Wikipedia censorship

I started to edit and add to the rather useless Wikipedia article on the Zitterbewegung. No mention of Hestenes or more recent electron models (e.g. Burinskii’s Kerr-Newman geometries. No mention that the model only works for electrons or leptons in general – not for non-leptonic fermions. It’s plain useless. But all the edits/changes/additions were erased by some self-appointed ‘censor’. I protested but then I got reported to the administrator ! What can I say? Don’t trust Wikipedia. Don’t trust any ‘authority’. We live in weird times. The mindset of most professional physicists seems to be governed by ego and the Bohr-Heisenberg Diktatur.

For the record, these are the changes and edits I tried to make. You can compare and judge for yourself. Needless to say, I told them I wouldn’t bother to even try to contribute any more. I published my own article on the Vixrapedia e-encyclopedia. Also, as Vixrapedia did not have an entry on realist interpretations of quantum mechanics, I created one: have a look and let me know what you think. 🙂

Zitterbewegung (“trembling” or “shaking” motion in German) – usually abbreviated as zbw – is a hypothetical rapid oscillatory motion of elementary particles that obey relativistic wave equations. The existence of such motion was first proposed by Erwin Schrödinger in 1930 as a result of his analysis of the wave packet solutions of the Dirac equation for relativistic electrons in free space, in which an interference between positive and negative energy states produces what appears to be a fluctuation (up to the speed of light) of the position of an electron around the median, with an angular frequency of ω = 2mc²/ħ, or approximately 1.5527×10²¹ radians per second. Paul Dirac was initially intrigued by it, as evidenced by his rather prominent mention of it in his 1933 Nobel Prize Lecture (it may be usefully mentioned he shared this Nobel Prize with Schrödinger):

“The variables give rise to some rather unexpected phenomena concerning the motion of the electron. These have been fully worked out by Schrödinger. 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.”^[1]

In light of Dirac’s later comments on modern quantum theory, it is rather puzzling that he did not pursue the idea of trying to understand charged particles in terms of the motion of a pointlike charge, which is what the Zitterbewegung hypothesis seems to offer. Dirac’s views on non-leptonic fermions – which were then (1950s and 1960s) being analyzed in an effort to explain the ‘particle zoo‘ in terms of decay reactions conserving newly invented or ad hoc quantum numbers such as strangeness^[2] – may be summed up by quoting the last paragraph in the last edition of his Principles of Quantum Mechanics:

“Now there are other kinds of interactions, which are revealed in high-energy physics. […] These interactions are not at present sufficiently well understood to be incorporated into a system of equations of motion.”^[3]

Indeed, in light of this stated preference for kinematic models, it is somewhat baffling that Dirac did not follow up on this or any of the other implications of the Zitterbewegung hypothesis, especially because it should be noted that a reexamination of Dirac theory shows that interference between positive and negative energy states is not a necessary ingredient of Zitterbewegung theories.^[4] The Zitterbewegung hypothesis also seems to offer interesting shortcuts to key results of mainstream quantum theory. For example, one can show that, for the hydrogen atom, the Zitterbewegung produces the Darwin term which plays the role in the fine structure as a small correction of the energy level of the s-orbitals.^[5] This is why authors such as Hestenes refers to it as a possible alternative interpretation of mainstream quantum mechanics, which may be an exaggerated claim in light of the fact that the zbw hypothesis results from the study of electron behavior only.

Zitterbewegung models have mushroomed^[6] and it is, therefore, increasingly difficult to distinguish between them. The key to understanding and distinguishing the various Zitterbewegung models may well be Wheeler‘s ‘mass without mass’ idea, which implies a distinction between the idea of (i) a pointlike electric charge (i.e. the idea of a charge only, with zero rest mass) and (ii) the idea of an electron as an elementary particle whose equivalent mass is the energy of the zbw oscillation of the pointlike charge.^[7] The ‘mass without mass’ concept requires a force to act on a charge – and a charge only – to explain why a force changes the state of motion of an object – its momentum p = m_γ·v(with γ referring to the Lorentz factor) – in accordance with the (relativistically correct) F = dp/dt force law.

History

As mentioned above, the zbw hypothesis goes back to Schrödinger’s and Dirac’s efforts to try to explain what an electron actually is. Unfortunately, both interpreted the electron as a pointlike particle with no ‘internal structure’.David Hestenes is to be credited with reviving the Zitterbewegung hypothesis in the early 1990s. While acknowledging its origin as a (trivial) solution to Dirac’s equation for electrons, Hestenes argues the Zitterbewegung should be related to the intrinsic properties of the electron (charge, spin and magnetic moment). He argues that the Zitterbewegung hypothesis amounts to a physical interpretation of the elementary wavefunction or – more boldly – to a possible physical interpretation of all of quantum mechanics: “Spin and phase [of the wavefunction] are inseparably related — spin is not simply an add-on, but an essential feature of quantum mechanics. […] A standard observable in Dirac theory is the Dirac current, which doubles as a probability current and a charge current. However, this does not account for the magnetic moment of the electron, which many investigators conjecture is due to a circulation of charge. But what is the nature of this circulation? […] Spin and phase must be kinematical features of electron motion. The charge circulation that generates the magnetic moment can then be identified with the Zitterbewegung of Schrödinger “^[8] Hestenes’ interpretation amounts to an kinematic model of an electron which can be described in terms of John Wheeler‘s mass without mass concept.^[9] The rest mass of the electron is analyzed as the equivalent energy of an orbital motion of a pointlike charge. This pointlike charge has no rest mass and must, therefore, move at the speed of light (which confirms Dirac’s en Schrödinger’s remarks on the nature of the Zitterbewegung). Hestenes summarizes his interpretation as follows: “The electron is nature’s most fundamental superconducting current loop. Electron spin designates the orientation of the loop in space. The electron loop is a superconducting LC circuit. The mass of the electron is the energy in the electron’s electromagnetic field. Half of it is magnetic potential energy and half is kinetic.”^[10]

Hestenes‘ articles and papers on the Zitterbewegung discuss the electron only. The interpretation of an electron as a superconducting ring of current (or as a (two-dimensional) oscillator) also works for the muon electron: its theoretical Compton radius r_C = ħ/m_μc ≈ 1.87 fm falls within the CODATA confidence interval for the experimentally determined charge radius.^[11] Hence, the theory seems to offer a remarkably and intuitive model of leptons. However, the model cannot be generalized to non-leptonic fermions (spin-1/2 particles). Its application to protons or neutrons, for example, is problematic: when inserting the energy of a proton or a neutron into the formula for the Compton radius (the r_C = ħ/mc formula follows from the kinematic model), we get a radius of the order of r_C = ħ/m_pc ≈ 0.21 fm, which is about 1/4 of the measured value (0.84184(67) fm to 0.897(18) fm). A radius of the order of 0.2 fm is also inconsistent with the presumed radius of the pointlike charge itself. Indeed, while the pointlike charge is supposed to be pointlike, pointlike needs to be interpreted as ‘having no internal structure’: it does not imply the pointlike charge has no (small) radius itself. The classical electron radius is a likely candidate for the radius of the pointlike charge because it emerges from low-energy (Thomson) scattering experiments (elastic scattering of photons, as opposed to inelastic Compton scattering). The assumption of a pointlike charge with radius r_e = α·ħ/m_pc) may also offer a geometric explanation of the anomalous magnetic moment.^[12]

In any case, the remarks above show that a Zitterbewegung model for non-leptonic fermions is likely to be somewhat problematic: a proton, for example, cannot be explained in terms of the Zitterbewegung of a positron (or a heavier variant of it, such as the muon- or tau-positron).^[13] This is why it is generally assumed the large energy (and the small size) of nucleons is to be explained by another force – a strong force which acts on a strong charge instead of an electric charge. One should note that both color and/or flavor in the standard quark–gluon model of the strong force may be thought of as zero-mass charges in ‘mass without mass’ kinematic models and, hence, the acknowledgment of this problem does not generally lead zbw theorists to abandon the quest for an alternative realist interpretation of quantum mechanics.

While Hestenes‘ zbw interpretation (and the geometric calculus approach he developed) is elegant and attractive, he did not seem to have managed to convincingly explain an obvious question of critics of the model: what keeps the pointlike charge in the zbw electron in its circular orbit? To put it simply: one may think of the electron as a superconducting ring but there is no material ring to hold and guide the charge. Of course, one may argue that the electromotive force explains the motion but this raises the fine-tuning problem: the slightest deviation of the pointlike charge from its circular orbit would yield disequilibrium and, therefore, non-stability. [One should note the fine-tuning problem is also present in mainstream quantum mechanics. See, for example, the discussion in Feynman’s Lectures on Physics.] The lack of a convincing answer to these and other questions (e.g. on the distribution of (magnetic) energy within the superconducting ring) led several theorists working on electron models (e.g. Alexander Burinskii^[14]^[15]) to move on and explore alternative geometric approaches, including Kerr-Newman geometries. Burinskii summarizes his model as follows: “The electron is a superconducting disk defined by an over-rotating black hole geometry. The charge emerges from the Möbius structure of the Kerr geometry.”^[16] His advanced modelling of the electron also allows for a conceptual bridge with mainstream quantum mechanics, grand unification theories and string theory: “[…] Compatibility between gravity and quantum theory can be achieved without modifications of Einstein-Maxwell equations, by coupling to a supersymmetric Higgs model of symmetry breaking and forming a nonperturbative super-bag solution, which generates a gravity-free Compton zone necessary for consistent work of quantum theory. Super-bag is naturally upgraded to Wess-Zumino supersymmetric QED model, forming a bridge to perturbative formalism of conventional QED.”^[17]

The various geometric approaches (Hestenes’ geometric calculus, Burinskii’s Kerr-Newman model, oscillator models) yield the same results (the intrinsic properties of the electron are derived from what may be referred to as kinematic equations or classical (but relativistically correct) equations) – except for a factor 2 or 1/2 or the inclusion (or not) of variable tuning parameters (Burinskii’s model, for example, allows for a variable geometry) – but the equivalence of the various models that may or may not explain the hypothetical Zitterbewegung still needs to be established.

The continued interest in zbw models may be explained because Zitterbewegung models – in particular Hestenes’ model and the oscillator model – are intuitive and, therefore, attractive. They are intuitive because they combine the Planck-Einstein relation (E = hf) and Einstein’s mass-energy equivalence (E = mc²): each cycle of the Zitterbewegung electron effectively packs (i) the unit of physical action (h) and (ii) the electron’s energy. This allows one to understand Planck’s quantum of action as the product of the electron’s energy and the cycle time: h = E·T = h·f·T = h·f/f = h. In addition, the idea of a centripetal force keeping some zero-mass pointlike charge in a circular orbit also offers a geometric explanation of Einstein’s mass-energy equivalence relation: this equation, therefore, is no longer a rather inexplicable consequence of special relativity theory.

The section below offers a general overview of the original discovery of Schrödinger and Dirac. It is followed by further analysis which may or may not help the reader to judge whether the Zitterbewegung hypothesis might, effectively, amount to what David Hestenes claims it actually is: an alternative interpretation of quantum mechanics.

Theory for a free fermion

[See the article: the author of this section does not seem to know – or does not mention, at least – that the Zitterbewegung hypothesis only applies to leptons (no strong charge).]

Experimental evidence

The Zitterbewegung may remain theoretical because, as Dirac notes, the frequency may be too high to be observable: it is the same frequency as that of a 0.511 MeV gamma-ray. However, some experiments may offer indirect evidence. Dirac’s reference to electron scattering experiments is also quite relevant because such experiments yield two radii: a radius for elastic scattering (the classical electron radius) and a radius for inelastic scattering (the Compton radius). Zittebewegung theorists think Compton scattering involves electron-photon interference: the energy of the high-energy photon (X- or gamma-ray photons) is briefly absorbed before the electron comes back to its equilibrium situation by emitting another (lower-energy) photon (the difference in the energy of the incoming and the outgoing photon gives the electron some extra momentum). Because of this presumed interference effect, Compton scattering is referred to as inelastic. In contrast, low-energy photons scatter elastically: they seem to bounce off some hard core inside of the electron (no interference).

Some experiments also claim they amount to a simulation of the Zitterbewegung of a free relativistic particle. First, with a trapped ion, by putting it in an environment such that the non-relativistic Schrödinger equation for the ion has the same mathematical form as the Dirac equation (although the physical situation is different).^[18]^[19] Then, in 2013, it was simulated in a setup with Bose–Einstein condensates.^[20]

The effective mass of the electric charge

The 2m factor in the formula for the zbw frequency and the interpretation of the Zitterbewegung in terms of a centripetal force acting on a pointlike charge with zero rest mass leads one to re-explore the concept of the effective mass of an electron. Indeed, if we write the effective mass of the pointlike charge as m_γ = γm₀ then we can derive its value from the angular momentum of the electron (L = ħ/2) using the general angular momentum formula L = r × p and equating r to the Compton radius:

$p=m_{\gamma }c={L \over r_{C}}={\hbar \over 2}{m_{e}c \over \hbar }={m_{e}c \over 2}\Longleftrightarrow m_{\gamma }={m_{e} \over 2}$

This explains the 1/2 factor in the frequency formula for the Zitterbewegung. Substituting m for m_γ in the ω = 2mc²/ħ yields an equivalence with the Planck-Einstein relation ω = m_γc²/ħ. The electron can then be described as an oscillator (in two dimensions) whose natural frequency is given by the Planck-Einstein relation.^[21]

The wavefunction of a zero-mass particle

Post scriptum note added on 11 July 2016: This is one of the more speculative posts which led to my e-publication analyzing the wavefunction as an energy propagation. With the benefit of hindsight, I would recommend you to immediately the more recent exposé on the matter that is being presented here, which you can find by clicking on the provided link. In fact, I actually made some (small) mistakes when writing the post below.

Original post:

I hope you find the title intriguing. A zero-mass particle? So I am talking a photon, right? Well… Yes and no. Just read this post and, more importantly, think about this story for yourself. 🙂

One of my acquaintances is a retired nuclear physicist. We mail every now and then—but he has little or no time for my questions: he usually just tells me to keep studying. I once asked him why there is never any mention of the wavefunction of a photon in physics textbooks. He bluntly told me photons don’t have a wavefunction—not in the sense I was talking at least. Photons are associated with a traveling electric and a magnetic field vector. That’s it. Full stop. Photons do not have a ψ or φ function. [I am using ψ and φ to refer to position or momentum wavefunction. You know both are related: if we have one, we have the other.] But then I never give up, of course. I just can’t let go out of the idea of a photon wavefunction. The structural similarity in the propagation mechanism of the electric and magnetic field vectors E and B just looks too much like the quantum-mechanical wavefunction. So I kept trying and, while I don’t think I fully solved the riddle, I feel I understand it much better now. Let me show you the why and how.

I. An electromagnetic wave in free space is fully described by the following two equations:

∂B/∂t = –∇×E
∂E/∂t = c²∇×B

We’re making abstraction here of stationary charges, and we also do not consider any currents here, so no moving charges either. So I am omitting the ∇·E = ρ/ε₀ equation (i.e. the first of the set of four equations), and I am also omitting the j/ε₀ in the second equation. So, for all practical purposes (i.e. for the purpose of this discussion), you should think of a space with no charges: ρ = 0 and j = 0. It’s just a traveling electromagnetic wave. To make things even simpler, we’ll assume our time and distance units are chosen such that c = 1, so the equations above reduce to:

∂B/∂t = –∇×E
∂E/∂t = ∇×B

Perfectly symmetrical! But note the minus sign in the first equation. As for the interpretation, I should refer you to previous posts but, briefly, the ∇× operator is the curl operator. It’s a vector operator: it describes the (infinitesimal) rotation of a (three-dimensional) vector field. We discussed heat flow a couple of times, or the flow of a moving liquid. So… Well… If the vector field represents the flow velocity of a moving fluid, then the curl is the circulation density of the fluid. The direction of the curl vector is the axis of rotation as determined by the ubiquitous right-hand rule, and its magnitude of the curl is the magnitude of rotation. OK. Next step.

II. For the wavefunction, we have Schrödinger’s equation, ∂ψ/∂t = i·(ħ/2m)·∇²ψ, which relates two complex-valued functions (∂ψ/∂t and ∇²ψ). Complex-valued functions consist of a real and an imaginary part, and you should be able to verify this equation is equivalent to the following set of two equations:

Re(∂ψ/∂t) = −(ħ/2m)·Im(∇²ψ)
Im(∂ψ/∂t) = (ħ/2m)·Re(∇²ψ)

[Two complex numbers a + ib and c + id are equal if, and only if, their real and imaginary parts are the same. However, note the −i factor in the right-hand side of the equation, so we get: a + ib = −i·(c + id) = d −ic.] The Schrödinger equation above also assumes free space (i.e. zero potential energy: V = 0) but, in addition – see my previous post – they also assume a zero rest mass of the elementary particle (E₀ = 0). So just assume E₀= V = 0 in de Broglie’s elementary ψ(θ) = ψ(x, t) = e⁻ⁱ^θ = a·e^{−i[(E₀+ p2/(2m) + V)·t − p∙x]/ħ} wavefunction. So, in essence, we’re looking at the wavefunction of a massless particle here. Sounds like nonsense, doesn’t it? But… Well… That should be the wavefunction of a photon in free space then, right? 🙂

Maybe. Maybe not. Let’s go as far as we can.

The energy of a zero-mass particle

What m would we use for a photon? It’s rest mass is zero, but it’s got energy and, hence, an equivalent mass. That mass is given by the m = E/c²mass-energy equivalence. We also know a photon has momentum, and it’s equal to its energy divided by c: p = m·c = E/c. [I know the notation is somewhat confusing: E is, obviously, not the magnitude of E here: it’s energy!] Both yield the same result. We get: m·c = E/c ⇔ m = E/c²⇔ E = m·c².

OK. Next step. Well… I’ve always been intrigued by the fact that the kinetic energy of a photon, using the E = m·v²/2 = E = m·c²/2 formula, is only half of its total energy E = m·c². Half: 1/2. That 1/2 factor is intriguing. Where’s the rest of the energy? It’s really a contradiction: our photon has no rest mass, and there’s no potential here, but its total energy is still twice its kinetic energy. Quid?

There’s only one conclusion: just because of its sheer existence, it must have some hidden energy, and that hidden energy is also equal to E = m·c²/2, and so the kinetic and hidden energy add up to E = m·c².

Huh? Hidden energy? I must be joking, right?

Well… No. No joke. I am tempted to call it the imaginary energy, because it’s linked to the imaginary part of the wavefunction—but then it’s everything but imaginary: it’s as real as the imaginary part of the wavefunction. [I know that sounds a bit nonsensical, but… Well… Think about it: it does make sense.]

Back to that factor 1/2. You may or may not remember it popped up when we were calculating the group and the phase velocity of the wavefunction respectively, again assuming zero rest mass, and zero potential. [Note that the rest mass term is mathematically equivalent to the potential term in both the wavefunction as well as in Schrödinger’s equation: (E₀·t +V·t = (E₀+ V)·t, and V·ψ + E₀·ψ = (V+E₀)·ψ—obviously!]

In fact, let me quickly show you that calculation again: the de Broglie relations tell us that the k and the ω in the eⁱ^{(kx − ωt)} = cos(kx−ωt) + i∙sin(kx−ωt) wavefunction (i.e. the spatial and temporal frequency respectively) are equal to k = p/ħ, and ω = E/ħ. If we would now use the kinetic energy formula E = m·v²/2 – which we can also write as E = m·v·v/2 = p·v/2 = p·p/2m = p²/2m, with v = p/m the classical velocity of the elementary particle that Louis de Broglie was thinking of – then we can calculate the group velocity of our eⁱ^{(kx − ωt)} = cos(kx−ωt) + i∙sin(kx−ωt) as:

v_g = ∂ω/∂k = ∂[E/ħ]/∂[p/ħ] = ∂E/∂p = ∂[p²/2m]/∂p = 2p/2m = p/m = v

[Don’t tell me I can’t treat m as a constant when calculating ∂ω/∂k: I can. Think about it.] Now the phase velocity. The phase velocity of our eⁱ^{(kx − ωt)} is only half of that. Again, we get that 1/2 factor:

v_p = ω/k = (E/ħ)/(p/ħ) = E/p = (p²/2m)/p = p/2m = v/2

Strange, isn’t it? Why would we get a different value for the phase velocity here? It’s not like we have two different frequencies here, do we? You may also note that the phase velocity turns out to be smaller than the group velocity, which is quite exceptional as well! So what’s the matter?

Well… The answer is: we do seem to have two frequencies here while, at the same time, it’s just one wave. There is only one k and ω here but, as I mentioned a couple of times already, that eⁱ^{(kx − ωt)} wavefunction seems to give you two functions for the price of one—one real and one imaginary: eⁱ^{(kx − ωt)} = cos(kx−ωt) + i∙sin(kx−ωt). So are we adding waves, or are we not? It’s a deep question. In my previous post, I said we were adding separate waves, but now I am thinking: no. We’re not. That sine and cosine are part of one and the same whole. Indeed, the apparent contradiction (i.e. the different group and phase velocity) gets solved if we’d use the E = m∙v² formula rather than the kinetic energy E = m∙v²/2. Indeed, assuming that E = m∙v² formula also applies to our zero-mass particle (I mean zero rest mass, of course), and measuring time and distance in natural units (so c = 1), we have:

E = m∙c² = m and p = m∙c²= m, so we get: E = m = p

Waw! What a weird combination, isn’t it? But… Well… It’s OK. [You tell me why it wouldn’t be OK. It’s true we’re glossing over the dimensions here, but natural units are natural units, and so c = c²= 1. So… Well… No worries!] The point is: that E = m = p equality yields extremely simple but also very sensible results. For the group velocity of our eⁱ^{(kx − ωt)} wavefunction, we get:

v_g = ∂ω/∂k = ∂[E/ħ]/∂[p/ħ] = ∂E/∂p = ∂p/∂p = 1

So that’s the velocity of our zero-mass particle (c, i.e. the speed of light) expressed in natural units once more—just like what we found before. For the phase velocity, we get:

v_p = ω/k = (E/ħ)/(p/ħ) = E/p = p/p = 1

Same result! No factor 1/2 here! Isn’t that great? My ‘hidden energy theory’ makes a lot of sense. 🙂 In fact, I had mentioned a couple of times already that the E = m∙v² relation comes out of the de Broglie relations if we just multiply the two and use the v = f·λ relation:

f·λ = (E/h)·(h/p) = E/p
v = f·λ ⇒ f·λ = v = E/p ⇔ E = v·p = v·(m·v) ⇒ E = m·v²

But so I had no good explanation for this. I have one now: the E = m·v²is the correct energy formula for our zero-mass particle. 🙂

The quantization of energy and the zero-mass particle

Let’s now think about the quantization of energy. What’s the smallest value for E that we could possible think of? That’s h, isn’t it? That’s the energy of one cycle of an oscillation according to the Planck-Einstein relation (E = h·f). Well… Perhaps it’s ħ? Because… Well… We saw energy levels were separated by ħ, rather than h, when studying the blackbody radiation problem. So is it ħ = h/2π? Is the natural unit a radian (i.e. a unit distance), rather than a cycle?

Neither is natural, I’d say. We also have the Uncertainty Principle, which suggests the smallest possible energy value is ħ/2, because ΔxΔp = ΔtΔE = ħ/2.

Huh? What’s the logic here?

Well… I am not quite sure but my intuition tells me the quantum of energy must be related to the quantum of time, and the quantum of distance.

Huh? The quantum of time? The quantum of distance? What’s that? The Planck scale?

No. Or… Well… Let me correct that: not necessarily. I am just thinking in terms of logical concepts here. Logically, as we think of the smallest of smallest, then our time and distance variables must become count variables, so they can only take on some integer value n = 0, 1, 2 etcetera. So then we’re literally counting in time and/or distance units. So Δx and Δt are then equal to 1. Hence, Δp and ΔE are then equal to Δp = ΔE = ħ/2. Just think of the radian (i.e. the unit in which we measure θ) as measuring both time as well as distance. Makes sense, no?

No? Well… Sorry. I need to move on. So the smallest possible value for m = E = p would be ħ/2. Let’s substitute that in Schrödinger’s equation, or in that set of equations Re(∂ψ/∂t) = −(ħ/2m)·Im(∇²ψ) and Im(∂ψ/∂t) = (ħ/2m)·Re(∇²ψ). We get:

Re(∂ψ/∂t) = −(ħ/2m)·Im(∇²ψ) = −(2ħ/2ħ)·Im(∇²ψ) = −Im(∇²ψ)
Im(∂ψ/∂t) = (ħ/2m)·Re(∇²ψ) = (2ħ/2ħ)·Re(∇²ψ) = Re(∇²ψ)

Bingo! The Re(∂ψ/∂t) = −Im(∇²ψ) and Im(∂ψ/∂t) = Re(∇²ψ) equations were what I was looking for. Indeed, I wanted to find something that was structurally similar to the ∂B/∂t = –∇×E and ∂E/∂t = ∇×B equations—and something that was exactly similar: no coefficients in front or anything. 🙂

What about our wavefunction? Using the de Broglie relations once more (k = p/ħ, and ω = E/ħ), our eⁱ^{(kx − ωt)} = cos(kx−ωt) + i∙sin(kx−ωt) now becomes:

eⁱ^{(kx − ωt)} = eⁱ^{(ħ·x/2 − ħ·t/2)/ħ} = eⁱ^{(x/2 − t/2)}= cos[(x−t)/2] + i∙sin[(x−t)/2]

Hmm… Interesting! So we’ve got that 1/2 factor now in the argument of our wavefunction! I really feel I am close to squaring the circle here. 🙂 Indeed, it must be possible to relate the ∂B/∂t = –∇×E and ∂E/∂t = c²∇×B to the Re(∂ψ/∂t) = −Im(∇²ψ) and Im(∂ψ/∂t) = Re(∇²ψ) equations. I am sure it’s a complicated exercise. It’s likely to involve the formula for the Lorentz force, which says that the force on a unit charge is equal to $E + v \times B$ $, with v the velocity of the charge. Why? Note the vector cross-product. Also note that \partial B /\partialt and \partial E /\partialt are vector-valued functions, not scalar-valued functions. Hence, in that sense, \partial B /\partialt and \partial E /\partialt and not like the Re (\partialψ/\partialt) and/or Im (\partialψ/\partialt) function. But\dots Well\dots For the rest, think of it:$ $E and B are orthogonal vectors, and that’s how we usually interpret the real and imaginary part of a complex number as well: the real and imaginary axis are orthogonal too!$

$So I am almost there.$ Who can help me prove what I want to prove here? The two propagation mechanisms are the “same-same but different”, as they say in Asia. The difference between the two propagation mechanisms must also be related to that fundamental dichotomy in Nature: the distinction between bosons and fermions. Indeed, when combining two directional quantities (i.e. two vectors), we like to think there are four different ways of doing that, as shown below. However, when we’re only interested in the magnitude of the result (and not in its direction), then the first and third result below are really the same, as are the second and fourth combination. Now, we’ve got pretty much the same in quantum math: we can, in theory, combine complex-valued amplitudes in four different ways but, in practice, we only have two (rather than four) types of behavior only: photons versus bosons.

Is our zero-mass particle just the electric field vector?

Let’s analyze that eⁱ^{(x/2 − t/2)}= cos[(x−t)/2] + i∙sin[(x−t)/2] wavefunction some more. It’s easy to represent it graphically. The following animation does the trick:

Animation

I am sure you’ve seen this animation before: it represents a circularly polarized electromagnetic wave… Well… Let me be precise: it presents the electric field vector (E) of such wave only. The B vector is not shown here, but you know where and what it is: orthogonal to the E vector, as shown below—for a linearly polarized wave.

Let’s think some more. What is that eⁱ^{(x/2 − t/2)}function? It’s subject to conceiving time and distance as countable variables, right? I am tempted to say: as discrete variables, but I won’t go that far—not now—because the countability may be related to a particular interpretation of quantum physics. So I need to think about that. In any case… The point is that x can only take on values like 0, 1, 2, etcetera. And the same goes for t. To make things easy, we’ll not consider negative values for x right now (and, obviously, not for t either). So we’ve got a infinite set of points like:

eⁱ^{(0/2 − 0/2)}= cos(0) + i∙sin(0)
eⁱ^{(1/2 − 0/2)}= cos(1/2) + i∙sin(1/2)
eⁱ^{(0/2 − 1/2)}= cos(−1/2) + i∙sin(−1/2)
eⁱ^{(1/2 − 1/2)}= cos(0) + i∙sin(0)
…

Now, I quickly opened Excel and calculated those cosine and sine values for x and t going from 0 to 14 below. It’s really easy. Just five minutes of work. You should do yourself as an exercise. The result is shown below. Both graphs connect 14×14 = 196 data points, but you can see what’s going on: this does effectively, represent the elementary wavefunction of a particle traveling in spacetime. In fact, you can see its speed is equal to 1, i.e. it effectively travels at the speed of light, as it should: the wave velocity is v = f·λ = (ω/2π)·(2π/k) = ω/k = (1/2)·(1/2) = 1. The amplitude of our wave doesn’t change along the x = t diagonal. As the Last Samurai puts it, just before he moves to the Other World: “Perfect! They are all perfect!” 🙂

graph imaginary graph real

In fact, in case you wonder how the quantum vacuum could possibly look like, you should probably think of these discrete spacetime points, and some complex-valued wave that travels as it does in the illustration above.

Of course, that elementary wavefunction above does not localize our particle. For that, we’d have to add a potentially infinite number of such elementary wavefunctions, so we’d write the wavefunction as ∑ a_je^−iθ_j functions. [I use the j symbol here for the subscript, rather than the more conventional i symbol for a subscript, so as to avoid confusion with the symbol used for the imaginary unit.] The a_jcoefficients are the contribution that each of these elementary wavefunctions would make to the composite wave. What could they possibly be? Not sure. Let’s first look at the argument of our elementary component wavefunctions. We’d inject uncertainty in it. So we’d say that m = E = p is equal to

m = E = p = ħ/2 + j·ħ with j = 0, 1, 2,…

That amounts to writing: m = E = p = ħ/2, ħ, 3ħ/2, 2ħ, 5/2ħ, etcetera. Waw! That’s nice, isn’t it? My intuition tells me that our a_jcoefficients will be smaller for higher j, so the a_j(j) function would be some decreasing function. What shape? Not sure. Let’s first sum up our thoughts so far:

The elementary wavefunction of a zero-mass particle (again, I mean zero rest mass) in free space is associated with an energy that’s equal to ħ/2.
The zero-mass particle travels at the speed of light, obviously (because it has zero rest mass), and its kinetic energy is equal to E = m·v²/2= m·c²/2.
However, its total energy is equal to E = m·v²= m·c²: it has some hidden energy. Why? Just because it exists.
We may associate its kinetic energy with the real part of its wavefunction, and the hidden energy with its imaginary part. However, you should remember that the imaginary part of the wavefunction is as essential as its real part, so the hidden energy is equally real. 🙂

So… Well… Isn’t this just nice?

I think it is. Another obvious advantage of this way of looking at the elementary wavefunction is that – at first glance at least – it provides an intuitive understanding of why we need to take the (absolute) square of the wavefunction to find the probability of our particle being at some point in space and time. The energy of a wave is proportional to the square of its amplitude. Now, it is reasonable to assume the probability of finding our (point) particle would be proportional to the energy and, hence, to the square of the amplitude of the wavefunction, which is given by those a_j(j) coefficients.

Huh?

OK. You’re right. I am a bit too fast here. It’s a bit more complicated than that, of course. The argument of probability being proportional to energy being proportional to the square of the amplitude of the wavefunction only works for a single wave a·e^−iθ. The argument does not hold water for a sum of functions ∑ a_je^−iθ_j. Let’s write it all out. Taking our m = E = p = ħ/2 + j·ħ = ħ/2, ħ, 3ħ/2, 2ħ, 5/2ħ,… formula into account, this sum would look like:

a₁eⁱ^{(x − t)(1/2)}+ a₂eⁱ^{(x − t)(2/2)}+ a₃eⁱ^{(x − t)(3/2)}+ a₄eⁱ^{(x − t)(4/2)}+ …

But—Hey! We can write this as some power series, can’t we? We just need to add a₀eⁱ^{(x − t)(0/2)}= a₀, and then… Well… It’s not so easy, actually. Who can help me? I am trying to find something like this:

power series

Or… Well… Perhaps something like this:

power series 2

Whatever power series it is, we should be able to relate it to this one—I’d hope:

Hmm… […] It looks like I’ll need to re-visit this, but I am sure it’s going to work out. Unfortunately, I’ve got no more time today, I’ll let you have some fun now with all of this. 🙂 By the way, note that the result of the first power series is only valid for |x| < 1. 🙂

Note 1: What we should also do now is to re-insert mass in the equations. That should not be too difficult. It’s consistent with classical theory: the total energy of some moving mass is E = m·c², out of which m·v²/2 is the classical kinetic energy. All the rest – i.e. m·c² − m·v²/2 – is potential energy, and so that includes the energy that’s ‘hidden’ in the imaginary part of the wavefunction. 🙂

Note 2: I really didn’t pay much attentions to dimensions when doing all of these manipulations above but… Well… I don’t think I did anything wrong. Just to give you some more feel for that wavefunction eⁱ^{(kx − ωt)}, please do a dimensional analysis of its argument. I mean, k = p/ħ, and ω = E/ħ, so check the dimensions:

Momentum is expressed in newton·second, and we divide it by the quantum of action, which is expressed in newton·meter·second. So we get something per meter. But then we multiply it with x, so we get a dimensionless number.
The same is true for the ωt term. Energy is expressed in joule, i.e. newton·meter, and so we divide it by ħ once more, so we get something per second. But then we multiply it with t, so… Well… We do get a dimensionless number: a number that’s expressed in radians, to be precise. And so the radian does, indeed, integrate both the time as well as the distance dimension. 🙂

Tag: the elementary wavefunction