With respect to Cl(2,6), you say "... now I have a way to connect my work with yours ...". The way I see that connection is described on my web page at ... clfpq2.html#physicspaths where I say that "... dimensional reduction changes Real M(16,R) of Cl(0,8), Cl(1,7), and Cl(4,4) to the Quaternionic M(8,Q) of Cl(2,6) or Cl(3,5) ...". In other words, in my view, at high energies where spacetime is really 8-dimensional the relevant Clifford algebra is Cl(1,7) = Cl(0,8) = the 16x16 real matrix algebra which has real/octonionic structure but when you to lower energies (where our experiments are done) the cooling forces a quaternionic structure to freeze out (or, in other language, real/octonionic symmetry is broken by choosing a particular quaternionic substructure) and the result is that at our lower energies things look like Cl(2,6) = the 8x8 quaternion matrix algebra, which has a Cl(2,4)= the 4x4 quaternion matrix algebra = = the conformal Lie algebra for gravity + Higgs and if you factor the 15-dim Spin(2,4) conformal Lie algebra out of the 28-dim Spin(2,6) Lie algebra of high-energy gauge bosons you see that the remaining 13 generators correspond to: 1 is a U(1) to be added to Spin(2,4) to get U(2,2) = U(1)xSU(2,2), and physically represents propagator complex phases; 1 is an electromagnetic U(1); 3 are weak SU(2); and 8 are color SU(3). The way those generators form those groups is based on the Weyl group root vector symmetries, and not as conventional group/subgroup structures. One reason that conventional physicists/mathematicians have a hard time understanding what I have done is that such Weyl/root vector stuff is not found in existing textbooks or papers. A description is on my web page at ... Sets2Quarks4a.html#WEYLdimredGB I should also note that the quaternionic structure of Cl(2,4) is inherited by its subalgebra Cl(1,3) = 2x2 quaternionic matrices which is consistent with the physical interpretation of fermions in terms of quaternionic half-spinors and with John Baez's statement "... fermions are quaternionic and bosons are real ..." on which I have elaborated on my web page at ... clfpq2.html#13vs31 Also, your comment that: "... Machian dream ...express[ing] the maximal force = F = m( Planck ) c^2 / L(Planck) as F = M ( Universe ) c^2 / R ( Hubble ) then we get one of the Dirac-Eddington's large number coincidences : M ( Universe ) / m ( Planck ) = R ( Hubble ) / L ( Planck ) = of the order of 10^{ 61 } . Since the Planck mass = 10^{ 19 } m ( proton ) you get the famous M ( Universe ) = 10^{ 80} m ( proton ) which is the Dirac-Eddington number. ..." is somewhat related to some comments that I put on my web page at ... LARGEsmall.html

with respect to infinite dimensional Clifford algebras, my view is to consider the conventional Hyperfinite II1 factor which is roughly an infinite-dimensional version of the spinor representation of Complex Clifford algebras, which have periodicity 2 and so are like an infinite limit of what John Baez calls "... the fermionic Fock space over C^(2n) ..." and then generalize it to the case of Real Clifford Algebras with periodicity 8 so that what you get is an infinite limit of a tensor product of a lot of copies of 256-dim Cl(8), as I have described on my web pages, including ... II1vNfactor.html Each Cl(8) would describe physics locally in the neighborhood of a given spacetime point, as described in my physics model which is summarized in papers and web pages such as ... 2002SESAPS.html All the Cl(8) things in the generalized Hyperfinite II1 factor (roughly an infinite tensor product) would be linked together to form (at the next higher energy level above our quaternionic 4-dim physical spacetime plus 4-dim CP2 internal symmetry space) a higher-energy real/octonionic 8-dim spacetime as described on my web pages, including ... ClifTensorGeom.html When you take quantum superpositions in the many-worlds quantum theory, you get things such as quantum loops/graphs of higher and higher order, whose description involves the prime numbers as described on my web pages, including ... Rzetazeta.html#qsohprime and which may be closely related to the p-adic geometry that you use, Matti. However, I have a hard time visualizing p-adic stuff, so I have not written much about it. I had been trying to read about it and understand it recently, but I have become depressed and discouraged about doing further work. I should note that my model also gives two significant sets of calculations not given on that web page. They are: 1 - WMAP ratio calculation at ... coscongraviton.html and 2 - neutrino mass and mixing angle calculations at ... snucalc.html#asno

... I see it as: The 256-dimensional Cl ( 8 ) has the graded structure : 1 + 8 + 28 + 56 + 70 + 56 + 28 + 8 + 1 the 1-dim ( Higgs ) , the 8-dim ( spacetime that breaks to 4-dim physical spacetime plus CP2 ) and the 28-dim ( gravity plus gauge bosons ) and it also has spinor structure in addition to graded structure. At high energies before dimensional reduction the Clifford algebra is Cl(0,8) = Cl(1,7) = 16x16 real matrices and the full spinor space is 8-dim which reduces to 8-dim half-spinor space for 8 fundamental fermion particles plus a mirror image 8-dim half-spinor space for 8 fundamental antiparticles. After dimensional reduction the Clifford algebra is Cl(2,6) = 8x8 quaternionic matrices and the full spinor space is 8x1 quaternionic left ideal which reduces to a half-spinor 4x1 quaternions which are 16-dim and so correspond to 8 Dirac fermion particles with two helicities (the neutrino right helicity is suppressed due to inherited structure from real/octonionic Cl(1,7)) plus a mirror image half-spinor 4x1 quaternions which are 16-dim and so correspond to 8 Dirac fermion antiparticles with two helicities (the antineutrino left elicity is suppressed due to inherited structure from real/octonionic Cl(1,7)). This structure is consistent with the quaternionic structure of conformal Cl(2,4) = 4x4 quaternionic matrices and with the quaternionic structure of Cl(1,3) = 2x2 quaternionic matrices, so the 4-dim physical spacetime has signature (1,3) and not signature (3,1) of Cl(3,1) = 4x4 real matrices. -------------------------------------------- ...[all Cl(8) multivectors]... were not completely encoded [by spinors] (and neither were grades 0 (Higgs) 1 (spacetime plus CP2) or 2 (gauge bosons) ) because the spinor fermions are only the left ideals, sort of the square root of the 16x16 = 256, which leaves the right ideals free to represent the gammas (this was the basic insight of David Hestenes with respect to 4-dim Clifford algebra in his spacetime algebra program). Of course, the gammas appear in the structure of the Lagrangian.

From my point of view, there are only 3 generations, as follows: The 16 8-dim spinors correspond to 8 first-generation fermion particles and 8 first-generation fermion antiparticles and they propagate in 8-dim spacetime, where there is only one generation (the first). The 8-dim propagation looks like this: origin * ------ > destination lying in 8-dim spacetime. However, after dimensional reduction, the 8-dim spacetime is no longer one layer ---------------------------------------------- but splits into two layers: ------------------------------------ 4-dim Internal Symmetry Space ------------------------------------ 4-dim physical spacetime and If the spinor propagator origin * and destination > both lie in the 4-dim physical spacetime, then the low-dim propagator looks like this: origin * ------ > destination lying in 4-dim physical spacetime and the spinor fermion particle (or antiparticle) looks like a 4-dim first-generation fermion, and is represented by a single octonion basis element corresponding to the origin * . However, if the origin * lies in the 4-dim Internal Symmetry Space then the low-dim propagator looks like this: -- origin * ------------------- 4-dim Internal Symmetry Space | | -- virtual origin (*) ------ > destination -- 4-dim physical spacetime and the spinor fermion particle (or antiparticle) looks like a 4-dim second-generation fermion, and is represented by a pair of octonion basis elements corresponding to the origin * plus the virtual origin (*) . and if the > destination lies in the 4-dim Internal Symmetry Space then the low-dim propagator looks like this: --------- (>*) intermediate virtual orgin/destination -- 4-dim Internal / \ Symmetry Space / \ -- origin * > destination -------------------------- 4-dim physical spacetime and the spinor fermion particle (or antiparticle) looks like a 4-dim second-generation fermion, and is represented by a pair of octonion basis elements corresponding to the origin * plus the intermediate virtual orgin/destination (>*) . The only other possible case is that both the origin and the > destination lie in the 4-dim Internal Symmetry Space in which case the low-dim propagator looks like this: ----------- origin * (>*) intermediate virtual orgin/destination -- 4-dim | / \ Internal | / \ Symmetry Space -- virtual origin (*) > destination -------------------------- 4-dim physical spacetime and the spinor fermion particle (or antiparticle) looks like a 4-dim third-generation fermion, and is represented by a triple of octonion basis elements corresponding to the origin * plus the virtual origin (*) plus the intermediate virtual orgin/destination (>*) . What this means is that physically there can only be 3 generations, and the first generation is represented by octonion basis elements; the second generation is represented by pairs of octonion basis elements; and the third generation is represented by triples of octonion basis elements. The single - pair - triple relationship gives combinatorial relationships among the masses of the various generations of fermions, which produces constituent masses that are roughly consistent with experiment, which is a reason that I think that my point of view is correct. It is (as far as I know) the only point of view that gives roughly equal constituent masses in the first generation quarks, but in the second generation charm is somewhat heavier than strange and in the third generation truth is a lot heavier than beauty, and using these calculated masses lets me calculate realistic Kobayashi-Maskawa parameters, also something that nobody else has done AFAIK. All this is in my web pages and even in my papers on arXiv (they are probably there because they were put up prior to arXiv blacklisting me or during the brief interval that arXiv allowed its endorsement system to work without interference from the blacklisting moderators). Also, there is experimental evidence that there are only 3 generations, such as the Z width experiments. According to a Particle Data Group Particle Properties review http://pdg.lbl.gov/ (quote from the August 2001 revision by D. Karlen): "... The most precise measurements of the number of light neutrino types, Nv, come from studies of Z production in e+e- collisions. The invisible partial width ... is determined by subtracting the measured visible partial widths, corresponding to Z decays into quarks and charged leptons, from the total Z width. The invisible width is assumed to be due to Nv light neutrino species each contributing the neutrino partial width ... as given by the Standard Model. ... The combined result from the four LEP experiments is Nv = 2.984 +/- 0.008 ...".

What I am doing is taking the Cl(8) Clifford algebra and making a Lagrangian by using different parts of Cl(8) in different ways: grade 0 is a Higgs term; grade 1 is a vector spacetime term (the manifold over which the Lagrangian density is integrated); grade 2 is a bivector gauge boson term (like a YM term); +half-spinors are fermion particles; and -half-spinors are fermion antiparticles. As I understand it, you want to ask WHY those parts should play those roles, and in order to do that you want to start with a supergravity in a Clifford superspace and see whether its various terms fall into the places where I put them. A way to look at superspace in terms of Clifford algebras is described in a paper by Doran, Lasenby and Gull at http://www.mrao.cam.ac.uk/~clifford/publications/abstracts/grass_mech.html in which they say (their term for "Clifford algebra" is "geometric algebra"): "... Given a set of n Grassmann generators ... we can map these into geometric algebra by introducing a set of n independent vectors ... and replacing the product of Grassmann variables by the exterior product ... In this way any combination of Grassmann variables can be replaced by a multivector ... ... Berezin calculus ... can be handled entirely within the algebra ... by introducing the reciprocal frame ...[and]... Integration is defined to be equivalent to right differentiation ... ... Thus we see that Grassmann calculus amounts to no more than Clifford contraction and the results of "Grassmann analysis" ...can all be expressed as simple algebraic identities for multivectors ... Furthermore these results are now given a firm geometric significance through the identification of Clifford elements with directed line, plane segments etc ...". I guess you want to apply the Doran-Lasenby-Gull viewpoint to high-dimensional supergravity. There is a paper cached in html form at http://64.233.161.104/search?q=cache:U7Cy3y87nb4J:www.cgtp.duke.edu/~drm/PCMI2001/notes.pdf+supergravity+geometric+algebra&hl=en&ie=UTF-8 by Savdeep Sethi that says: "... The aim of this lecture is to overview the general structure and features of supergravity ... Why 11 dimensions? ... ... restrict to D = 11 ... This representation, as in all these cases, must have an equal number of bosons and fermions. Each application ... changes a boson to a fermion, and vice-versa. In this case, we get 128 bosons and 128 fermions. ... ... a gravitino ... is often called a spin 3/2 particle (a vanilla spinor is typically called a spin 1/2 particle). There are no fermions beyond the gravitino in 11 dimensions ... ... For D > 11, if you impose SUSY, and follow the same construction, you will find higher spin fields. Massless fields with spin greater than 2 cannot (in a way that we currently understand) consistently be coupled to gravity. There is a strong belief that such theories do not make sense. In this respect, 11-dimensional supergravity is quite unique. It is the minimal and maximal supergravity possible in 11 dimensions, and we believe that it gives a good description of M theory on large smooth spaces ...". Therefore, it seems to me that if you describe 11-dim supergravity in Clifford terms, you will NOT end up with my model, because my model does NOT have spin 3/2 gravitinos and my model does NOT have a 1-1 supersymmetric correspondence of 128 bosons with 128 fermions. What I have is a more subtle supersymmetry in the 8-dim structure, whereby the 28 gauge bosons are the 8/\8 = 28 bivectors and the there are 8 +half-spinor fermion particles and 8 -half-spinor fermion antiparticles. Due to triality, the vector 8 = the +half-spinor 8 = the -half-spinor 8 so that you can identify the 28 bosons with the /\ product of the 8 fermions with the 8 antifermions. If you look carefully at the spinor fermion term in the 8-dim Lagrangian, you see that the dimensionality of each half-spinor factor is 7/2 (compare the dimensionality of 3/2 in conventional 4-dim Lagrangians). Therefore, since the dimensionality of a gauge boson is 1, the total dimensionality of the bosons = 28 x 1 = 28 which is exactly balanced by the total dimensionality of the fermions = 8 x 7/2 = 28, and you should get nice cancellations in my model at the 8-dim level. In other words, my subtle supersymmetry does the nice things of 11-dim supergravity but does not have its unrealistic particle content of 128 bosons and 128 fermions including spin 3/2 gravitinos.

Here is how my model contains parts of the graded structure of Cl(8), 1 8 28 56 35+35 56 28 8 1 where I have written the middle 70 as 35+35 because it is self-dual under Hodge duality. By Hodge duality, the 1 8 28 56 35 is dual to the 35 56 28 8 1 As I have discussed on my web site at ... clfpq2.html#clifstructure Dennis Marks has shown that the 1 8 28 56 35 correspond to physical stuff in the coordinate representation while the 35 56 28 8 1 correspond to physical stuff in the momentum representation. As he says "... Complementarity between space-time and momentum-energy is achieved by bit inversion, which interconverts between position representation and momentum representation ...". That means that the physical interpretation of Cl(8) graded stuff is: coordinate rep of Higgs scalar 1 1 momentum rep of Higgs scalar coordinate rep of spacetime 8 8 momentum rep of spacetime coordinate rep of gauge bosons 28 28 momentum rep of gauge bosons coordinate rep of trivectors 56 56 momentum rep of trivectors coordinate rep of 4-vectors 35 35 momentum rep of 4-vectors In my Lagrangian, I usually do not write out the trivector and 4-vector terms, because they are not physically effective at low energies after dimensional reduction. However, here is what they are (in my model) at high energies: In the full 8-dimensional theory, the 56 trivectors are related to the structure of 1+3=4-dimensional subspaces of 1+7=8-dimensional spacetime that are connected with the E8 HyperDiamond lattice links that are (normalized) sums of 4 of the basis octonions. To reduce the dimension of spacetime to 1+3=4 dimensions, an associative 3-form is used to pick a particular quaternionic substructure of the 8-dim octonionic structure. This effectively fixes a particular trivector, so the 56 trivectors do not play a dynamical role in my model at low energies after dimensional reduction. Here and now, we do not have the technology to do experiments that could test the structure of the full 56-dimensional trivector sector. The 70 4-vectors in the 4-grade subspace are reducible to two sets of 35 4-vectors each. They are the symmetric parts of 8x8 = 8/\8 + 8sym8 = 28 + 36 after taking out the 1-dimensional scalar 0-vector Higgs scalar. The 1-dimensional scalar 0-vector representing the Higgs scalar can be thought of as the trace of the full symmetric 1+35=36-dimensional space of symmetric 8x8 real matrices. The 35 4-vectors are the traceless symmetric 8x8 matrices. They are related to the coassociative 4-form that is fixed in the dimensional reduction process to determine the internal symmetry space. At our low energy levels, below the Planck-scale at which dimensional reduction occurs in the D4-D5-E6 model, the 35 4-vectors do not play a dynamical role that we can test experimentally here and now. However, they show that the Higgs mechanism is related fundamentally to BOTH the particles and fields of the internal symmetry space and the spacetime of conformal MacDowell-Mansouri gravity. Therefore, my model does use all the graded parts of Cl(8), and also the spinor structure of Cl(8), but the 56 and 35 parts are not physically effective at low energies after dimensional reduction, so usually I don't write them explicitly in my Lagrangian, which I usually use to calculate force strengths, particle masses, etc in the energy region where we do experiments now.

it might be useful for me to say how I see that spinors are related to the graded structure of 256-dim Cl(8), based on the material on my web site including ... 8idempotents.html As Pertti Lounesto says in his book Spinor Valued Regular Functions in Hypercomplex Analysis (Report-HTKK-MAT-A154 (1979) Helsinki University of Technology) (in this quote I have changed his notation for a Clifford algebra from R_(p,q) to Cl(p,q)) at pages 40-42: "... To fix a minimal left ideal V of Cl(p,q) we can choose a primitive idempotent f of Cl(p,q) so that V = Cl(p,q) f . By means of an orthonormal basis { e_1 , e_2 , ... , e_n } for [the grade-1 vector part of Cl(p,q)] Cl^1(p,q) we can construct a primitive idempotent f as follows: Recall that the 2^n elements e_A = e_a_1 e_a_2 ... e_a_k , 1 < a_1 < a_2 < ... < s_k < n constitute a basis for Cl(p,q). ... dim_R V = 2^X , where X = h or X = h + 1 according as p - q = 0, 1, 2 mod 8 or p - q = 3, 4, 5, 6, 7 mod 8 and h = [ n / 2 ] . Select n - X elements e_A, e_A^2 = 1 , so they are pairwise commuting and generate a group of order 2^( n - X ) . then the idempotent ... f = (1/2)( 1 + e_A_1) (1/2)( 1 + e_A_2 ) ... (1/2)( 1 + e_A_( n - X ) ) is primitive ... ... Cl(0,8) has a primitive idempotent f = (1/2)( 1 + e_1248 )(1/2)( 1 + e_2358 )(1/2)( 1 + e_3468 )(1/2)( 1 + e_4578 ) = =(1/16)( 1 + + e_1248 + e_2358 + e_3468 + e_4578 + e_5618 + e_6728 + e_7138 - - e_3567 - e_4671 - e_5712 - e_6123 - e_7234 - e_1345 - e_2456 + + e_J ) with four factors [and where J = 12345678 ] ...". Physically (refer to my recent earlier messages for details), in my model I identify the first scalar and 7 4-vectors with the Higgs and the 7 diagonal elements of traceless symmetric 35 in their coordinate representations and I identify the last pseudoscalar and 7 4-vectors of the dual 35 with them in their momentum-space representations. In other words, 8 of the symmetric 36 of 8x8 = 8/\8 + 8sym8 = 28 + 36 = 28 + 35 + 1 correspond to 8 elements of the Primitive Idempotent, 1 for scalar Higgs and 7 for 4-vector 35. The other 8 terms of the 16-term Primitive Idempotent correspond to the 1 pseudoscalar and 7 of the 4-vector dual 35. Since Primitive Idempotent produces Spinors, the Higgs and Spinors are connected, which gives a Cl(8) structure interpretation for Yukawa couplings, something that is not clearly motivated in the usual Standard Model. Of course, the Higgs/electroweak coupling in my model, based on electroweak gauge boson connection with Higgs scalar, is the same as that of the usual Standard Model. In short, the spinors in my model come from a Primitive Idempotent living in the scalar, the 35+35 4-vector, and pseudoscalar parts of the Cl(8) graded structure.

Carlos, you are correct that I have been imprecise in saying things like "... a 35 of the 4-vector is like the 35 in 8x8 = 8/\8 + 8sym8 = 28 + 36 = 28 + 35 + 1 ...". However, there is a structural sense in which it is a valid way to visualize the elemets of Cl(8). Consider ...[this]... image of a 16x16 array of 256 elements.

The two diagonal 8x8 parts represent the even subalgebra. The 1+1 red dots are the scalar and its dual pseudoscalar. The 28+28 blue dots are the bivectors and their dual 6-vectors. The 35+35 white dots are the 4-vectors. As you can see, the 64+64 elements of the even subalgebra can be thought of as: 64 = 28 bivectors (8/\8) plus 1 scalar and 35 4-vectors (8sym8), with the other 64 being interpreted similarly in dual grades. The scalar and 7+7 of the 35+35 and the pseudoscalar have yellow dots, and they correspond to the 16 terms of the Primitive Idempotent. As you can see, they lie on the diagonals of the two 8x8s. The odd parts (the two off-diagonal 64-element 8x8s) of Cl(8) are: the 8+8 green dots are vectors and pseudovectors and the 28+28+28+28 = 56+56 gold dots are the trivectors and their dual 5-vectors. I hope this clarifies to some degree the way that I visualize Cl(8). Please note that details of this nice visualization are peculiar to Cl(8), which has exceptional and unusual symmetries and structure, and is also a fundamental building block of all real Clifford algebras, due to real 8-periodicity. For instance, if you were to look at Cl(16) = Cl(2x8) = Cl(8) (x) Cl(8) you would see a 2-level tensor-nesting of the 16x16 structure, and the Cl(16) diagonal would be made up of 16 dots each of which looked like an entire Cl(8) 16x16 with its 16-element diagonal, so that Cl(16) spinors would look like 16x16 = 256 elements, with 128 +half-spinors and 128 -half-spinors, which is indeed the case.

...[this is from part of]... my web page at ... Jordan.html#3x3trless4x4 "... The 28-real-dimensional degree-4 quaternionic Jordan algebra J4(Q) of 4x4 Hermitian matrices over the Quaternions p D B A D* q E C B* E* r F A* C* F* t where * denotes conjugate and p,q,r,t are in the reals R and A,B,C,D,E,F are in the quaternions Q ... The 4x28 = 112-real dimensional Quaternification of J4(Q) can be represented as the Symmetric Space E8 / E7 x SU(2) ... J4(Q) contains the traceless 28-1 = 27-dimensional subalgebra J4(Q)o that "has the unique structure of" the 27-dimensional exceptional Jordan algebra J3(O) of 3x3 Hermitian matrices over the Octonions p B A B* q C A* C* r where * denotes conjugage and p,q,r are in the reals R and A,B,C are in the Octonions O. The 2x27 = 54-real dimensional Complexification of J3(O) = J4(Q)o can be represented as the Symmetric Space E7 / E6 x U(1) ... J3(O) contains a traceless 27-1 = 26-dimensional subalgebra J3(O)o that can be represented as the Symmetric Space E6 / F4 ...". In other words, the chain 26, 27, 28 gives you Jordan algebra structures J3(O)o, J3(O) = J4(Q)o, J4(Q) which in turn give you the Lie algebra structures E6, E7, E8 which Lie algebras are all either E8 or E8 subgroups. To connect with (real) Clifford algebras, here are some more parts of that same web page: "... how to fit E8 into Clifford structure ... use 248-dim E8 = 120-dim bivector adjoint of D8 + 128-dim D8 half-spinor and so embed E8 in the Clifford algebra Cl(16), with graded structure 1 16 120 560 1820 4368 8008 11440 12870 11440 8008 4368 1820 560 120 16 1 and total dimension 2^16 = 65,536 = (128+128)(128+128) Since Cl(16) = Cl(2x8) = Cl(8)xCl(8) ( For example, 120 = 1x28 + 8x8 + 28x1 and 128 = 8x8 + 8x8. ) E8 can be represented in a tensor product of Cl(8) algebras ...".