In December 2001, Gary Ford sent me an e-mail message (with copies to Jack Sarfatti and to Neil Boyd) forwarding a question from Kklingon:

I replied, based on my D4-D5-E6-E7-E8 VoDou Physics model, and several messages were exchanged. Thanks to Gary Ford for forwarding to me many messages from Jack Sarfatti that my e-mail server did not receive directly, and for also including some others, such as Saul-Paul Sirag and R. M. Kiehn. Jack Sarfatti asked a lot of provocative questions, such as:

- Interesting calculation Tony, but it needs motivation. What is the physical idea here?
- Can you give some pedagogical discussion on Shilov Boundaries?
- In Jack Sarfatti's theory the "cosmological constant" /\ becomes a locally variable and controllable cosmological field /\(x). Can you get that?
- In post quantum theory, IT
and BIT co-determine each other.
- The "lattice" is "IT", i.e discrete pre-geometry that is the generalization of a Bohm "particle" or "classical field configuration".
- IT is piloted by a BIT Bohm "super potential" (for continuum field theory) and vice versa.

In trying to respond to Jack Sarfatti's questions and comments, my writings evolved into this (sometimes a bit oversimplified for clarity) exposition of my D4-D5-E6-E7-E8 VoDou Physics model.

- Fermion Mass Ratios
- Shilov Boundaries
- Gravity, Higgs, and Gauge Bosons
- Clifford Algebra Structure
- Lagrangian
- String, M, and F
Theories
- Geometric and Physical Interpretations
- Universes, Branes, and Dimension Size
- Gravity at Various Scales

During the e-mail exchanges, R. M. Kiehn noted that "... Sir A. Eddington computed the proton to electron mass ratio to 5 decimal places before WW II ...",

which led me to comment that Eddington (who died in 1944) did not know of quarks or the Standard Model, and was therefore at a disadvantage in trying to build a fundamental physics model.

However, C. W. Kilmister, on page 192 of his book Eddington's Search for a Fundamental Theory (Cambridge 1994), says: "... Eddington was entranced then by what I have come to think of as the 'magic' of the Clifford algebras and he continued to be so. I mean by 'magic' the perplexing way in which the algebra seems to provide a unifying raison d'etre for a diverse collection of results. ...",

which leads me to comment that if Eddington had had the benefit of observations and theory of the Standard Model, those observations and theoretical ideas might have forced him (as they did me) to look beyond the small Clifford algebras of Pauli or Dirac matrices to higher-level Clifford algebras, and to realize that periodicity implies that Cl(8) is a good fundamental building block for a physics model.

After reading the exchange of e-mail messages, I was very pleased to receive from R. M. Kiehn a message saying:

"... Tony ... I have always had the gut feeling that the folks who do quark theory, etc., ... are whistling in the dark unless they can produce the mass ratio of the electron to the proton. I am very impressed that you have determined a way to do this. ...".

Why is a proton 1836 times more massive than an electron?

For details of my calculations for the down (and up) quark constituent mass ratio to the the electron mass, click here. The result is

6 V(S^7 x RP^1) = 2 pi^5 = 612.04

where V(S^7 x RP1) is the volume of the 7-sphere times RP1, where RP1 is real projective 1-space. S^7 x RP1 is the Shilov boundary of a bounded complex domain that is a fermion representation space in the D4-D5-E6-E7-E8 VoDou Physics model. In the D4-D5-E6-E7-E8 VoDou Physics model the proton has 3 valence constituent quarks, so that the proton mass is 3 times the mass of up and down quarks (the up and down quark constituent masses being equal), and the proton/electron mass ratio is calculated to be

3 x 2 pi^5 = 6 pi^5 = 1,836.12

Please do NOT confuse the constituent mass of quarks with the current mass of quarks. Often, in text-books and such, only the current masses are given, which can be quite different from constituent masses. (For example, the current masses of up and down quarks might be stated to be only a few MeV, and not equal to each other.) To see some of my discussions about current vs. constituent masses, click here. Further, the distinction is described in some standard texts, such as for example, Introduction to High Energy Physics, by Donald H. Perkins (4th edition, Cambridge 2000) at pages 24-25.

There are two main ideas used in my model for such calculations:

1 - The representation space for first-generation fermion particles has a corresponding compact geometrical structure that looks like S1 x S7 (which is topologically equivalent to RP1 x S7). Since the electron has only electric charge, it only "fills up" or "uses up" part of that geometrical space. Since the quarks (up and down) have both electric charge and color charge, they "fill up" or "use" more of that space. The mass ratio of up/down quark to electron is the ratio of the volumes "used" or "filled up" by each. As to the neutrino, it has no electric or color charge, so its "tree-level" mass is zero (although effectively it can get a small mass by higher-order processes).

2 - As to second and third generations, their relative masses are determined by combinatorial rules that are based on the idea that

8-dim high-energy spacetime "reduces" at our energies to 4-dim physical spacetime plus 4-dim internal symmetry space.

A first generation fermion * lives only in 4-dim physical space, in that its path begins and ends only in 4-dim physical space,

4-dim internal symmetry space *---* 4-dim physical spacetime

and its path has only one link from beginning 4-dim physical spacetime point to ending 4-dim physical spacetime point the 8 first generation fermion particles correspond to the 8 octonion basis elements as follows:

Octonion Fermion Basis Element Particle 1 e-neutrino i red up quark j green up quark k blue up quark E electron I red down quark J green down quark K blue down quark

Since the path of a first generation fermion particle has only one link, it looks like a single octonion basis element. A second generation fermion * lives partly in 4-dim spacetime, and partly in 4-dim internal symmetry space, in that its path begins in 4-dim physical space and ends in 4-dim internal symmetry space with a connection to our 4-dim physical space, the connection being shown below as the vertical link (or vice versa).

* 4-dim internal symmetry space / | / | * * 4-dim physical spacetime

Since the path of a second generation fermion particle has 2 links, a second generation fermion particle looks like a pair of octonion basis elements. A third generation fermion * lives in 4-dim internal symmetry space, in that its path begins and ends in 4-dim internal symmetry space with connections to our 4-dim physical space, the connection being shown below as the vertical link.

*---* 4-dim internal symmetry space | | | | * * 4-dim physical spacetime

Since the path of a third generation fermion particle has 3 links, a third generation fermion particle looks like a triple of octonion basis elements. For example, the single octonion basis element {i} corresponds to a red up quark and the single octonion basis element {I} corresponds to a red down quark. Since up quarks and down quarks each correspond to one single octonion element, their masses are equal. the 3 pairs {1,J} and {J,1} and {J,J} make up the green strange quark, and the 51 mixed pairs such as {J,k} make up charm quarks. Since there are 51/3 = 17 mixed pairs for each color of charm quark, the charm quark is 17/3 = 5.67 times as massive as the strange quark. the 7 triples {1,K,1} and {K,K,1} and {K,K,K} etc. make up the blue beauty quark and the 483 mixed triples such as {K,j,I} make up truth quarks. Since there are 483/3 = 161 mixed pairs for each color of truth quark, the truth quark is 161/7 = 23 times as massive as the beauty quark.

For a web page with more details, click here.

Again, please note that the quark masses are calculated as constituent masses, NOT as current masses. To see some of my discussions about current vs. constituent masses, click here. Further, the distinction is described in some standard texts, such as for example, Introduction to High Energy Physics, by Donald H. Perkins (4th edition, Cambridge 2000) at pages 24-25.

Also, please note that I prefer the terms beauty and truth for the third generation quarks, instead of the terms bottom and top preferred by Fermilab etc.

Can you give some pedagogical discussion on Shilov boundaries? ...". Shilov boundaries are best understood intuitively by considering a very simple example: the unit disk. Consider the 2-real-dim 1-complex-dim unit disk. Its topological boundary is the 1-real-dim unit circle, and the 1-real-dim unit circle is also its Shilov boundary. When you go to more complicated bounded domains, often the Shilov boundary is only a subset of the topological boundary. The property of the unit circle that you need to generalize to get a Shilov boundary is NOT "topological boundary", but is the minimal "thing" on which, if you know the values of an analytic function f(z) on the "thing", you can find the values of f(z) on ALL the disk by using the Poisson kernel and integrating over the "thing". In other words, the values of an analytic function on the ENTIRE space (unit disk) is completely determined by its values on the Shilov boundary (circle). You can readily see that the Shilov boundary is usually a proper subset of the topological boundary by considering the Cartesian product of two unit disks D1 x D1. Its Shilov boundary is the Cartesian product of two circles S1 x S2, which is clearly a proper subset of the topological boundary. The relevance to physics is that the unit disk, and other bounded complex domains, have natural Laplacian-type operators that operate on them in natural ways. My "big high-energy spacetime" has 8 complex dimensions, represented in compact form by

Spin(10) / Spin(8) x U(1)

where Spin(8) is a local symmetry with 28 generators, which, after reduction to 4-dim spacetime, become 16 which will give a local U(2,2) = Spin(2,4) symmetry (that is where the math similarity to Penrose twistors comes in) that gives conformal-group gravity plus a Higgs mechanism and 12 which will give the gauge bosons of the SU(3)xSU(2)xU(1) standard model. The 8-complex-dimensional "big high-energy spacetime" has an 8-real-dimensional Shilov boundary that looks like S1 x S7. The Shilov boundary S1 x S7 is the geometry of the 8-real-dimensional vector representation of Spin(8). Recall that 28-real-dim Spin(8) is the bivector part of the real Clifford algebra Cl(8), with graded structure

1 8 28 56 70 56 28 8 1

The first 8 corresponds to the vector representation S1 x S7. To get fermions, you have to look at the two half-spinor representations of Spin(8), which each have dimension (1/2) sqrt(1+8+28+56+70+56+28+8+1) = (1/2) sqrt(256) = (1/2) 16 = 8. By Triality (full Triality is unique to Cl(8)), the vector S1 x S7 is isomorphic to each of the 8-real-dim half-spinor reps. Therefore, the first-generation fermion particles are represented by S1 x S7 of one of the half-spinor reps, and the antiparticles by another copy of S1 x S7. That is how the 8-real-dim S1 x S7 comes to represent the neutrino, electron, and red,blue, and green up and down quarks. The 8-real-dim S1 x S7 of "big high-energy spacetime" is reduced (at our low energies) to two 4-dim spaces:

S1 x S3 for 4-real-dim physical spacetime and CP2 for 4-real-dim internal symmetry space (CP2 is sort of like S4, but with structure at infinity that is consistent with the complex projective plane CP2) on which SU(3) and SU(2) and U(1) are represented as consistent with CP2 = SU(3) / SU(2)xU(1).

All this complex math structure is actually necessary and useful (not usless fancy-schmanzy math) because, among other things, it gives

a nice prescription for writing down an 8-dim Lagrangian in "big high-energy spacetime" whose components are:

integral over the S1 x S7 8-real-dim vector spacetime of gauge boson curvature term using covariant derivative and the 28 gauge bosons of Spin(8) plus fermion term using the fermion particle and antiparticles acted on by the Dirac operator coming from the Clifford algebra Cl(8).

EACH COMPONENT OF THE 8-DIM LAGRANGIAN IS UNIQUELY AND CLEARLY DETERMINED BY THE GEOMETRY - NOTHING IS AD HOC.

After dimensional reduction, that 8-dim Lagrangian is transformed, in a clear (but a lot of work - all on my web site and papers) way into a Lagrangian over 4-dim physical spacetime whose components are EXACTLY those of

Einstein-Hilbert-Cartan gravity with torsion and a cosmological constant

and the Standard Model including the Higgs mechanism.

When you use the struture to calculate particle masses and force strength constants,

YOU GET WHAT WE OBSERVE, WITH THE CORRECT MASSES (as to T-quark, I contend my 130 Gev is really correct) and STRENGTHS and NO UN-OBSERVED PARTICLES OF NAIVE 1-1 SUPERSYMMETRY.

WHAT YOU SEE IN THE MODEL IS WHAT YOU SEE IN THE REAL WORLD OF EXPERIMENTS, NO MORE AND NO LESS.

My model does have a subtle triality supersymmetry, based on the structure of the "big high-energy spacetime", in which the supersymmetry is NOT 1-1, but is a relationship between the 28 gauge bosons there, and the 8 + 8 fermion particles and antiparticles there, such that

the gauge boson term in the "big high-energy spacetime" Lagrangian EXACTLY CANCELS the fermion term,

which gives you ultraviolet finiteness of the high-energy 8-dim theory, therefore insuring nice behavior of the model at our low energies with 4-dim spacetime.

......