Matti Pitkanen (email@example.com)
Sun, 4 Apr 1999 13:11:31 +0300 (EET DST)
On Sat, 3 Apr 1999, Ben Goertzel wrote:
> >Also we had very interesting and fruitful discussions with Tony
> >Smith and I have studied Tony's homepage. One one results on my side was
> >the realization that 8-dimensional
> >imbedding space H of TGD allows octonionic structures as tangent space
> >structure: one could say that H is locally a number field.
> This is remarkable and interesting....
> But, what role does the 8-dim imbedding space play in your theory?
> (I do plan to study your theory closely, but have not found the time yet)
Much the same role as 10-dimensional imbedding space in super string
models. Difference with respect to string models is that this
8-dimensional space is fixed uniquely both by physical and (rather
general) mathematical considerations. It is NOT dynamical!
a) The basic symmetries of theory are symmetries of imbedding space
rather than spacetime as in GRT. Elementary particle quantum numbers
dictate H= M^4_+xCP_2 as the only possible choice. CP_2 codes standard
model symmetries in its isometries and generalized vielbein gauge group.
Classical gauge fields allow geometrization
in terms of submanifold geometry.
b) In the construction configuration space geometry 8-dimensionality of H
has crucial role: in dimension 8 the dimension of spinors of given
chirality is 8 and
and this makes supersymmetry possible. The boundary of M^4_+ (moment
of big bang) is metrically 2-dimensional since it has one null direction
and this allows to generalized two-dimensional conformal structure to
the light cone boundary: this conformal structure makes possible Kahler
structure of configuration space and Super Virasoro symmetries.
Just the assumption that internal space S allows Kahler structure (it
must have it) and is symmetric space with maximal isometry group (laws
of physics do not depend on point of CP_2) fix S to CP_2.
> >Your idea is interesting also from my point of view. I am pondering
> >analogous problem: spacetime sheets decomposes in QFT limit to regions
> >with different p-adic primes: the physics in different regions are
> >described by different p-adic number fields. How to relate these
> >p-adic physics to each other?: this is the basic problem. I believe that
> >this is possible.
> If we could map the p-adic stuff onto the octonionic stuff in a clear way, this
> would help my intuition and build a bridge between several theories.
I think that octonions and quaternions and p-adicity are separate aspects
of theory. I do not know whether it is possible to define octonions
and quaternions obeying p-adic topology (components
would be regarded as p-adic numbers). Some delicacies might appear.
At least p mod 4=3 guaranteing that -1 does not have square root as
ordinary p-adic number does not exist.
There is also p-adic magic associated with dimensions 4 and 8.
Square roots of real numbers tend to appear in basic formulas of Riemann
geometry and group representations theory (Glebch-Gordans) and QM
(normalization factors). Hence it is desirable to be able to define the
square root of p-adic number
by performing algebraic extension of p-adics so that square roots of
ordinary p-adics exist. The algebraic extension allowing square roots
for all ordinary p-adic numbers (but not for those belonging to
extension) is 4-dimensional for p>2 and 8-dimensional for p=2. The
original idea was that the generalization of conformal invariance to 4-
and 8-dimensional cases must be crucial for quantum TGD and that quantum
TGD could be regarded as conformal field theory in dimension 4. Rather
disappointingly, I have found no place for this idea
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