David Finkelstein, Heinrich Saller, Tang Zhong, and I are working on structures we call MetaClifford algebras. As of now (6 February 1997) we are only just beginning, so what we have is more an outline of a program than a finished product. For instance, as you can see, some of the fundamental points are still conjectures as opposed to theorems. Also, since different people may visualize the same structures in different ways, David, Heinrich, and Zhong probably don't see things the same way that I see them, so I should make it clear that these rough notes are just my interpretation of MetaClifford algebras, and not necesssarily theirs.
That having been said, here is how I view MetaClifford algebras: The point is to start with the empty set, then use a clear process to "expand" the empty set into structures that model our physical universe. Perhaps, from a philosophical point of view, such a process would be a model of creation of our physical universe from nothing. Begin with the empty set PHI and the operator IOTA that creates a set containing a given set. For example, IOTA(PHI) = {PHI}. By repeating the process, in a standard way, you can create all the NATURAL NUMBERS. By introducing the set theoretical operators of union \/ and intersection /\ and relative complement --- you can, again in a standard way, construct the operations of addition corresponding to union \/ and subtraction corresponding to --- /\ Using subtraction the NATURAL NUMBERS can be extended to the INTEGERS Z. Then multiplication can be introduced as repeated addition, and division can be introduced by the inverse of multiplication. Using division, the INTEGERS Z can be extended to the RATIONAL NUMBERS (Here I do not use Q as a symbol for the rational numbers, because I use Q for quaternions.) Then, by taking limits, the RATIONAL NUMBERS can be extended to the REAL NUMBERS R. Now that we have the REAL NUMBERS R we can look again at the empty set PHI and the operator IOTA that creates a set containing a given set, with IOTA(PHI) = {PHI}. Let PHI represent a 0-dimensional point. Let {PHI} represent another 0-dimensional point, and use the REAL NUMBERS R to construct a 1-dimensional real vector space with 0 corresponding to PHI and 1 corresponding to {PHI}. Denote that vector space by V(0,1). We can also construct a 1-dimensional real vector space with 0 corresponding to PHI and -1 corresponding to {PHI}. Denote that vector space by V(1,0). Here, notice that if we had used any pair of distinct numbers rather than the pairs (0,1) and (-1,0) we would have gotten the "same" two structures in the sense that they would be equivalent under translation and scale transformation. The only distinction that "matters" is the relative ordering of the origin and the other point. I do NOT want to say that vector spaces are equivalent under reflection, because I DO want to "see" orientation. Now that we have vector spaces, introduce the exterior and interior products: exterior product /\ corresponding to set theoretical union \/ and interior product |_ corresponding to set theoretical intersection \/ My notation /\ for the exterior product and |_ for interior product corresponds to Grassmann's progressive and regressive products, which are denoted by \/ and /\ respectively in Barnabei, Brini, and Rota (J. Alg. 96 (1985) 120-160). David Finkelstein consistently uses the better notation \/ and /\ but I use /\ and |_ (worse) because it is more in line with common usage. Also, my interior product is NOT the same as the standard "interior product", as my interior product is based on the geometry of vector spaces, NOT on dual spaces. To my taste, the dual spaces, functions, etc., that are used in the standard "interior product" are overcomplicated and unintuitive, a point on which I agree with Barnabei, Brini, and Rota. We can use the exterior product /\ to construct higher-dimensional vector spaces V(p,q) with p dimensions of negative signature and q dimensions of positive signature. The vector subspaces of V(p,q) are called multivectors, and k = p+q is called the grade of a multivector. Now, define the Clifford product . for a vector space with orthonormal basis {e1, e2, e3, ... , eN} and vector a and multivector B of grade k to be a.B = a /\ B - a |_ B Then, construct in the standard way the real Clifford algebras. In the case V(0,0) of a zero-dimensional point vector space, it is conventional to get Cl(0,0) = R so that we go from 0-dim to 1-dim. The graded structure of Cl(0,0) is 1 with total dimension 1 From the real 1-dimensional vector space, we can construct Cl(0,1) = C = complex numbers and (where + denotes direct sum) Cl(1,0) = R + R The graded structure of Cl(0,1) is 1 1 with total dimension 2 Now is where the MetaClifford, as opposed to Clifford, structure appears. Define the MetaClifford algebra MCl(Cl(p,q)) of a Clifford algebra Cl(p,q) to be constructed by using the standard Clifford algebra construction process applied, not to a vector space, but to the Clifford algebra Cl(p,q) By ignoring the graded structure of the Clifford algebra Cl(p,q) and making Euclidean the signature of the vector space V(p,q) you get the EUCLIDEAN RESTRICTION of the MetaClifford algebra MCl(Cl(p,q)) to be the Clifford algebra Cl(0,p+q) MCl(Cl(p,q)) --restricts Euclidean to-- Cl(0,2^(p+q)) Next, construct the MetaClifford algebra MCl(Cl(0,1)) MCl(Cl(0,1)) --restricts Euclidean to-- Cl(0,2^1) = Cl(0,2) = Q where Q denotes the quaternions. The graded structure of Cl(0,2) is 1 2 1 with total dimension 4 Define the MetaClifford algebra MCl(MCl(Cl(p,q))) of a MetaClifford algebra of a Clifford algebra using similar construction except that you pay attention to the graded structure that MCl(Cl(p,q)) inherits from Cl(p,q) and proceed inductively to define MCl(MCl...(Cl(p,q))...) Now, construct the MetaClifford algebra MCl(MCl(Cl(0,1))) MCl(MCl(Cl(0,1))) --restricts Euclidean to-- Cl(0,2^2) = Cl(0,4) = M2(Q) where M2(Q) denotes 2x2 matrices of quaternions. The graded structure of Cl(0,4) is 1 4 6 4 1 with total dimension 16 Next, construct the MetaClifford algebra MCl(MCl(MCl(Cl(0,1)))) MCl(MCl(MCl(Cl(0,1)))) --restricts Euclidean to-- Cl(0,2^4) = Cl(0,16) The graded structure of Cl(0,16) is 1+16+120+560+1820+4368+8008+11440+12870+ +11440+8008+4368+1820+560+120+16+1 with total dimension 65,536 = 256 x 256 Now, using Bott period-8 periodicity for real Clifford algebras, we can FACTOR Cl(0,16) into Cl(0,8) x Cl(0,8) the tensor product of two Clifford algebras over the same 8-dimensional space I use in the D4-D5-E6 physics model. Therefore MCl(MCl(MCl(Cl(0,1)))) --restricts Euclidean to-- Cl(0,8) x Cl(0,8) where Cl(0,8) = M16(R), the 16x16 matrix algebra over the reals. The graded structure of Cl(0,8) is 1 8 28 56 70 56 28 8 1 with total dimension 256 and I CONJECTURE that: If you go to the next stage by forming MCl(MCl(MCl(MCl(Cl(0,1))))) then MCl(MCl(MCl(MCl(Cl(0,1))))) --restricts Euclidean to-- Cl(0,2^8) x Cl(0,2^8) = = Cl(0,256) x Cl(0,256) = = Cl(0,32x8) x Cl(0,32x8) = = Cl(0,8) x ...(64)... x Cl(0,8) = = 64 copies of Cl(0,8) SO THAT instead of going to larger, more complicated Clifford algebras over very high-dimensional vector spaces, all you get in future steps based on Cl(0,8) is a lot of copies of the Clifford algebra of the D4-D5-E6 physics model, each which can be regarded as the local structure of a (possibly curved or topologically complicated) universe that made up of a lot of (flat) neighborhoods with the local D4-D5-E6 physics model on each (flat) neighborhood. If the restricted Euclidean structure gives the D4-D5-E6 physics model, then WHAT IS THE POINT OF USING THE METACLIFFORD ALGEBRA STRUCTURE? The point is that, I conjecture, the MetaClifford structure of the 8-dimensional vector space of the Cl(0,8) includes a 4-dimensional subspace corresponding to Cl(0,4) = M2(Q) so that the METACLIFFORD STRUCTURE DEFINES THE PHYSICAL 4-DIMENSIONAL SPACETIME and so defines the dimensional reduction mechanism of 8-dimensional spacetime to 4 dimensions as required by the D4-D5-E6 physics model. Notice that the Clifford algebra Cl(0,8) = M16(R) the even subalgebra of Cl(0,8) = Cl(0,7) = M8(R) + M8(R) the even subalgebra of Cl(0,7) = Cl(0,6) = M8(R) the even subalgebra of Cl(0,6) = Cl(0,5) = M4(C) and Cl(0,5) = M4(C) is the complexification of Cl(0,4) Cl(4,0) = Cl(1,3) = M2(Q) and of Cl(2,2) = Cl(3,1) = M4(R) and therefore is the physically useful complexification of the Dirac gamma matrices.
Note that much of the above is conjectural. In fact, since I have not done the proofs, it is even conjectural that MetaClifford Algebras are well-defined.
Compare these MetaClifford Algebras to my Simplex Physics and to my Quantum Sets, as well as the construction of Clifford Algebras from Set Theory.
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