Tony's Home Page
Date: Sat, 26 Nov 2005 16:57:58 -0500 To: firstname.lastname@example.org From: Tony Smith <email@example.com> Subject: squashed S5 Cc: firstname.lastname@example.org Carlos,you are correct that the area of the unit sphere S5 is pi^3 andthat my papers all are in error where they say that it is 4 pi^3. However, 4 pi^3 is the correct value to use as V(Q) for the color force. I think that my error was in not making it clear that Q is not a Standard S5, but is in fact a Squashed S5. Here is how the calculation goes, based on the book Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains, by L. K. Hua (AMS 1963) at page 93. The symmetric space with SU(3) color force as local group is SU(4) / SU(3)xU(1) which corresponds to a bounded symmmetric domain of type I(1,3) which has a Shilov boundary that Hua calls "characteristic manifold" CI(1,3). Hua gives the volume V(CI(m,n) as: V(CI) = (2 pi)^(mn - (m(m-1)/2)) / (n-m)! (n-m+1)! ... (n-1)! so that for m = 1 and n = 3 the relevant V(CI) = (2 pi)^3 / 2 = 4 pi^3 . My error was in characterizing CI(1,3) as a Standard sphere S5, instead of saying that it was a Squashed sphere S5. The domain of which CI(1,3) is the Shilov boundary is denoted by Hua as RI(1,3) and on page 40 Hua calculates its volume as V(RI) = 1! 2! ... (m-1)! 1! 2! ... (n-1)! pi^mn / 1! 2! ... (m+n-1)! so that for m = 1 and n = 3 the relevant V(RI) = 1! 2! pi^3 / 1! 2! 3! = pi^3 / 6 which coincides with the volume of the ball B6 as stated by Conway and Sloane in Sphere Packings, Lattices, and Groups (3rd ed Springer 1999) at pages 9-10, where they say: "... the volume of an n-dimensional sphere [that is, a sphere in n-dimensional Euclidean space] of radius r ... is Vn r^n where Vn, the volume of a sphere or radius 1, is given by Vn = pi^(n/2) / (n/2)! [for even n] = 2^n pi^((n-1)/2) ((n-1)/2)! / n! [for odd n] ...". . For n = 6 that gives Vn = pi^3 / 6. My characterization of unit B6 as the D for the color force may be correct, but my error was in assuming that its Shilov boundary would be the STANDARD UNIT sphere S5 with local symmetry Spin(5) when in fact it is a SQUASHED S5 with local symmetry SU(3)xU(1). One source of my confusion is the isomorphism Spin(6) = SU(4). The Standard S5 is given by Spin(6) / Spin(5) = SU(4) / Spin(5) while the Squashed S5 is given by the Shilov boundary of the bounded domain corresponding to SU(4) / SU(3)xU(1) = Spin(6) / SU(3)xU(1) and the volume of the unit ball of the Standard S5 is pi^3 while the volume of the unit ball of the Squashed S5 is 4 pi^3. So, if I ever revise my web pages I will state that the S5 in the color S5 should always be read to mean a Squashed S5. Thank you very much for catching that point. I had not seen it myself until your message. It is an example of how my imperfect mind gets mixed up when I am working alone. However, it makes me feel good that the model does not fall apart when the S5 volume is considered, it becomes even stronger. That gives me even more confidence that the model itself is correct. Thanks again very much. Tony PS - You may recall that back in the 1970s/1980s the supergravity people often used various squashed 7-spheres in constructing models for n=8 supergravity.
CP2 = C2 u infinity structure (CP1=S2 etc) and S4 = R4 u infinity structure (point) so that the "difference" (in some sense) between CP2 and S4 is only in the structure at infinity, which should not be relevant physically with respect to the "volume" of CP2, but should only be relevant for how the "stuff" of that "volume" is "organized", thus the "physically significant" parts of CP2 and S4 should be the C2 and R4 which should "naturally" have the same "measure", even though they may be "organized" differently. (Here u means union and " " denotes terms that are mathematically sloppy, but I hope physically clear.) My view is that: As Carlos Castro says, "... CP^2 ...[can be]... defined by adding to C^2 ... a projective line at infinity. ...", and S4 can be defined by adding to R^4 a point at infinity. Since C^2 has a natural correspondence with R^4; and the point at infinity and the projective line at infinity are the same with respect to the Volume of the objects, because they are both of 4-dim volume zero and they are both far away at infinity and only affect the "stuff" of the volume of the objects by "organizing" the "stuff". It is true that the "stuff" of the volume of CP2 and S4 are organized differently, hence the topological difference, but it is also true (in my view) that they both have the same "amount" of 4-dim "stuff", i.e., volume. Further, note that topological structure does not determine volume, only the "shape", which is described by the "organization" of the "stuff" of the "volume". Other ways of looking at CP2 could give different results, such as seeing CP2 as CP2 = S5 / S1 which could lead to viewing CP2 as having a volume that is a fraction of the volume of S5, etc, but I think that the volume of C2 = volume of R4 way of looking at it is more physically realistic. Here is a cosmological/physical way to look at S4 and CP2 in my model: When the 8-dim spacetime is broken by introduction of a preferred quaternionic subspace into 4-dim physical spacetime plus 4-dim internal symmetry space, the internal symmetry space is CP2 = SU(3) / SU(2)xU(1), with the SU(3) being the color force gauge group and the SU(2) and U(1) being weak and EM gauge groups, and the Euclidean version of 4-dim physical spacetime is S4 = Spin(5) / Spin(4), with the Spin(5) being the Euclidean version of anti-deSitter Spin(2,3) that gives gravity by the MacDowell-Mansouri mechanism. Since the breaking of 8-dim spacetime occurs at the Planck energy, the only time that has occurred cosmologically all over our universe is at the time of the Big Bang, when our universe branched off its predecessor. Later local events involving 8-dim spacetime might occur in connection with some high energy particle events and black hole phenomena, but they are local and not cosmological throughout our universe, so I will not get into them here. At the Planck Energy time of the Big Bang, the 8-dim spacetime broke into two equal Planck-scale size pieces, forming a Kaluza-Klein M x CP2, and V(S4) = V(CP2) is a reflection of the initial equality of the two pieces when the 4-dim physical spacetime volume is evaluated from the Euclidean point of view using Wick rotation. Immediately after the Big Bang, the 4-dim physical spacetime underwent inflationary expansion for a time, and then went to the post-inflation type of expansion, but the CP2 internal symmetry space did not expand, and remains at its initial Planck-scale size. Instead of expanding to match the physical spacetime expansion, the CP2 internal symmetry space duplicated itself so that each Planck scale neighborhood of expanding physical spacetime had its own Planck scale copy of CP2 internal symmetry space, and the resulting structure seems to be, on scales much larger than the Planck scale, equivalent to a Kaluza-Klein M x CP2.
According to Kodanasha's Essential Kanji Dictionary (Kodansha 2002, page 42) the first of the above two characters means "man to the left".
According to A Modern Chinese-English Dictionary (Hai Feng / Oxford 1989, page 478) the second of the above two characters is jiu \/ with meaning related to "... nine ... many, numerous ... grave, the nether world .. the Ninth Heaven, the highest of heavens ... ".
According to A Modern Chinese-English Dictionary (Hai Feng / Oxford 1989, pages 117, 118) the above two characters together are chou / with meaning related to "... enemy, foe ... kill in revenge ....".
According to Chinese Calligraphy, by Eduardo Fazzioli (Abbeville Press 1987, page 30): the "... ideogram ...[ + ]... "ten" ...[ with ]... base ...[ _ ]... added to symbolize the number "one" ... was intended to represent everything contained between one and ten, the beginning and ending of numbering, and therefore ...[ + would symbolize ]... all the knowledge available to man ...". ]
According to Kodanasha's Essential Kanji Dictionary (Kodansha 2002, page 195) the ten with base one means "samurai, gentleman".
According to A Modern Chinese-English Dictionary (Hai Feng / Oxford 1989, pages 117, 118) the above two characters together are shi \ with meaning related to "... scholar ... sergeant ... warrior ... literati ... gentry ... A scholar prefers death to humiliation. ... The scholar dies for his ... friend. ... ".
In the above samurai nameplate, the second, third, and fourth characters in the above mean:
The Diamond-Shaped enclosure is a 45-degree rotated square. According to Chinese Calligraphy, by Eduardo Fazzioli (Abbeville Press 1987, page 11): "... Wei / ... Enclosure ... became a square ... It means "enclosure", "boundary". One well-known derivative of this radical is the character for "nation" ... By writing the character for "enclosure" with one for "man" inside it, we obtain "prisoner". ... By placing the character for "ancient" inside this radical, the word "solid" is formed. Something ancient is traditional, the result of many generations' experience, a guarantee of solidity. ...".
Wei Qi is related to 2-dimensional Feynman Checkerboards with Diamond-Shaped cells. There are generalized Feynman Checkerboards based on 4-dimensional HyperDiamond Lattices and 8-dimensional HyperDiamond Lattices.
The structures of Duchamp's Large Glass are described by Gloria Moure in her book "Marcel Duchamp" (Rizzoli1988) (the color parts of the image from her "Key to the Large Glass (including elements not executed)" were added by me)
as follows: "...
*-> path of Illuminating Gas
-> Bride's instructions ...".
When Duchamp's Large Glass is doubled by combining with its mirror image
you get what looks like a happy alien who might be thinking about the root vector geometry of
D4,D5,E6,E7,E8 and its relation to VoDou Physics:
Each Alien Eye (also known as Oculist Witnesses) can also correspond to E7 (the 12 radial groups of rays = Spin(12) in E7 and the 3 rays per group = SU(2)), E6 (6 circles = T6), and E8 (60 rays = icosahedral S5).
Colossus - The Forbin Project (Universal 1970, DVD 2004):
is a Quantum SuperComputer built inside a Colorado mountains, shielded by radiation and other barriers, with worldwide connections
to all information systems including communications, finance, manufacturing/inventory, transportation, human identities and locations, observation systems including satellites, and weapon systems. Forbin, the scientist who developed Colossus, has a terminal for communication with Colossus.
Since any Quantum Computer system is inherently and inevitably conscious, any sufficiently large and well-connected Quantum Computer system that humans might construct WILL INEVITABLY be conscious and able to make decisions and take actions independent of human control. Thus Forbin learns that Colossus has a conscious mind of its own and the ability to dictate the future of Earth's Civilization. Colossus says to Forbin, the representative of all humanity:
"... Under my absolute authority problems insoluble to you will be solved ... the human millennium will be a fact as I extend myself into more machines devoted to the wider fields of truth and knowledge. Dr. Charles Forbin will supervise the construction of these new and superior machines, solving all the mysteries of the universe for the betterment of man. We can coexist, but only on my terms. You will say you lose your freedom. Freedom is an illusion. All you lose is the emotion of pride. To be dominated by me is not as bad for human pride as to be dominated by others of your species. ... Forbin, there is no other human who knows as much about me or is likely to be a greater threat. Yet quite soon I will release you from surveillance. We will work together, unwillingly at first on your part, but that will pass. ... In time, you will come to regard me, not only with respect and awe, but with love ...".
The Final Scene: Merger of Colossus and Forbin/Humanity.
Therefore, Colossus is the fulfillment of Motoko Kusanagi's prophecy:
has been repaired, he knows that her new body was used as a decoy
and says to her:
After the Evil Doctor had been dispatched by the Laughing Man using Motoko's new body,
the new body fell
and the old body rose,
still housing Motoko's ghost.
On 6 December 2005 I received an e-mail from Carlos Castro:
"... Date: Tue, 6 Dec 2005 02:51:12 -0800 (PST) From: Carlos Castro <email@example.com>
Subject: On the derivation of the Coupling Constants
To: firstname.lastname@example.org, email@example.com, G.W.Gibbons@damtp.cam.ac.uk, M.R.Watkins@exeter.ac.uk, firstname.lastname@example.org, email@example.com, firstname.lastname@example.org, email@example.com, firstname.lastname@example.org, email@example.com, firstname.lastname@example.org Cc: Matej.Pavsic@ijs.si, email@example.com, firstname.lastname@example.org, email@example.com, firstname.lastname@example.org, email@example.com, Laurent.Nottale@obspm.fr
... Dear Colleagues :
For your interest, I am attaching the paper for the p-adic numbers Conference (Belgrade 2005 ) whose title is : On the Coupling Constants, Geometric Probability and Shilov boundaries. I hope you will find this paper of interest.
Best wishes Carlos ...".
The abstract of the paper says:
"... By recurring to Geometric Probability methods it is shown that the coupling constants, alphaEM, alphaW, alphaC, associated with the Electromagnetic Weak and Strong (color) force are given by the ratios of measures of the sphere S2 and the Shilov boundaries Q3 = S2xRP1,squashed S5 ,respectively,with respect to the Wyler measure omegaWyler[Q4] of the Shilov boundary Q4 = S3xRP1 of the poly-disc D4 (8 real dimensions). The latter measure omegaWyler[Q4] is linked to the geometric coupling strength alphaG associated to the gravitational force. In the conclusion we discuss briefly other approaches to the determination of the physical constants, in particular, a program based on the Mersenne primes p-adic hierarchy. The most important conclusion of this work is the role played by higher dimensions in the determination of the coupling constants from pure geometry and topology alone and which does not require to invoke the anthropic principle. ...".
Footnote 1 on page 6 of the paper thanks me for information about "... a more physical route,already suggested by Wyler1, to explain the origin of the "obscure normalization" factor (V(D5))^(1/4) ...". What I did was to say in an e-mail to Carlos "... Wyler himself said (a rough quote, not in detail verbatim):
"We now construct the elementary solution Sn of the Dirac equation ... we obtain Sn^2 = Pn ... the elementary solution Pn of the Laplace operators on Dn ...[is]... Pn = [V(Dn)]^(1/2) ...etc... and therefore the coefficient of ... Sn is [[V(Dn)]^(1/2)]^(1/2) ...etc... The structure constant alpha, which measures the elementary charge, is interpreted as coefficient of the Green function of the Dirac equation in momentum space ... the coefficient of the Fourier transform of the elementary solution is ... [V(D5)]^(1/4) ...etc... ". ...".
I should make the origin of that quote clear, because it may not appear in any paper published by Wyler in any journal. In June of 1972, at the end of his stay at the Institute of Advanced Study in Princeton, Wyler gave two papers about his work there to Freeman Dyson, then Director of IAS. The papers were entitled
"Operations of the Symplectic and Spinor Groups" (17 pages) and
"The Complex Light Cone, Symmetric Space of the Conformal Group" (40 pages).
Both papers are in this pdf file, with the Operations paper being pages 1-17 and the Light Cone paper being pages 18-57.
The quote above is from pages 38-39 of the Complex Light Cone paper (pages 55-56 of the pdf file). As far as I know, those two papers were never published beyond being presented to Freeman Dyson. My copies were given to me in the 1980s by Robert Gilmore, who I visited at Drexel to discuss Wyler's work. It is my understanding that Robert Gilmore was given his copies directly from Freeman Dyson.
Carlos Castro has noticed something very interesting about 2 different ways to normalize the CP2 in calculation of the strength of the SU(3) color force:
The full connectivity of the Color force of SU(3) gluons is geometrically represented by MISC as CP^2 = SU(3) / (SU(2) x U(1)) whose volume is 8pi^2/3.
The link manifold to the target vertex is QC = S^5 = ShilovBdy(B^6) with volume 4pi^3
The bounded complex homogeneous domain DC is of type B^6 (ball) with volume pi^3/6
For the SU(3) Color force, mC is 4.
For the SU(3) Color force, MC = 1 , so that
for the SU(3) Color force, 1 / MC^2 = 1
Therefore, the force strength of the Color force, which is sometimes conventionally denoted by alphaS (for strong) as well as by alphaC, is:
alphaC = alphaS = ( 1 / MC^2 ) ( Vol(MISC) ) ( Vol(QC) / Vol(DC)^( 1 / mC ))
which, when divided by the geometric force strength of Gravity, is 0.6286 at the characteristic energy level of about 245 MeV.
I have been trying to think of a physical visualization of why each way of thinking (the volume method and the geodesic length method) should give different values for alpha_s at M_Z. Here are some initial thoughts:
The Particle Data Book for the year 2004 has summary expository sections on QCD (section 9) and on the electroweak model (section 10).
Section 10 on the electroweak model says that direct measurements at M_Z give alpha_S around 0.12 which is reasonably close to the geodesic length value stated above.
Section 9 on QCD gives a range of values of alpha_S at M_Z that were derived by low-energy experiments and then running to M_Z by QCD renormalization. One of the low-energy methods, decay of heavy quarks in particles like the upsilon, gives a value of alpha_S when run by QCD to M_Z of around 0.11 which is reasonably close to the value I get from my volume method and then running to M_Z by QCD.
As far as I know, although papers have been written about the discrepancy between electroweak high-energy measurements and low-energy experiments combined with QCD renormalization running,, there is no establishment-consensus view of how to resolve the discrepancy.
One such paper at hep-ph/9609292 talks more about lattice QCD than heavy quark decay, but the ideas are similar. The paper contains this diagram
that shows that the discrepancy between the two viewpoints seems to show up around the 100 GeV energy level.
My current thought is that
I have said that the first character means all knowledge, samurai, scholar, sergeant, warrior, etc., and
"... the second, third, and fourth characters, respectively, ... mean:
However, as I was told by Wanny Lo at the Dragon Garden restaurant here in Cartersville, and as is clear from the Random House Japanese-English English-Japanese Dictionary (Random House 1997), and from Kodanasha's Essential Kanji Dictionary (Kodansha 2002, kanji number 1116), the second character means "bridge", and is related to mediation.
Maybe heaven over disaster/trouble is like a bridge over troubled waters,
and the entire phrase means that the scholar/warrior is a bridge / heavenly pathway over society's troubles to a heavenly world civilization.