Back to Blogging, Uploaded a paper I wrote in 1986 about Preonic Grand Unification

It has been almost 3 years since my last Blog post.  Much of my time has been diverted into a condo hotel project in Longboat Key Florida, and the focus I need to do good physics has been impossible to come by.  Then, the other day, Ken Tucker, a frequent participant at sci.physics.foundations, emailed me about some new research showing that electrons have constituent substructure.  That brought me back immediately to the half a year I spent back in 1986 developing a 200-page paper about a preonic substructure for quarks and leptons, which culminated six years of study from 1980 to 1986.  I finished that paper in August 1986, and then took an 18 year hiatus from physics, resuming again in late-2004.

Ken’s email motivated me to dig out this 1986 paper which I manually typed out on an old-fashioned typewriter, scan it into electronic form, and post it here.  Links to the various sections of this paper are below.  This is the first time I have ever posted this.

Keep in mind that I wrote this in 1986.  I tend to study best by writing while I study, and in this case, what I wrote below was my “study document” for Halzen and Martin’s book “Quarks and Leptons” which had just come out in 1984 and was the first book to pull together what we now think of as modern particle physics and the (then, still fairly new) electroweak unification of Weinberg-Salam.

What is in this paper that I still to this day believe is fundamentally important, and has not been given the attention it warrants, is the isospin redundancy between (left-chiral) quarks and leptons.  This to me is an absolute indication that these particles have a substructure, so that a neutrino and an up quark both have contain the same “isospin up” preon, and an electron and a down quark both contain the same “isospin down” preon.  Section 2.11 below is the key section, if you want to cut to the chase with what I was studying some 26 years ago.  I did post about this in February 2008 at https://jayryablon.wordpress.com/2008/02/02/lab-note-4-an-interesting-left-chiral-muliplet-perhaps-indicative-of-preonic-structure-for-fermions/, though that post merely showed a 1988 summary I had assembled based on my work in 1986, at the behest of the late Nimay Mukhopadhyay, who at the time was teaching at RPI and had become a good friend and one of my early sources of encouragement.  This is the first time I am posting all of that early up-to-1986 work, in complete detail.

Lest you think me crazy, note that seventeen years later, G. Volovik, in his 2003 book “The Universe in a Helium Droplet,” took a very similar tack, see Figure 12.2 in this excerpt: Volovik Excerpt on Quark and Lepton Preonic Structure.

The other aspect of this 1986 paper that I still feel very strongly about, is taking the Dirac gamma-5 as a fifth-dimension indicator.  I know I have been critiqued by technical arguments as to why this should not be taken as a sign of a fifth dimension, but this fits seamlessly with Kaluza Klein which geometrizes the entirely of Maxwell’s theory and is still the best formal unification of classical electromagnetism and gravitation ever developed.  For those who maintain skepticism of Kaluza-Klein and ask “show me the fifth dimension,” just look to chirality which is well-established experimentally.  Why do we have to assume that this fifth dimension will directly manifest in the same way as space and time, if its effects are definitively observable in the chiral structure of fermions?  Beyond this, I remain a very strong proponent of the 5-D Space-Time-Matter Consortium, see http://astro.uwaterloo.ca/~wesson/, which regards matter itself as the most direct manifestation of a fifth physical dimension.  Right now, most folks think about 4-D spacetime plus matter.  These folks correctly think about 5-D space-time-matter, no separation.  And Kaluza-Klein, which historically predated Dirac’s gamma-5, is the underpinning of this.

After my hiatus of the past couple of years, I am going to try in the coming months to write some big-picture materials about physics, which will pull together all I have studied so far in my life.  I am thinking of doing a “Physics Time Capsule for 2100” which will try to explore in broad strokes, how I believe physics will be understood at the end of this century, about 88 years from now.

Anyway, here is my entire 1986 paper:

Preonic Grand Unification and Quantum Gravitation: Capsule Outline and Summary

Abstract and Contents

Section 1.1: Introduction

Section 1.2: Outline and Summary

Section 2.1: A Classical Spacetime Introduction to the Dirac Equation, and the Structure of Five-Dimensional Spacetime with a Chiral Dimension

Section 2.2: Particle/Antiparticle and Spin-Up/Spin-Down Degrees of Quantum Mechanical Freedom in Spacetime and Chirality, Gauge Invariance and the Dirac Wavefunction

Section 2.3: Determination and Labeling of the Spinor Eigensolutions to the Five-Dimensional Dirac Equation, and the High and Low Energy Approximations

Section 2.4: The Fifth-Dimensional Origin of Left and Right Handed Chiral Projections and the Continuity equation in Five Dimensions: Hermitian Conjugacy, Adjoint Spinors, and the Finite Operators Parity (P) and Axiality (A)

Section 2.5: Conjugate and Transposition Symmetries of the Dirac Equation in Five Dimensions, the Finite Operators for Conjugation (C) and Time Reversal (T), and Abelian Relationships Among C, P, T and A

Section 2.6: Charge Conjugation, and the Definitions and Feynman Diagrams for “Electron” and “Positron” Spinors

Section 2.7: Simple Unpolarized s,t,u Scattering Channels with a Covariant Propagator, and the Covariant (Real and Virtual) Polarization States of Massive and Massless Vector Bosons

Section 2.8: Prelude to Preons: The Spinor Decomposition of Four Real Spacetime Dimensions ct,x,y,z into Two Complex Spinor Dimensions Using the Covariant Polarization States of Vector Bosons

Section 2.9: Introduction to Isospin Preons in Electroweak Theory: The Preonic Decomposition of Four Real Electroweak Bosons A, W+, W-, Z into Two Complex Preons Denoting “Isospin Up” and “Isospin Down”

Section 2.10: Summarization of Prior Discussion, and on the Fundamental Importance of Preons in Particle Physics

Section 2.11: The Four-Preon Flavor SU(4) Unification of the Electromagnetic, Weak and Colorless Strong Interactions Excluding Quantum Gravitation; and the Colorless Flavor Classification of Left Handed Real Fermion and Boson Chiral Projections, for a Single Fermion Generation

Section 2.12: The Four-Preon Flavor SU(4)xU(1) Unification of Electromagnetic, Weak, Colorless Strong and Quantum Gravitational Interactions; and the Colorless Flavor Classification of Left and Right Handed Real Fermion and Boson Chiral Projections, for a Single Fermion Generation

Addendum to Section 2.12

Section 2.13: The Six-Preon Unification of Flavor SU(4)xU(1) with High Energy Color SU(4)xU(1) and Two Overlapping Degrees of Freedom; the Flavor and Color Classification of Real Fermions and Vector Bosons for a Single Generation; and the Derivation of Electroweak and Strong/Hyperweak Massless and Massive Neutral Current Vector Bosons

Section 2.14: On the Replication of Fermion Generations: Four Generational Grand Unification with Eighteen Preons and Nine Independent Flavor/Color/Generation Degrees of Freedom, and a Preonic Discussion of Mesons and Meson Decay

References and Bibliography

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Finite Amplitudes Without +i\epsilon

To all,

I have now completed a paper at the link below, which summarizes the work I have been doing for the past two months (and in a deeper sense, for much of my adult life) to lay a foundation for understanding and calculating particle masses:

finite-amplitudes-without-i-epsilon

I have also taken the plunge and submitted this for peer review. ;-)?

The abstract is as follows:

By carefully reviewing how the invariant amplitude M is arrived at in the simplest Yang-Mills gauge group SU(2), we show how to arrive at a finite, pole-free amplitudes without having to resort to the “+i prescription.”  We first review how gauge boson mass is generated in the SU(2) action via spontaneous symmetry breaking in the standard model, and then carefully consider the formation of finite, on-shell amplitudes, without +i.

Comments are welcome, and I wish everyone a happy holiday and New Year!

Jay.

Foldy-Wouthuysen, continued

Just for the heck of it, I did a calculation of what happens to the mass matrix  during the transformation from the Dirac-Pauli representation to the Newton-Wigner representation via Foldy-Wouthuysen.  This is shown in:

https://jayryablon.files.wordpress.com/2008/06/foldy-wouthuysen.pdf

Not sure where to go from there, but I’ll be away the rest of the week on vacation, so I’ll take another look when I return.

Interested in any further thoughts anyone may have.

Jay

Might Foldy-Wouthuysen Transformations Contain a Hidden Fermion Mass Generation Mechanism?

I have been looking over the following three links for the Foldy-Wouthuysen transformation from the Dirac-Pauli to the Newton-Wigner representation of Dirac’s equation:

The first shows the calculation itself of this transformation:

I: http://www.physics.ucdavis.edu/~cheng/230A/RQM7.pdf.

The second, an excellent and lucid exposition of the physics (why this is of interest), is to be found at:

II: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.27.3209&rep=rep1&type=pdf.

The third, dealing with Zitterbewegung motion and the velocity operator in the Dirac-Pauli representation, is at:

III: http://en.wikipedia.org/wiki/Zitterbewegung.

What I would like to discuss, for the purpose of getting your reactions as to whether I am on a sensible track, is the possibility that a mechanism for generating fermion mass may be hidden in all of this.

I say this in particular because in the Dirac-Pauli representation, the velocity operator is given by:

(1)

where , see reference III.  Further, the eigenvalues of this velocity operator constrain the velocity of the Fermion of be the speed of light, see reference II in the middle of page 3.  This means that the fermion must be massless and luminous, in the Dirac-Pauli representation.  Why this is so, has long been a mystery, and is thought not to make any sense, for obvious reasons.

Now, transform into the Newton-Wigner representation via Foldy-Wouthuysen.  The velocity operator in Newton-Wigner now takes the classical form:

   (2)

where  is the position operator.  But even more importantly, Newton-Wigner permits a range of eigenvalues less than the speed of light, and so, the fermions permitted by Newton-Wigner are massless and sub-luminous.

Following this to its logical conclusion, this seems to suggest that somewhere hidden in the Foldy-Wouthuysen transformation, we have gone from a fermion which is massless and luminous, to one which has a finite, non-zero rest mass and travels at sub-luminous velocity.  It seems, then, that it would be important to specifically trace how the velocity operator (1) of the Dirac-Pauli representation with eigenvalues transforms into the velocity operator (2) of Newton-Wigner which allows a continuous, sub-luminous velocity spectrum, and at the same time, to trace through how the rest mass goes from necessarily zero (with decoupled chiral components), to non-zero with chiral couplings.

By doing so, perhaps one would find a mechanism for generating fermion masses.

One contrast to make here: think about how vector boson masses are generated.  One starts with a Lagrangian in which the boson mass term is omitted entirely.  Then, via a well-knows technique, one breaks the symmetry and reveals a boson mass.  Perhaps the mystery of luminous velocity eigenvalues in the Dirac-Pauli representation is telling us a similar thing: Start out with a Dirac-Pauli Lagrangian in which the mass of the fermion is zero, i.e., without a mass term.  Then, the +/- c velocity eigenvalues make sense.  Transform that into the Newton-Wigner representation.  Somewhere along the line, a mass must appear, because a subliminous velocity appears.

I will, of course, try to pinpoint how this all happens, if it does indeed happen.  But I would for now like some reactions as to the tree up which I am barking.

Thanks,

Jay.

Heisenberg Uncertainty and Schwinger Anomaly: Two Sides of the Same Coin?

 In section 3 of Heisenberg Uncertainty and Schwinger Anomaly: Two Sides of the Same Coin?, I have posted a calculation which shows why the Schwinger magnetic anomaly may in fact be very tightly tied to the Heisenberg inequality .  The bottom line result, in (3.11) and (3.12), is that the gyromagnetic “g-factor” for a charged fermion wave field with only intrinsic spin (no angular momentum) is given by:

  (3.11)

It is also helpful to look at this from the standpoint of the Heisenberg principle as:

  (3.12)

The draft paper linked above has details of how I got here.  Below, are some ways to think about this:

 First, if (3.11) is true, then the greater than or equal to inequality of Heisenberg says, in this context, that the magnitude of the intrinsic g-factor of a charged wavefunction is always greater than or equal to 2.  That is, the inequality becomes another way of stating a parallel inequality .  We know this to be true for the charged leptons, which have , , and respectively. [The foregoing data is extracted from W.-M. Yao et al., J. Phys. G 33, 1 (2006)]

 Secondly, the fact that the charged leptons have g-factors only slightly above 2, suggests that these a) differ from perfect Gaussian wavefunctions by only a very tiny amount, b) the electron is slightly more Gaussian than the muon, and the muon slightly more-so than the tauon.  The three-quark proton, with , is definitively less-Gaussian the charged leptons.  But, it is intriguing that the g-factor is now seen as a precise measure of the degree to which a wavefunction differs from a perfect Gaussian.

 Third, (3.11) states that the magnetic moment anomaly via the g-factor is a precise measure of the degree to which exceeds .  This is best seen by writing (3.11) as (3.12).

Thus, for the electron, , to give an exact numerical example.  For a different example, for the proton, .

 Fourth, as a philosophical and historical matter, one can achieve a new, deeper perspective about uncertainty.  Classically, it was long thought that one can specify position and momentum simultaneously, with precision.  To the initial consternation of many and the lasting consternation of some, it was found that even in principle, one could at best determine the standard deviations in position and momentum according to .  There are two aspects of this consternation:  First, that one can never have as in classical theory.  Second, that this is merely an inequality, not an exact expression, so that even for a particle with , we do not know for sure what is its exact value of .  This latter issue is not an in-principle limitation on position and momentum measurements; it is a limitation on the present state of human knowledge.

 Now, while is a lower bound in principle, the question remains open to the present day, whether there is a way, for a given particle, to specify the precise degree to which its exceeds , and how this would be measured.  For example, one might ask, is there any particle in the real world that is a perfect Gaussian, and therefore can be located in spacetime and conjugate momentum space, down to exactly .  Equation (3.12) above suggests that if such a particle exists, it must be a perfect Gaussian, and, that we would know it was a perfect Gaussian, if its g-factor was experimentally determined to be exactly equal to the Dirac value of 2.  Conversely, (3.12) tells us that it is the g-factor itself, which is the direct experimental indicator of the magnitude of for any given particle wavefunction.  The classical precision of comes full circle, and while it will never return, there is the satisfaction of being able to replace this with the quantum  mechanical precision of (3.12), , rather than the weaker inequality of .

 Fifth, if (3.12) is correct, then since it is independently known from Schwinger that , this would mean that we would have to have:

  (3.13)

Thus, from the perturbative viewpoint, the degree to which exceeds would have to be a function of the running coupling strength in Heaviside-Lorentz units.  As Carl Brannnen has explicitly pointed out to me, this means that a Gaussian wavepacket is by definition non-interacting; as soon as there is an interaction, one concurrently loses the exact Gaussian.

 Sixth, since deviation of the g-factor above 2 would arise from a non-Gaussian wavefunction such as , the rise of the g-factor above 2 would have to stem from the term in this non-Gaussian wavefuction.  In this regard, we note to start, that , for a non-Gaussian wavefunction, versus for a perfect Gaussian.

 Finally, to calculate this all out precisely, one would need to employ a calculation similar to that shown at http://en.wikipedia.org/wiki/Uncertainty_principle#Wave_mechanics, but for the non-Gaussian rather than the Gaussian, to arrive at the modified bottom line equation of this Wiki section.  That is the next calculation I plan, but this is enough, I believe, to post at this time.

A Possible Kaluza-Klein Experiment

It has been suggested — appropriately so — that I consider whether there might be one or more experiments which can be designed to validate or falsify some of the Klauza-Klein results which I have been posting of late. I believe that one possible experiment resides in the non-symmetric energy tensor of trace matter derived in (11.6) of my latest posted paper. Thus, I have added a new section 15 to this paper, and reposted the entire paper, with this new section 15, at Kaluza-Klein Theory and Lorentz Force Geodesics Rev. 6.0 Because this is of particular interest as it may open some new experimental windows, I have posted section 15 below as well. Please note: the specific discussion of the connection between the compactified fifth dimension, and intrinsic spin, is not updated in this paper, and the latest discusssion I have written up on this topic, is at Intrinsic Spin and the Kaluza-Klein Fifth Dimension.

  Section 15: At this juncture, we have enough information to propose an experiment to validate or falsify some of the results derived thus far.  We turn for this purpose to the stress energy tensor of matter (11.6), which we raise into contravariant notation as follows:

. (15.1)

The Maxwell tensor is, of course, a symmetric tensor.  But the added trace matter term is not necessarily symmetric, that is, there is no a priori reason why must be equal to .  The origin of this non-symmetry was discussed earlier in Section 9.

 With an eye toward conducting an experiment, let us now consider (15.1) in the linear approximation of (13.6) where .  In the linear approximation, as used to reach (13.3), (12.11) reduces to , and (15.1) becomes:

, (15.2)

where we have also used and , and divided out .  The transpose of this non-symmetric energy tensor is:

, (15.3)

 Now, it is known that a non-symmetric energy tensor, physically, is indicative of a non-zero spin density.  In particular, using (15.2) and (15.3), the non-symmetry of the energy tensor is related to a non-zero spin density tensor according to: [A good, basic discussion of the spin tensor is at http://en.wikipedia.org/wiki/Spin_tensor.]

. (15.4)

For such a non-symmetric tensor, the “energy flux” is not identical to the “momentum density, as these differ by (15.4), for , and vice versa.  If the spin density , then  in this special case, (15.4) yields:

. (15.5)

So, for , (15.3) may be written using (15.5) as the explicitly-symmetric tensor:

. (15.6)

 Now, let’s consider a experiment which is entirely classical.  The “Poynting” components of (15.4), (15.6) represent the energy flux across a two-dimensional area, for a flux of matter which we will take to be a stream of electrons, while the components represent the momentum density.  The proposed experiment, then, will be to fire a stream of a very large number of electrons thereby constituting an electron “wave,” and to detect the aggregate flux of energy across a two-dimensional surface under various spin preparations, in precisely the same manner that one might test the flow of luminous energy across a surface when using light waves rather than electron waves.  Specifically, we propose in test I to fire electrons without doing anything to orient their spins, so that, statistically, the number of electrons flowing through the flux surface with positive helicity is equal to the number with negative helicity and so the spin density is zero, and (15.6) applies.  In test II, we fire electrons, but apply a magnetic field before detecting the flux, to ensure that all of the electrons are aligned to positive helicity.  In this event, the spin density, by design, is non-zero, and one of (15.2) or (15.3) will apply.  In test III, we do the same, but now apply the magnetic field to ensure that all of the electrons have negative helicity, before detecting the flux.

Lab Note 2 Intermezzo: Change of View to a Spacelike Fifth Dimension, as the Geometric Foundation of Intrinsic Spin

(A PDF version of this Lab Note may be downloaded from Lab Note 2 Intermezzo.) 

 Those who have followed the development of this lab note know that I have been working with a Kaluza-Klein theory which regards the fifth dimension as timelike, rather than spacelike.  After reviewing some key literature in the field including a Sundrum Lecture recommended by Martin Bauer and several articles by Paul Wesson linked over at The 5-D Space-Time-Matter Consortium, I have undergone a conversion to the view that the fifth dimension needs to be spacelike – not timelike – and specifically, that it needs to be a compact, spacelike hypercylinder.  In this conversion, I am motivated by the following reasoning, which gives a geometric foundation to intrinsic spin: 

Lab Note 3, Part 2: Unification of Particle, Nuclear and Atomic Phenomonology

This lab note will be brief.

On April 28, 2007, I posted a paper which went from baryons and confinement to strings to particle phenomenology to atomic physics and deuterons and a whole range of phenomenology including fermion generation replication which appeared to lend itself to a common, underlying explanation based on the work I have previously discussed with respect to baryons and confinement in particular.  The underlying thread throughout, is to connect spacetime symmetry to internal symmetry using the Pauli fermionic exclusion principle.   I am afraid, however, that this paper may have been buried amidst all of the other postings, so I want to specifically call it to your attention, at the link below:

On The Natural Origin of Baryons, Short-Range Mesons, and QCD Confinement, from Maxwell’s Magnetic Equations for a Yang-Mills Field

In the spirit of “Lab Notes” which are a scientific diary of theoretical explorations, I ask you in particular to look at the second half of this paper, starting at section 6.  In football, there is something known as a “Hail Mary” pass where the quarterback throws the ball all the way down the field hoping for a touchdown.  The second half of the above paper is just that.  While certainly speculative, it seems to me that this ties together a very diverse range of observable phenomenology which has not previously been tied together.   It is probably the most audacious piece of physics writing I have done, and I don’t want it to get lost in the shuffle.

So, if nothing comes of it, so be it.  But, it may well be that someone in the end zone will catch this long pass, and physics will come to rest in a different place from where it rests today.  That is why it is so important to take good lab notes!

Lab Note 4: An Interesting Left-Chiral Muliplet Perhaps Indicative of Preonic Structure for Fermions

I return in this brief Lab Note to the underlying spirit of one of the basic premises of this Weblog, which is that these are a series of “Lab Notes.”  We often tend these days to think and speak about “theories,” rather than “notes,” and certainly, many of the “Lab Notes” which I am presenting here are intended to be thought of as “theories,” or “theories-under-development,” as much as “lab notes.”

But when one talks about “Lab Notes,” what should be the prevailing thought is that sometimes, in the course of research, one uncovers an interesting “data point.”   Perhaps that data point goes nowhere; perhaps, when one looks hard at that data point and follows it up carefully, it leads in a whole new direction and changes many things in physics.

It is in this spirit that I present Lab Note 4, “An Interesting Left-Chiral Muliplet Perhaps Indicative of Preonic Structure for Fermions,” which I offer in the spirit of an interesting theoretical “data point” I uncovered back in 1988.  No more, no less.

For some history, I had been auditing some physics courses at nearby Rensselaer Polytechnic Institute (which my daughter Paula is now attending as a Freshman Chemical Engineer), and in the course of my studies there began a dialogue and I later became friends with the tragically-late Professor Nimai Mukhopadhyay, who was a particle physicist. 

At the time, particularly because of the “isospin redundancy” between quarks and leptons — which is my way of saying that both quarks and leptons can exist in both an “isospin up” and an “isospin down” state — I began to realize that there are really two distinct “attributes” which specify the “flavor” of an “elementary” fermion, within each generation.   First: is it a “quark” or a “lepton”?  Second, is its isospin “up” or “down.”  And, this, I began to suspect, was indicative of a preon substructure for the fermions — part of which provided the quark versus lepton aspect of flavor, the other part of which provided the up versus down isospin aspect of flavor. 

Following this thinking, I found that if one were to consider the flavor quantum numbers for only the left-chiral fermions, it turned out that the simple gauge group SU(4) could be used to represent the flavor symmetry of these left-chiral fermions, and that four preons, simply, A, B, C, D so as to avoid any preconceptions at all, could be used in pairs so as to construct the left-chiral fermion flavors.  I wrote this up for the 1988 “Excited Baryons” conference at RPI, and Dr. Mukhopadhyay included the writeup with the conference program.  For your consideration as a “lab note,” i.e., as an interesting piece of research data to keep in one’s mind, I link to a copy of that 1988 writeup below:

Left-Chiral Flavor Muliplet 

At this point in time, 20 years later, it is clear to me that I am not the only person to have thought in this way.  For example, G. Volovik, in his 2003 book “The Universe in a Helium Droplet,” includes an excellent discussion in section 12.2, which I have uploaded to the following link:  

Volovik Excerpt on Quark and Lepton Preonic Structure

Volovik makes a separation very similar to what I was going after in 1988, does so very clearly, and also, nicely handles the right-chiral states which drove me to fits back in 1988.  In fact, this excerpt from Volovik is another very important “lab note,” in and of itself, and I commend it to the reader.   The main problem which I perceive with this excerpt from Volovik, however, is his handling of the spins, which motivates the use of “holons” (spin 0) and “spinons” (spin 1/2) to construct a spin 1/2 fermion out of two preons.  In my view, it would be preferred for each of these preons to have spin 1/2, and when combined, for the resulting particle to also have spin 1/2.  That is, we need to find a way to have 1/2 = 1/2 + 1/2.

How we do this is another story which utilizes the fact that a fermion is a four-component Dirac spinor, and that left- and right-chirality each occupy two of the four components.  Thus, for the left- and right-chiral components of a fermion , each of which has spin 1/2, one can combine those into the whole four-component fermion — which still has spin 1/2.  That is, may be a way to implement 1/2 = 1/2 + 1/2 within the context of Volovik, thereby avoiding the seemingly-artificial (to me) holons and spinons.  But, that is a topic for another lab note, and this approach clearly exploits pre-existing chiral properties of fermions. 

In any event, please take a look at my 1988 publication linked above, look also at the linked Volovik excerpt, think about his and my separation of the fermion flavor attributes into quark/lepton and isospin up/down, think about the inelegance of using holons and spinons (at least if you and I have the same sense of elegance), and think about the chiral properties of fermions.  I do continue to believe the believe that the clearly-established “isospin redundancy” between quarks and leptons is the best evidence we have, of a preonic substructure for fermions which is waiting to be better understood, and cast into a suitable formal pedagogical structure.

That concludes this lab note.