Stephen P. King (stephenk1@home.com)
Fri, 03 Sep 1999 13:03:09 -0400
Dear matti et al,
A few questions...
Matti Pitkanen wrote:
snip
> [MP] In TGD closed spacetime surfaces are possible absolute minima
> of Kaehler action. They have necessarily finite extension in both
> time and spatial directions. One can imagine of constructing them
> by taking two finite pieces of Minkowski space in M^4_+xCP_2, slightly
> deforming them in CP_2 directions, gluing along their boundaries,
> smoothening resulting edges, and requiring that absolute minimum
> of Kaehler action is in question. These surfaces decompose
> to regions with Minkowskian signature and Euclidian signature
> or have global Euclidian signature. They are not locally vacua:
> globally they are: classical energies, momenta, etc are
> of opposite sign on the two sheets. These surfaces are analogous
> to vacuum bubbles appearing in perturbative quantum field
> theories and contributing nothing to S-matrix elements.
This is what I have been looking for! Thank you! Now, a question: You
are saying that the cognitive and material space-time sheets have
"opposite signs" of their "classical energies, momenta, etc". Would
there be a CPT symmetry violation predicted from TGD to account for the
statistics of neutral (?) kaons?
> Physical considerations pose no restrictions on the density of these
> purely 'mindlike' objects (using the terminology of TGD inspired
> theory of consciousness): they represent pure thought.
> I have told about amusing structural isomorphism between the properties of
> asymptotic selves (selves for which all subsystems have vanishing p-adic
> entanglement) and between the closed Euclidian 4-surfaces.
> I would identify them as liberated Buddhas! (Ontogeny recapitules
> phylogeny.)
> *****************
But, Matti, these are mere geometric objects and do not follow the
requirements of minds (e.g. boolean lattice-like structures). I see the
"vacuum bubbles" are the objects of observation by other LSs. You see,
all geometry, is cast as the content of an observation with the Chu
space duality paradigm. The mind that does the observations is modeled
in terms of the boolean (for 2-nary relations) structures that act as
the "labels" of the geometries. This results from a consideration that
the class of labels (or meanings) that can be attached to any given
geometric object can not be trivially mapped to a geometry itself,
unless one is considering the singleton case (it is self-dual, e.g. MIND
= BODY for the Universe as a whole).
One the other hand, the phylogenic hierarchical ordering of the sheets
is very important! I see it as the "vertical" organization of a self,
just as you think of it. But, it is a mistake to assume that the mind is
geometries or fields of matter/energy. (Bohm has pointes out that they
are the same thing really!)
> We might imagine that a flat, closed, unbounded universe of this
> type would tend to collapse if it contained any matter, unless a
> non-zero cosmological constant is assumed. On the other hand, I'm
> not sure what "collapse" would mean in this context. It might
> mean that the R parameters would shrink, but R is not a dynamical
> parameter of the model. The 4D field equations operate only on
> x,y,z,t. Also, any "change" in R would imply some meta-time
> parameter T, so that all the R coefficients in the embedding
> formulas would actually be functions R(T).
All of this applied, but to the "what" is observed by an LS. The
difference in properties between the "inside" and "outside" of an LS are
very important! Umm, the idea of mapping (or identifying) the perceived
behaviour of objects in a given observer's universe (which is a finite
subset of The Universe) with the "internal" dynamics of the quantum
propagator is what I call "clocking and gauging"...
The key notion is that we must understand that we can not ever observe
The Universe, only "our version of it"!
> It seems that the flatness of the 4-space is independent of the
> value of R(T), and if the field equations are satisfied for one
> value of R they would be satisfied for any value of R.
This, I see as the "free fall" frame of the observer. The "forces" are
given when comparisons of pairs of frames like Bill's NOW/PAST pairs!
> But I'm not sure how the meta-time T would relate to the internal time
> t for a given observer. It might require some "meta field
> equations" to relate T to the internal parameters x,y,z,t.
> Possibly these meta-equations would allow (require?) the value
> of R to be "increasing" versus T, and therefore indirectly
> versus our internal time t = f(T), in order to achieve stability.
How could meta-time T be observable? Why is it even necessary to
consider such an idea? Is "stability" really necessary to assume? (think
of dissipative systems!)
> [MP] In critical Robertson-Walker cosmology one has somewhat
> similar situation. 3-space is Euclidian and could be compactified to
> 3-torus. Allowing R to on t one obtains curved 4-space as is clear from
> the fact that mass density (G^tt component of Einstein tensor) is
> nonvanishing. In special case one would have the
>
> ds^2= t^2(dt^2 -dx^2-dy^2-dz^2)
>
> I checked from general formula for mass density in RW cosmology
> that mass density goes like 1/t^4, that is scales. This is to be expected
> since line element is Weyl equivalent to flat metric. It seems
> that this is nothing but the radiation dominated critical cosmology which
> is scale invariant (massless particles dominate).
> In this case R indeed increases with t linearly.
So this is a quantification of "what can be observed" given a
particular "mass density"?
>
> **********************
>
> On the general method of embedding a locally flat n-dimensional
> space in a flat 2n-dimensional space, I wonder if every orthogonal
> basis in the n-space maps to an orthogonal basis in the 2n-space
> according to a set of formulas formally the same as those shown
> above. If not, is there a more general mapping that applies to
> all bases?
>
> By the way, the above totally-cylindrical spacetime has a natural
> expression in terms of "octonion space", i.e., the Cayley algebra
> whose elements are two ordered quaterions
>
> x1 = i R1 cos x/R1 x2 = i R1 sin x/R1
> x3 = j R2 cos y/R2 x4 = j R2 sin y/R2
> x5 = k R3 cos z/R3 x6 = k R3 sin z/R3
> x7 = R4 cos t/R4 x8 = R4 sin t/R4
>
> Thus each point (x,y,z,t) in 4D spacetime represents two quaterions
>
> q1 = x1 + x3 + x5 + x7
>
> q2 = x2 + x4 + x6 + x8
>
> To determine the absolute distances in this 8D space we again consider
> the eight coordinate differentials, exemplified by
>
> d x1 = i R1 (-sin(x/R1)) (1/R1) (dx)
>
> (using the rule for total differentials) so the squared differentials
> are exemplified by
>
> (d x1)^2 = - sin^2(x/R1) (dx)^2
>
> Adding up the eight squared differentials to give the square of the
> absolute differential interval leads again to the locally Lorentzian
> 4D metric
>
> (ds)^2 = (dt)^2 - (dx)^2 - (dy)^2 - (dz)^2
>
> [MP] It is often believed that quaternions and octonions are inherently
> Euclidian objects. This is actually not true as I discovered for year or
> two ago. The point is that one can define inner product as
> real part Re(z1z2) of z1z2: the resulting inner product is standard
> Minkowskian inner product. If one defines inner product as Re(z1^*z2)
> one obtains the usual Euclidian inner product.
What is the relationship between Re(z1z2) and Re(z1^*z2)? Are they
orthogonal?
Later,
Stephen
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