, think of the scalar Higgs field here), but we could also think of it as an oscillation of the spacetime fabric itself: a tiny gravitational wave, in effect. All we need to do here is to associate the sine and cosine component with a physical dimension. The analogy here is the electromagnetic field vector, whose dimension is force per unit charge (newton/coulomb). So we may associate the sine and cosine components of the wavefunction with, say, the force per unit mass dimension (newton/kg) which, using Newton’s Law (F = m·a) reduces to the dimension of acceleration (m/s2), which is the dimension of gravitational fields. I’ll refer to this interpretation as the field interpretation of the matter wave (or wavefunction).

(2) The other interpretation is what I refer to as the flywheel interpretation of the electron. If you google this, you won’t find anything. However, you will probably stumble upon the so-called Zitterbewegung interpretation of quantum mechanics, which is a more elaborate theory based on the same basic intuition. The Zitterbewegung (a term which was coined by Erwin Schrödinger himself, and which you’ll see abbreviated as zbw) is, effectively, a local circulatory motion of the electron, which is presumed to be the basis of the electron’s spin and magnetic moment. All that I am doing, is… Well… I think I do push the envelope of this interpretation quite a bit. 🙂

The first interpretation implies our rotating arrow is, effectively, some field vector. In contrast, the second interpretation implies it’s only the tip of the rotating arrow that, literally, matters: we should look at it as a pointlike charge moving around a central axis, which is the direction of propagation. Let’s look at both.

The flywheel interpretation

The flywheel interpretation has an advantage over the field interpretation, because it also gives us a wonderfully simple physical interpretation of the interaction between electrons and photons—or, further speculating, between matter-particles (fermions) and force-carrier particles (bosons) in general. In fact, Feynman shows how this might work—but in a rather theoretical Lecture on symmetries and conservation principles, and he doesn’t elaborate much, so let me do that for him. The argument goes as follows.

A light beam—an electromagnetic wave—consists of a large number of photons. These photons are thought of as being circularly polarized: look at those animations above again. The Planck-Einstein equation tells us the energy of each photon is equal to E = ħ·ω = h·f. [I should, perhaps, quickly note that the frequency is, obviously, the frequency of the electromagnetic wave. It, therefore, is not to be associated with a matter wave: the de Broglie wavelength and the wavelength of light are very different concepts, even if the Planck-Einstein equation looks the same for both.]

Now, if our beam consists of photons, the total energy of our beam will be equal to W = N·E = N·ħ·ω. It is crucially important to note that this energy is to be interpreted as the energy that is carried by the beam in a certain time: we should think of the beam as being finite, somehow, in time and in space. Otherwise, our reasoning doesn’t make sense.

The photons carry angular momentum. Just look at those animations (above) once more. It doesn’t matter much whether or not we think of light as particles or as a wave: you can see there is angular momentum there. Photons are spin-1 particles, so the angular momentum will be equal to ± ħ. Hence, then the total angular momentum Jz (the direction of propagation is supposed to be the z-axis here) will be equal to JzN·ħ. [This, of course, assumes all photons are polarized in the same way, which may or may not be the case. You should just go along with the argument right now.] Combining the W = N·ħ·ω and JzN·ħ equations, we get:

JzN·ħ = W/ω

For a photon, we do accept the field interpretation, as illustrated below. As mentioned above, the z-axis here is the direction of propagation (so that’s the line of sight when looking at the diagram). So we have an electric field vector, which we write as ε (epsilon) so as to not cause any confusion with the Ε we used for the energy. [You may wonder if we shouldn’t also consider the magnetic field vector, but then we know the magnetic field vector is, basically, a relativistic effect which vanishes in the reference frame of the charge itself.] The phase of the electric field vector is φ = ω·t.

RH photon

Now, a charge (so that’s our electron now) will experience a force which is equal to F = q·ε. We use bold letters here because F and ε are vectors. We now need to look at our electron which, in our interpretation of the elementary wavefunction, we think of as rotating about some axis. So that’s what’s represented below. [Both illustrations are Feynman’s, not mine. As for the animations above, I borrowed them from Wikipedia.]

electron

Now, in previous posts, we calculated the radius based on a similar argument as the one Feynman used to get that JzN·ħ = W/ω equation. I’ll refer you those posts and just mention the result here: r is the Compton scattering radius for an electron, which is equal to:

radius formula

An equally spectacular implication of our flywheel model of the electron was the following: we found that the angular velocity v was equal to vr·ω = [ħ·/(m·c)]·(E/ħ) = c. Hence, in our flywheel model of an electron, it is effectively spinning around at the speed of light. Note that the angular frequency (ω) in the vr·ω equation is not the angular frequency of our photon: it’s the frequency of our electron. So we use the same Planck-Einstein equation (ω = E/ħ) but the energy E is the (rest) energy of our electron, so that’s about 0.511 MeV (so that’s an order of magnitude which is 100,000 to 300,000 times that of photons in the visible spectrum). Hence, the angular frequencies of our electron and our photon are very different. Feynman casually reflects this difference by noting the phases of our electron and our photon will differ by a phase factor, which he writes as φ0.

Just to be clear here, at this point, our analysis here diverges from Feynman’s. Feynman had no intention whatsoever to talk about Schrödinger’s Zitterbewegung hypothesis when he wrote what he wrote back in the 1960s. In fact, Feynman is very reluctant to venture into physical interpretations of the wavefunction in all his Lectures on quantum mechanics—which is surprising. Because he comes so tantalizing close at many occasions—as he does here: he describes the motion of the electron here as that of a harmonic oscillator which can be driven by an external electric field. Now that is a physical interpretation, and it is totally consistent with the one I’ve advanced in my recent posts. Indeed, Feynman also describes it as an oscillation in two dimensions—perpendicular to each other and to the direction of motion, as we do— in both the flywheel as well as the field interpretation of the wavefunction!

This point is important enough to quote Feynman himself in this regard:

“We have often described the motion of the electron in the atom as a harmonic oscillator which can be driven into oscillation by an external electric field. We’ll suppose that the atom is isotropic, so that it can oscillate equally well in the x– or y- directions. Then in the circularly polarized light, the x displacement and the displacement are the same, but one is 90° behind the other. The net result is that the electron moves in a circle.”

Right on! But so what happens really? As our light beam—the photons, really—are being absorbed by our electron (or our atom), it absorbs angular momentum. In other words, there is a torque about the central axis. Let me remind you of the formulas for the angular momentum and for torque respectively: L = r×p and τr×F. Needless to say, we have two vector cross-products here. Hence, if we use the τr×F formula, we need to find the tangential component of the force (Ft), whose magnitude will be equal to Ft = q·εtNow, energy is force over some distance so… Well… You may need to think about it for a while but, if you’ve understood all of the above, you should also be able to understand the following formula:

dW/dt = q·εt·v

[If you have trouble, remember is equal to ds/dt = Δs/Δt for Δt → 0, and re-write the equation above as dW = q·εt·v·dt = q·εt·ds = Ft·ds. Capito?]

Now, you may or may not remember that the time rate of change of angular momentum must be equal to the torque that is being applied. Now, the torque is equal to τ = Ft·r = q·εt·r, so we get:

dJz/dt = q·εt·v

The ratio of dW/dt and dJz/dt gives us the following interesting equation:

Feynman formula

Now, Feynman tries to relate this to the JzN·ħ = W/ω formula but… Well… We should remind ourselves that the angular frequency of these photons is not the angular frequency of our electron. So… Well… What can we say about this equation? Feynman suggests to integrate dJz and dW over some time interval, which makes sense: as mentioned, we interpreted W as the energy that is carried by the beam in a certain time. So if we integrate dW over this time interval, we get W. Likewise, if we integrate dJz over the same time interval, we should get the total angular momentum that our electron is absorbing from the light beam. Now, because dJz = dW/ω, we do concur with Feynman’s conclusion: the total angular momentum which is being absorbed by the electron is proportional to the total energy of the beam, and the constant of proportionality is equal to 1/ω.

It’s just… Well… The ω here is the angular frequency of the electron. It’s not the angular frequency of the beam. Not in our flywheel model of the electron which, admittedly, is not the model which Feynman used in his analysis. Feynman’s analysis is simpler: he assumes an electron at rest, so to speak, and then the beam drives it so it goes around in a circle with a velocity that is, effectively, given by the angular frequency of the beam itself. So… Well… Fine. Makes sense. As said, I just pushed the analysis a bit further along here. Both analyses raise an interesting question: how and where is the absorbed energy being stored? What is the mechanism here?

In Feynman’s analysis, the answer is quite simple: the electron did not have any motion before but does spin around after the beam hit it. So it has more energy now: it wasn’t a tiny flywheel before, but it is now!

In contrast, in my interpretation of the matter wave, the electron was spinning around already, so where does the extra energy go now? As its energy increases, ω = E/ħ must increase, right? Right. At the same time, the velocity v = r·ω must still be equal to vr·ω = [ħ·/(m·c)]·(E/ħ) = c, right? Right. So… If ω increases, but r·ω must equal the speed of light, then must actually decrease somewhat, right?

Right. It’s a weird but inevitable conclusion, it seems. I’ll let you think about it. 🙂

To conclude this post—which, I hope, the reader who triggered it will find interesting—I would like to quote Feynman on an issue on which most textbooks remain silent: the two-state nature of photons. I will just quote him without trying to comment or alter what he writes, because what he writes is clear enough, I think:

“Now let’s ask the following question: If light is linearly polarized in the x-direction, what is its angular momentum? Light polarized in the x-direction can be represented as the superposition of RHC and LHC polarized light. […] The interference of these two amplitudes produces the linear polarization, but it has equal probabilities to appear with plus or minus one unit of angular momentum. [Macroscopic measurements made on a beam of linearly polarized light will show that it carries zero angular momentum, because in a large number of photons there are nearly equal numbers of RHC and LHC photons contributing opposite amounts of angular momentum—the average angular momentum is zero.]

Now, we have said that any spin-one particle can have three values of Jz, namely +101 (the three states we saw in the Stern-Gerlach experiment). But light is screwy; it has only two states. It does not have the zero case. This strange lack is related to the fact that light cannot stand still. For a particle of spin which is standing still, there must be the 2j+1 possible states with values of Jz going in steps of from j to +j. But it turns out that for something of spin j with zero mass only the states with the components +j and j along the direction of motion exist. For example, light does not have three states, but only two—although a photon is still an object of spin one.”

In his typical style and frankness—for which he is revered by some (like me) but disliked by others—he admits this is very puzzling, and not obvious at all! Let me quote him once more:

“How is this consistent with our earlier proofs—based on what happens under rotations in space—that for spin-one particles three states are necessary? For a particle at rest, rotations can be made about any axis without changing the momentum state. Particles with zero rest mass (like photons and neutrinos) cannot be at rest; only rotations about the axis along the direction of motion do not change the momentum state. Arguments about rotations around one axis only are insufficient to prove that three states are required. We have tried to find at least a proof that the component of angular momentum along the direction of motion must for a zero mass particle be an integral multiple of ħ/2—and not something like ħ/3. Even using all sorts of properties of the Lorentz transformation and what not, we failed. Maybe it’s not true. We’ll have to talk about it with Prof. Wigner, who knows all about such things.”

The reference to Eugene Wigner is historically interesting. Feynman probably knew him very well—if only because they had both worked together on the Manhattan Project—and it’s true Wigner was not only a great physicist but a mathematical genius as well. However, Feynman probably quotes him here for the 1963 Nobel Prize he got for… Well… Wigner’s “contributions to the theory of the atomic nucleus and elementary particles, particularly through the discovery and application of fundamental symmetry principles.” 🙂 I’ll let you figure out how what I write about in this post, and symmetry arguments, might be related. 🙂

That’s it for today, folks! I hope you enjoyed this. 🙂

Post scriptum: The main disadvantage of the flywheel interpretation is that it doesn’t explain interference: waves interfere—some rotating mass doesn’t. Ultimately, the wave and flywheel interpretation must, somehow, be compatible. One way to think about it is that the electron can only move as it does—in a “local circulatory motion”—if there is a force on it that makes it move the way it does. That force must be gravitational because… Well… There is no other candidate, is there? [We’re not talking some electron orbital here—some negative charge orbiting around a positive nucleus. We’re just considering the electron itself.] So we just need to prove that our rotating arrow will also represent a force, whose components will make our electron move the way it does. That should not be difficult. The analogy of the V-twin engine should do the trick. I’ll deal with that in my next post. If we’re able to provide such proof (which, as mentioned, should not be difficult), it will be a wonderful illustration of the complementarity principle. 🙂

However, just thinking about it does raise some questions already. Circular motion like this can be explained in two equivalent ways. The most obvious way to think about it is to assume some central field. It’s the planetary model (illustrated below). However, that doesn’t suit our purposes because it’s hard – if possible at all – to relate it to the wavefunction oscillation.

planetary model

The second model is our two-spring or V-twin engine model (illustrated below), but then what is the mass here? One hypothesis that comes to mind is that we’re constantly accelerating and decelerating an electric charge (the electron charge)—against all other charges in the Universe, so to speak. So that’s a force over a distance—energy. And energy has an equivalent mass.

V-2 engineThe question which remains open, then, is the following: what is the nature of this force? In previous posts, I suggested it might be gravitational, but so here we’re back to the drawing board: we’re talking an electrical force, but applied to some mass which acquires mass because of… Well… Because of the force—because of the oscillation (the moving charge) itself. Hmm…. I need to think about this.

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