**Matti Pitkanen** (*matpitka@pcu.helsinki.fi*)

*Sun, 4 Apr 1999 13:11:31 +0300 (EET DST)*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Matti Pitkanen: "[time 135] Re: [time 121] RE: [time 115] On Pratt's Duality [time 81] Entropy, wholeness, dialogue, algebras"**Previous message:**Matti Pitkanen: "[time 133] Re: [time 114] Re: prugovecki"**In reply to:**Stephen P. King: "[time 114] Re: prugovecki"**Next in thread:**Ben Goertzel: "[time 149] Re: [time 108] Re: [time 81] Entropy, wholeness, dialogue, algebras"

On Sat, 3 Apr 1999, Ben Goertzel wrote:

*> hi,
*

*>
*

*> >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)
*

*>
*

[MP]

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.
*

[MP]

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

in TGD.

MP

**Next message:**Matti Pitkanen: "[time 135] Re: [time 121] RE: [time 115] On Pratt's Duality [time 81] Entropy, wholeness, dialogue, algebras"**Previous message:**Matti Pitkanen: "[time 133] Re: [time 114] Re: prugovecki"**In reply to:**Stephen P. King: "[time 114] Re: prugovecki"**Next in thread:**Ben Goertzel: "[time 149] Re: [time 108] Re: [time 81] Entropy, wholeness, dialogue, algebras"

*
This archive was generated by hypermail 2.0b3
on Sun Oct 17 1999 - 22:31:51 JST
*