The Phase of the Schoedinger Wave Function and Spin: An Improper Transformation

This is a mildly and constructively critical analysis of the latest paper in Ulf Klein’s series on the Foundations of Quantum Theory.

In a series of recent papers, Ulf Klein sheds tremendous light on the previously hidden structure of non-relativistic Quantum Theory (QT) by developing a theory of probabilistic classical mechanics on phase space, which he calls Hamilton-Liouville-Lie-Kolmogorov theory (HLLK), and then deriving QT from HLLK with a projection onto configuration space. In Alain Aspect, John Clauser, Anton Zeilinger and Bohr’s Correspondence Principle: A Myth Dispelled. | by Wes Hansen | Dec, 2022 | Medium, motivated by, Pioneering Quantum Physicists Win Nobel Prize in Physics, a terribly biased article from Quanta magazine, I summarize two of Professor Klein’s earlier papers which produced several key elements of QT, but not all; in particular, the massive particles were absent the key characteristic of spin. In this article I am mildly and constructively critical of, currently, the last paper in the series, A Reconstruction of Quantum Theory for Spinning Particles [1], which is entirely focused on explicating quantum spin. In particular, I demonstrate a counter-example to his position stated, among other places, in Section 6.3, titled, The Magnetic Moment of the Electron, page 18:

“[L]et us now ask for possible interpretations of the ‘magnetic moment of the electron.’ In the particle picture it was suggested by Uhlenbeck and Goudsmit that an ‘intrinsic’ angular momentum of the electron, called spin, is responsible for the observed effects. The magnitude of the spin vector is assumed to be a constant equal to ℏ/2. As already mentioned, the form of the relevant coupling term is compatible with this assumption, but not the magnitude of the prefactor (g = 2 instead of g =1). The second, much more serious shortcoming is the universally accepted fact that such a classical intrinsic spin cannot exist because it is in conflict with fundamental physical principles. While the theories of Dirac, Eberlein, and Levy-Leblond provide an explanation for the value g = 2, the fundamental second difficulty remains. In a relativistic world the construction of a classical model for a spinning electron is just as impossible as in the non-relativistic case. Thus, while the existence of quantum spin is experimentally extremely well confirmed, a classical counterpart of this property — if interpreted as a property of individual electrons — does not exist. Nevertheless, the intuitive ideas of Uhlenbeck and Goudsmit dominate our thinking about spin even today. The reason is that there is no better explanation, at least in the framework of the individuality interpretation (particle picture) of QT. This represents a painful gap in our understanding of nature, especially in view of the fundamental importance of spin for the stability of matter.”

The counter-example to this position that I have in mind, there are possibly others, is David Hestenes’ Zitter Model of the Electron extended, by Oliver Consa, to a Helical Solenoid Model with a toroidal moment. This semi-classical (or semi-quantal) model with intrinsic spin conforms remarkably well to Professor Klein’s analysis and rather elegantly explains a couple of his open questions: Why is the phase factor of 4π required to return the Schroedinger wave function to its initial state?; and, Why is the magnetic moment of the electron anomalous?

To answer the first question, as David Hestenes explains in his brief FQXI essay, Electron Time, Mass, and Zitter [2]:

1. In the Dirac picture, the electron spin and phase are inseparably related; page 3,
2. Electron spin is a measure of helical orientation; page 7,
3. The phase of the Schroedinger wave function describes zitter phase shifts; page 8.

From this it follows that we are dealing with an improper transformation, where a phase transformation of 2π induces a reversal in helical orientation.

To answer the second question, in Oliver Consa’s Helical Toroidal Electron Model [3] there is a g-factor entirely dependent on the geometry, which he calls the helical g-factor,

g′ = √1+(rN/R)^2

where R is the radius of the torus, r its thickness, and N ∈ Z^+ the number of turns. Then, from pages 84 and 85:

“In calculating the angular momentum, the rotational velocity decreases in the same proportion as the equivalent radius increase, compensating for the helical g-factor. However, in the calculation of magnetic moment, the rotational velocity decreases by a factor of g′, while the equivalent radius increases by a factor approximately equal to g′ squared. This is the cause of the electron’s anomalous magnetic moment.”

To get right to it, what IS interesting here is how Professor Klein uses these “Clebsch potentials” from fluid dynamics to convert a vorticity tensor defined on momentum fields, which “are generally not functionally independent from each other,” to a vorticity tensor defined on “suitable functionally independent quantities” called vortical variables. This vorticity tensor is then conveniently replaced by a vorticity vector and these vectors describe a vorticity field which “is by definition solenoidal” (emphasis mine). This, in turn, constrains the topology. From Section 4.1, page 10:

“The mapping of R^3 in the space of the Clebsch variables P, Q, which is suitable for describing the quantum mechanical ensembles occurring in nature, belongs to a topologically non-trivial class, with linked vortex lines and non-vanishing helicity. It is given by the so-called Hopf map or Hopf Bundle. We use the complex form of the Hopf map, which also provides us with an appropriate definition for the new (spinorial) state variable, which will later be used to perform the transition to QT.”

This is somewhat confusing to me and it seems that it could be simplified and clarified with Geometric Algebra/Calculus (for clarification see, Bundles, Bundles, Bundles, Connections and the Yang-Mills Field | by Hassaan Naeem | Medium). At any rate, we have these points z, described by a pair of complex numbers, on the three-sphere S_3 embedded in R^4 and the Hopf map takes these to the two-sphere describing the “Clebsch variables” Q, P (the vorticity vector fields). These points obviously describe spinors and, importantly, from page 11:

“This relation show that χ may be interpreted as an angle of rotation around an axis determined by ϑ and φ. [I]n particular, z changes its sign when χ changes by 2π and returns to its original value only when χ changes by 4π. An equivalent form of a ‘spinor,’ as a directed quantity that describes a rotation, was derived by Payne using intuitive geometric methods.”

The ϑ and φ come about due to a change to spherical coordinates for describing the points z on S_3 and it, S_3, has the structure of a fiber bundle. These technical maneuvers are carried out for topological reasons and they certainly seem to conform to what Professor Hestenes does in his paper, Zitterbewegung Structure in Electrons and Photons [4] (see page 6), following Ranada. Ranada used the same Hopf map to study toroidal solutions to Maxwell’s classical field equations and this, together with Oliver Consa’s toroidal extension, allows Professor Hestenes to develop his semi-classical Maxwell-Dirac Theory, a unification of quantum and classical theories of electromagnetism. What is obvious here is, this χ corresponds to the Schroedinger wave function phase and z to spin, i. e. they are encoded in these “Clebsch potentials.” Certainly this is so, because the new state variable is spinorial. From page 11:

“If we accept z, as defined by (54), as our new state variable (apart from an amplitude which will be introduced later) then it is obvious to associate the two-component quantity in (54), with components u_1, u_2, with the new rotational degrees of freedom Q, P. As a consequence we identify the prefactor e^{iχ/2} with the earlier phase factor e^{iS/ℏ}, associated with the irrotational momentum fields studied in III.

[T]he topological meaning of the ‘canonical Clebsch potentials’ defined by (59) was clarified by Kutnetsov and Mikhailov. These authors studied ideal fluids which are, however, described by essentially the same mathematical structure as the present problem. In their work the constant ℏ/2 is replaced by an undetermined constant — let us recall that we were only able to fix the value of this constant by anticipating the quantization problem.”

The “canonical Clebsch potentials” are

P = ℏ/2 cosϑ, Q = φ

from the transition to spherical coordinates. In Section 6.1 we learn that this Hopf map was developed because it describes the topology of rotations on the Hilbert space H^2. From page 16 (SU(2) is the spin group):

“The spinorial geometric object ψ associated with SU(2) is essentially (except for a subsequently added amplitude) given by the Hopf map, defined in section 4.1. The introduction of the Hopf map was based on the assumption that the statistical ensembles occurring in nature belong to a topological class describing linked vortex lines. This choice, which may have seemed a bit arbitrary, is now justified by the fact that the Hopf map describes the topology of rotations in H^2. The property of linked vortex lines seems to correspond to a particular topological property of rotations, namely the fact that ψ does not return to the starting point until it has been rotated by 4π.”

And here we have emphasis on this phase factor of 4π again. As Professor Hestenes has pointed out years ago, this is already implicit in the zitterbewegung Schroedinger discovered in the Dirac equation: electron spin and phase are inseparably linked. This can only be described with an improper transformation in group theory.

If you read Professor Klein’s paper, what we are really dealing with is the Euclidean Group, E(3), of transformations which preserve Euclidean distance and, more specifically in this case I would think, that subset called isometries, i. e. rigid motions; these are the transformations which leave a two-dimensional or three-dimensional figure invariant.

• Definition 1: a two-dimensional isometry preserves orientation if the clockwise direction around a circle before the transformation is applied is still the clockwise direction after the transformation is applied;
• Definition 2: a two-dimensional isometry which preserves orientation is called proper.

Okay, a two-dimensional isometry is either a rotation, a translation, or a glide-reflection, where glide-reflections include simple reflections. Rotations and translations are proper isometries, but glide-reflections are improper (this should be obvious). If two isometries are composed, i. e. applied in series, then the result will be proper if both isometries are proper, proper if both isometries are improper, and improper if one isometry is proper and the other is improper.

With that out of the way, let’s assume David Hestenes’ Zitter model of the electron and see if we can make geometric sense out of things. This image of the Zitter electron is from [2]:

In the Zitter model the electron is a point charge which traces out a helical orbit centered on a streamline through spacetime. In Professor Hestenes’ Geometric Algebra language, electron spin is represented by a 1-vector summed with a 2-vector, this last, geometrically speaking, being an oriented plane. The helix in the image is the electron’s field (“pilot” wave) and the 2-vector is perpendicular to the helical axis. Spin is a measure of helical orientation. From [2] (emphasis mine):

Basic features of the zitter model can now be summarized as follows:

The spacetime history of electron is a lightlike helix.

Electron mass (≈ zitter frequency) is a measure of helical curvature.

Electron phase (≈ zitter angle) is a measure of helical rotation.

Electron spin is a measure of helical orientation.

Electron zitter generates a static magnetic dipole and rotating electric dipole!

Here is an image of the two possibilities corresponding to spin +1/2 and spin - 1/2:

Okay, so the transformation is actually a composition of a three-dimensional glide-rotation, which is proper and represents a phase shift, and a two-dimensional plane reflection, which is improper. Hence, a phase shift of 2π actually corresponds to an improper isometry which, when composed with its inverse, also a 2π phase shift, becomes proper, i. e. preserves or restores helical orientation (spin).

And this zitterbewegung is also related, I believe, to the Quantitative Geometrical Thermodynamics (QGT) of Michael Parker and Chris Jeynes. In their paper, Maximum Entropy (Most Likely) Double Helical and Double Logarithmic Spiral Trajectories in Space-Time, they show that the double helix has maximum geometric entropy and from nature, the plant kingdom in particular, it would seem that evolution shows that the helix alone has maximum geometric entropy (they also study this). In his Section 8, Professor Klein uses statistical considerations to transition from HLLK to QT. From page 24:

“We need a second assumption, presumably of a statistical nature. A most fundamental statistical principle says that all states that are unknown must occur with the same probability. In statistical mechanics (a type 2 theory) this principle is implemented through the requirement for maximum entropy. In the problem at hand, we are faced with the task of determining certain terms in a differential equation in accordance with this principle.”

As Professor Klein states himself, QT lives on configuration space and massive spin 1/2 particles come about due to the three dimensions of space, hence, it seems unavoidable that the entropy in question is geometric in nature (especially considering the above QGT). And what do we find? From page 25:

“A new adjustable parameter appears on the right-hand sides of these expressions which has been identified with ℏ^2/2m. Let us recall here that two different adjustable parameters appeared in the course of our developments both of which were identified with ℏ. The first ℏ was associated with the length of the spin vector s. The second ℏ is associated with the quantum-mechanical principle of maximal disorder. The physical meaning of these two adjustable parameters is different, but they must both be identified with Planck’s constant in order to enable the transition to QT.”

The physical meaning of these two adjustable parameters may very well be different, one clearly geometrical and the other statistical, but the work of Professors Parker and Jeynes strongly suggests to me that they are intimately related through the helical geometry of the Toroidal Solenoid Electron Model. And this semi-classical (or semi-quantal) model can also address Professor Klein’s second concern. From page 18:

“This insight automatically leads to the next question. If the deviation of the gyromagnetic factor from 1 is a purely topological effect why should we then need QT to derive it? The answer given by the present theory is that we actually do NOT need QT to derive it; according to the present derivation, it may be classified as a semi-classical (or semi-quantal) effect.”

He’s suggesting here that the deviation is purely topological because he earlier suggested that it was tempting to relate this deviation to the 4π phase factor, i. e. 2 = 4π/2π. But in Professor Consa’s model it’s purely geometric and arises due to his helical g-factor. Of course it’s a fine line that separates geometry and topology, the topology describes the inseparability of spin and phase while the geometry describes the topology! And the Toroidal Solenoid Electron Model is a semi-classical (or semi-guantal) model which has already been extended by David Hestenes to a semi-classical theory of electromagnetism called Maxwell-Dirac theory. From [4], page 12:

“Though a properly tuned magnetic resonance measurement may activate toroidal zitter in the electron, external fields are not necessary to maintain it. If the toroidal zitter can be quenched or activated at will, then the electron has at least two distinguishable internal states and might thereby serve as the ultimate magnetic storage device. On the other hand, if toroidal zitter is a persistent intrinsic property of the electron, then circular zitter should simply be regarded as an approximation.”

That the electron has two distinct states has already been shown in Magnetic Monople Field Exposed by Electrons, in which they simulate a magnetic monopole with a nano-needle, tip of which is poised over an aperture. Electrons which interact with the monopole field change state and they call these altered state electrons “vortex electrons!”

Finally, in his Section 9.2, Professor Klein checks the limit of the derived Pauli-Schroedinger equation as the reduced Planck’s constant goes to zero and finds that not all spin variables are destroyed. From pages 27 and 28:

“The survival of spin variables in the case ℏ = 0 is no surprise in our theory, since we have identified the vertical components of the momentum field as origin of quantum spin. In all works in which Eqs. (152), (153) were derived so far, the starting point was the quantum mechanical Pauli-Schoedinger equation (151), which was then rewritten, using a representation like (75) (see [38], [26], [61]). The limiting case ℏ = 0 was rarely dealt with in those theories, see however [61]. One reason for this might be that this limiting case is not compatible with the prevailing interpretation of spin as a purely quantum mechanical phenomenon. Due to this interpretation, all spin variables (or the corresponding terms in a Lagrangian function) should disappear from the theory in the limit ℏ = 0. The fact that this is not the case led Yahalom to the conclusion that the Pauli theory ‘has no standard classical limit’ [61]. In fact, one could have concluded from this fact that spin cannot be a purely quantum mechanical phenomenon.”

His idea seems to be that quantum spin has its origins in his theory QA, which is in the borderland between the classical and the quantum; it’s an ensemble phenomenon with origins in these momentum fields or, more precisely, the “Clebsch potentials” defined on these momentum fields. What he fails to take into consideration here is that in Professor Hestenes Zitter Model the momentum decomposes the spin into a vector component and a bivector component. From [2], page 7 (parentheses mine):

“The momentum determines an intrinsic decomposition of the spin into a spatial part is (the i here is the unit psuedoscalar and the s the spin vector) specifying the tube cross section and a temporal part mr specifying the temporal pitch of the helix.”

If you take Plack’s constant to zero, all that disappears is the bivector component describing the spatial part; the vector component survives. This spatial component disappears for the same reason position/momentum uncertainty does. From Professor Hestenes’ paper Quantum Mechanics from Self-Interaction, page 9:

“If we wish to localize a free electron, the zbw implies that the best we can do is confine it to a circular orbit of radius r = ℏ/mc with a fixed center. Therefore, the x-coordinate of the electron in the orbital plane will fluctuate with a range Δx = ℏ/mc. At the same time, since the electron travels at the same time, and since the electron travels at the speed of light with a zeropoint kinetic energy mc^2/2, the x-component of its momentum fluctuates with a range Δp_x = mc/2. Thus, we obtain the minimum uncertainty relation

ΔxΔp_x = ℏ/2

We now see the uncertainty relations as consequences of a zero-point motion with a fixed zero-point angular momentum, the spin of the electron. This explains why the limiting constant ℏ/2 in the uncertainty relations is exactly equal to the magnitude of the electron spin.”

It seems as though Ulf Klein is completely oblivious to zitterbewegung and, more generally, Professor Hestenes’ entire life’s work. Is this possible? David Hestenes is well-known due to his development of Geometric Algebra/Calculus from Clifford Algebra but, as he himself points out, well, from [2], page 3 and then pages 1 and 2:

“Schoedinger [5] coined the term zitterbewegung (trembling motion) to describe oscillations in free particle solutions of the Dirac equation. It’s putative physical interpretation has been clearly described by Huang [6]:

‘The well-known Zitterbewegung may be looked upon as a circular motion about the direction of the electron spin with radius equal to the Compton wavelength of the electron divided by 2π. The intrinsic spin of the electron may be looked upon as the orbital angular momentum of this motion. The current produced by the Zitterbewegung is seen to give rise to the intrinsic magnetic moment of the electron.’

Dirac himself concurred with this interpretation [7]. No doubt the weight of Dirac’s authority accounts for its reiteration in textbooks on relativistic quantum mechanics and field theory to this day, despite the fact that it has not been subjected to a single experimental test. Indeed, the zitterbewegung concept has served as no more than metaphorical window dressing on abstract formalism, while its staggering theoretical implications remain unexamined!

[d]e Broglie went further to propose that a wave of the same frequency was associated with the motion of an electron [1]. As everyone knows, this wave hypothesis was immediately extended by Schroedinger to create his famous wave equation that has become a paradigm of quantum mechanics. Ironically, de Broglie’s clock hypothesis has been ignored or forgotten in the physics literature since. Besides, how could one read time on a clock with a period of 10^{21} seconds?

Many decades passed before French experimental physicist Michel Gouanere resolved to search for the electron clock.

[A]s he related it to me, Gouanere discussed various experimental alternatives with his colleague M. Spighel until they seized on electron channeling as a feasible possibility.

[A] prediction of the resonant energy is easily calculated.

[T]hey knew that funding for such an offbeat experiment would be impossible to secure, so they organized a research team and wrote a proposal to study ‘Kumakhov radiation’ in the 54–110 MeV region on the linear accelerator at Saclay. It was not until the project was up and running that they informed other members of the team about what they really wanted to do.

[T]heir results were published in Annales de la Fondation Louis de Broglie in 2005. Predictably, the impact was nil, as that journal attracts few readers. To get more visability, Gouanere submitted a slightly modified account to Physical Review Letters. It was rejected in January 2007. The majority of reviewers regarded the reported results as physically implausible. Their response reminds me of Eddington’s ironic remark, “I won’t believe the experiment until it is confirmed by theory!” However, one reviewer suggested that the effect might be explained by Schroedinger’s zitterbewegung. Gouanere had never heard of zitterbewegung, so he Googled it and found an article of mine [2], which argues that zitterbewegung is fundamental for interpretation of the Dirac equation and a fortiori for interpretation of quantum mechanics.

[I] jumped at the chance to explain his data quantitatively. The results could not be more satisfactory.”

You can read his papers and decide for yourself. Whatever the case may be, as shown above, Professor Klein’s reconstruction is not only compatible with but remarkably supportive of Professor Hestenes’ Zitter Model and, more generally, his Maxwell-Dirac theory with its Toroidal Solenoid Electron curtesy of Oliver Consa. Why do we never see these works highlighted in the popular physics media, SciAm, Physics World, Quanta Magazine? All we see is wormholes, strings, branes, and other nonsense crackpottery.