Lab Notes for a Scientific Revolution (Physics)

April 22, 2012

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

May 7, 2009

Inferring Electrodynamic Gauge Theory from General Coordinate Invariance

I am presently working on a paper to show how electrodynamic gauge theory can be directly connected to generally-covariant gravitational theory.  In essence, we show how there is a naturally occurring gauge parameter in gravitational gemometrodynamics which can be directly connected with the gauge parameter used in electrodynamics, while at the same time local gauge transformations acting on fermion wavefunctions may be synonymously described as general coordinate transformations acting on those same fermion wavefunctions.

This is linked below, and I will link updates as they are developed.

Inferring Electrodynamic Gauge Theory from General Coordinate Invariance

If you check out sci.physics.foundations and sci.physics.research, you will see the rather busy path which I have taken over the last month to go from baryons and confinement to studying the Heisenberg equation of motion and Ehrenfest’s theorem, to realizing that there was an issue of interest in the way that Fourier kernels behave under general coordinate transformations given that a general coordinate x^u is not itself a generally-covariant four vector.  Each step was a “drilling down” to get at underlying foundational issues, and this paper arrives at the most basic, fundamental underlying level.

Looking forward to your feedback.

Jay

June 30, 2008

Foldy-Wouthuysen, continued

Just for the heck of it, I did a calculation of what happens to the mass matrix M\equiv \beta m 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

June 29, 2008

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:

v^{k} =\alpha ^{k} (1)

where \alpha ^{k} = \gamma ^{0} \gamma^{k} , 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:

v^{k} =dx^{k} /dt   (2)

where x^{k}  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 \pm c 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.

June 19, 2008

A New Lab Note: Commutation of Linear Rest Mass with Canonical Position

It has been awhile since my last blog entry, but if you want to check out some my recent wanderings through physicsland, check out sci.physics.foundations, relativity, and research.

Here, I would like to show a rather simple calculation, which may cast a different light on how one needs to think about the canonical commutation relationship \left[x_{j} ,p_{k} \right]=i\eta _{jk} ;\; j,k=1,2,3.  I would very much like your comments in helping me sort this through.  You may download this in pdf form at https://jayryablon.files.wordpress.com/2008/06/linear-mass-commutator-calculation.pdf.

I.  A Known Square Mass Commutation Calculation

 Consider a particle of mass m as a single particle system.  Consider canonical coordinates x_{\mu } , and that at least the space coordinates x_{j} ;\; j=1,2,3 are operators.  If we require that the mass m must commute with all operators, then we must have \left[x_{\mu } ,m\right]=0, and by easy extension, \left[x_{\mu } ,m^{2} \right]=0.  It is well known that the commutation condition \left[x_{\mu } ,m^{2} \right]=0, taken together with the on-shell mass relationshipm^{2} =p^{\sigma } p_{\sigma } and the single-particle canonical commutation relationship \left[x_{j} ,p_{k} \right]=i\eta _{jk} ;\; j,k=1,2,3, where diag\left(\eta _{\mu \nu } \right)=\left(-1,+1,+1,+1\right) is the Minkowski tensor, leads inexorably to the commutation relationship:

\left[x_{k} ,p_{0} \right]=-ip_{k} /p^{0} =-iv_{k}    (1.1)

where v_{k} is the particle velocity (in c=1 units) along the kth coordinate.  I leave the detailed calculation as an exercise for the reader not familiar with this calculation, and refer also to the sci.physics.research thread at http://www.physicsforums.com/archive/index.php/t-142092.html or http://groups.google.com/group/sci.physics.research/browse_frm/thread/d78cbfecf703ff6a.

 I would ask for your comments on the following calculation, which is totally analogous to the calculation that leads to (1.1), but which is done using the linear mass m rather than the square mass m^{2} , and using the Dirac equation written as m\psi =\gamma ^{\nu } p_{\nu } \psi , in lieu of what is, in essence, the Klein Gordon equation m^{2} \phi =p^{\sigma } p_{\sigma } \phi that leads to (1.1).

2.  Maybe New?? Linear Mass Commutation Calculation

 Start with Dirac’s equation written as:

m\psi =\gamma ^{\nu } p_{\nu } \psi .  (2.1) 

Require that:

\left[x_{\mu } ,m\right]=0   (2.2)

 Continue to use the canonical commutator \left[x_{j} ,p_{k} \right]=ig_{jk} .  Multiply (2.1) from the left by x_{\mu } noting that \left[\gamma ^{\nu } ,x_{\mu } \right]=0 to write:

x_{\mu } m\psi =\gamma ^{\nu } x_{\mu } p_{\nu } \psi =\gamma ^{0} x_{\mu } p_{0} \psi +\gamma ^{j} x_{\mu } p_{j} \psi .  (2.3) 

This separates into:

 \left\{\begin{array}{c} {x_{0} m\psi =\gamma ^{0} x_{0} p_{0} \psi +\gamma ^{j} x_{0} p_{j} \psi } \\ {x_{k} m\psi =\gamma ^{0} x_{k} p_{0} \psi +\gamma ^{j} x_{k} p_{j} \psi } \end{array}\right. .  (2.4)

  Now, use the canonical relation \left[x_{j} ,p_{k} \right]=i\eta _{jk} to commute the space (k) equation, thus:

 \begin{array}{l} {x_{k} m\psi =\gamma ^{0} x_{k} p_{0} \psi +\gamma ^{j} x_{k} p_{j} \psi =\gamma ^{0} x_{k} p_{0} \psi +\gamma ^{j} \left(p_{j} x_{k} +i\eta _{jk} \right)\, \psi } \\ {=\gamma ^{0} x_{k} p_{0} \psi +\gamma ^{j} p_{j} x_{k} \psi +i\gamma _{k} \psi } \\ {=\gamma ^{0} x_{k} p_{0} \psi +mx_{k} \psi -\gamma ^{0} p_{0} x_{k} \psi +i\gamma _{k} \psi } \end{array} .  (2.5)

In the final line, we use Dirac’s equation written as mx_{\mu } \psi =\gamma ^{\nu } p_{\nu } x_{\mu } \psi =\gamma ^{0} p_{0} x_{\mu } \psi +\gamma ^{j} p_{j} x_{\mu } \psi , and specifically, the \mu =k component equation \gamma ^{j} p_{j} x_{k} \psi =mx_{k} \psi -\gamma ^{0} p_{0} x_{k} \psi .

 If we require that \left[x_{\mu } ,m\right]=0, which is (2.2), then (2.5) reduces easily to:

 \gamma ^{0} \left[x_{k} ,p_{0} \right]\psi =-i\gamma _{k} \psi ,  (2.6)

Finally, multiply from the left by \gamma ^{0} , and employ \gamma ^{0} \gamma _{k} \equiv \alpha _{k} and \gamma ^{0} \gamma ^{0} =1 to write:

\left[x_{k} ,p_{0} \right]\, \psi =-i\alpha _{k} \psi .  (2.7) 

If we contrast (2.7) to (1.1) written as \left[x_{k} ,p_{0} \right]\phi =-iv_{k} \phi , we see that the velocity p_{k} /p^{0} =v_{k} has been replaced by the Dirac operator \alpha _{k} , that is, v_{k} \to \alpha _{k} .

3.  Questions

 Here are my first set of questions:

 1)  Is the calculation leading to (2.7) correct, and is (2.7) a correct result, or have I missed something along the way?

2)  If (2.7) is correct, has anyone seen this result before?  If so where?

3)  Now use the plane wave \psi =ue^{ip^{\sigma } x_{\sigma } } so that we can work with the Dirac spinors u\left(p^{\mu } \right), and rewrite (2.7) as:

\left\{\begin{array}{c} {\left(\alpha _{k} -\lambda \right)\, u=0} \\ {\lambda =i\left[x_{k} ,p_{0} \right]} \end{array}\right.  

The upper member of (3.1) is an eigenvalue equation.  Reading out this equation, I would say that the commutators \lambda =i\left[x_{k} ,p_{0} \right] are the eigenvalues of the Dirac \alpha _{k} matrices, which are:

{\bf \alpha }=\left(\begin{array}{cc} {0} & {{\bf \sigma }} \\ {{\bf \sigma }} & {0} \end{array}\right) and {\bf \alpha }=\left(\begin{array}{cc} {-{\bf \sigma }} & {0} \\ {0} & {{\bf \sigma }} \end{array}\right) ,  (3.2)

in the respective Pauli/Dirac and Weyl representations, and that the u are the eigenvectors associated with these eigenvalues \lambda =i\left[x_{k} ,p_{0} \right].  Am I wrong?  If not, how would one interpret this result?  Maybe the commutators \left[x_{j} ,p_{k} \right]=i\eta _{jk} can be discussed in the abstract, but it seems to me that the commutators \lambda =i\left[x_{k} ,p_{0} \right] can only be discussed as the eigenvalues of the matrices \alpha _{k} with respect to the eigenstate vectors u.  This, it seems, would put canonical commutation into a somewhat different perspective than is usual.

Just as Dirac’s equation reveals some features that cannot be seen strictly from the Klein Gordon equation, the calculation here seems to reveal some features about the canonical commutators that the usual calculation based on \left[x_{\mu } ,m^{2} \right]=0 and m^{2} =p^{\sigma } p_{\sigma } cannot, by itself, reveal.

I’d appreciate your thoughts on this, before I proceed downstream from here.

Thanks,

Jay.

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