# MetaClifford Algebras

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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:

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
and subtraction corresponding to --- /\

Using subtraction the NATURAL NUMBERS
can be extended to the INTEGERS   Z.

Then multiplication can be introduced
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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