Matti Pitkanen (matpitka@pcu.helsinki.fi)
Mon, 23 Aug 1999 13:34:26 +0300 (EET DST)
On Mon, 23 Aug 1999 WDEshleman@aol.com wrote:
> Matti,
> You have proposed a 4-dimensional surface in an
> 8-dimensional space...what made you arrive there
> instead of, say, a 4-dimensional surface on (in) a
> 5-dimensional space, (or 6 or 7)?
The reasons are two-fold.
a) The requirement that the symmetries (isometries) of imbedding space
explain elementary particle quantum numbers, force the option
M^4_+xCP_2 uniquely. Standard cosmology forces future lightcone
M^4_+ instead of M^4. Spacetime as surface geometrizes not
only particle quantum numbers but also classical fields. The mere
requirement that internal space S (actually CP_2) gives
rise to at least 4 degrees of freedom corresponding to those
of classical em vector potential forces S to be at least 4-dimensional.
b) H= M^4_+xCP_2 is forced by purely mathematical considerations.
The Kahler geometry and spinor structure of the configuration space of
3-surfaces require this option. Maximal symmetries
are required for H= M^n xS. These symmetries are 'lifted' to
symmetries of the space of 3-surfaces
b1) Super Virasoro invariance is crucial element making many
theoreticians to believe that string models are the TOE: the boundary
of the FOUR-dimensional lightcone has generalized conformal
structure and provides configuration space with Kahler structure
and Super Virasoro invariance becomes symmetry of the theory
despite the fact that basic objects are 3-surfaces rather than strings.
b2) Configurations space spinor structure requires super symmetry.
The number of spinor components of H-spinors of fixed chirality
must be same as dimension of H:
D= 2^((D/2)-1), D=8 is the only solution to the condition.
b3) Thus H is product of 4-dimensional future lightcone and space
S with Kahler structure and maximal group of isometries (S is symmetric
space, whose all points are geometrically equivalent, in terminology of
Cartan). S=CP_2 is the only possibility. S= S^2xS^2 does not possess
so large isometry group and leads to unphysical particle
spectrum.
> The way you did
> this, by doubling, is appropriate to a conversion to
> a statement of worlds instead of dimensions, without
> altering what you have already done. That is,
> rewording we might say, 4-worlds confining
> (or confined by) 8-worlds. Now, if we define a world
> as a classical volume, the infinite structure of
> reality (the total model for TGD) might be proposed
> as 1-world confining 2-worlds confining 4-worlds
> confining 8-worlds confining 16-worlds, etc.
> Or as you would say it: a 1-dimensional surface in
> a 2-dimensional space in a 4-dimensional space in
> an 8-dimensional space in a 16-dimensional space, etc.
>
One might think hierarchy of this kind.
The construction of configuration space geometry however
relies crucially on 4-dimensionality of basic dynamical
objects (4-surfaces) and on the fact that the geometry
of imbedding space is *non-dynamical* and possesses the high
symmetries it does. The generalization you propose would
require totally new approach to the quantization of the
theory. It is difficult to say whether this kind of
theory exists without spending 20 years or so trying to
construct it!(;-)
The progress in this kind of work is very slow and requires
infinite patience: only quite
recently I finally realized that Poincare invariance
is not lost at S-matrix level although future lightcone
is used instead of entire Minkowski space and what is the
precise definition of S-matrix in TGD.
> In relation to my own work, you have done what I could
> not do; you solved the geometry and discovered rules for
> the problem of 4-worlds confined by 8-worlds, but at the
> same time you have left my speculation undone. That is,
> that the worlds increase as 2^N in abundance at levels N =
> 0 to infinity. Finishing up with 1-2, 2-4 and to infinity
> by induction would be very interesting to me. And who
> knows, maybe some of those "embarrassing" predictions
> may disappear on the completion of the geometry.
>
Your scenario is interesting but it would require
reformulation of quantum theory. There have
also been suggestions that universe might be kind of
soup of surfaces of all possible dimensions in infinite
dimensional space intersecting each other.
Best,
MP
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