Stephen P. King (stephenk1@home.com)
Sat, 24 Jul 1999 13:33:59 -0400
Dear Matti,
I am sending these in order... and snipping to conserve space and
labeling who wrote what for the others in the List...
Matti Pitkanen wrote:
> > > Parallel transport is extremely general concept: 'connection' in fiber
> > > bundle with structure group G. Very abstractly. Metric (Riemann)
> > > connection special case: in this case inner product defined by Riemann
> > > metric is conserved in parallel transport. For Weyl connection only
> > > angles between vectors are conserved. In gauge theories one just
> > > postulates some structure group G (say standard model group SU3 times U2)
> > > for connection.
[SPK]
> > Could we discuss the notion of connections separately? To me it is a
> > key notion that distinguishes Hitoshi's LS theory from other models. My
> > ideas have developed independently from Hitoshi until I read his paper
> > and found a wonderful mathematical expression of my thoughts.
> > A key question: Could we construct at least two almost disjoint
> > 4-dimensional Riemannian manifolds from selected "pieces" of a
> > n-dinensional manifold with Weyl geometry?
> >
[MP]
> All depends on what you mean with almost disjoint. Thinking in terms
> of surface one can quite easily consider variants of the notion. Surfaces
> simply intersect in some lower dimensional set. N-1,....,0-dimensional.
> One can defined metric and induced quantities for these objects.
> In intersection points connections and metric are many value since
> one can move along two intersecting branches of the surface.
I have but a vague notion of the symbols needed to represent this
"almost disjoint" notion. It will probably take a few back and forth
messages to get it straightened out. :-) Ok, here are possible
definitions:
1) two (or more) spaces are almost disjoint if they are "neighboring" in
an embedding space that contains both as some generalized notion of a
curve.
2) two (or more) observations are almost disjoint if they are had by
observers that can predict each other's behavior up to the \epsilon of
accuracy. (I assume that uncertainty follows from Hitoshi's definition,
it follows from the asymptotic way that limit m -> \infinity)
3) two (or more) events are almost disjoint if their is at most one
point in the intersection of their framings. This notion is related to
the idea of a fixed point in a given group of transformations. If two
manifolds (smooth or Cantor dust-like) have regions (balls?) neighboring
a point that is mapped to another point by an automorphism (?) of the
ball, the points are almost disjoint iff the identification is only up
to \epsilon. An identification that is exact (\epsilon = 0) then the
points are strictly disjoint.
The problem that I am having is that I am a philosopher trying to
generalize the notion of continuity used by mathematicians! The ordinary
definition of continuity assumes that disjoint points have disjoint
neighborhoods. I am trying to work out the idea that the neighborhoods
of points can have a non-empty intersection. This notion comes from
fuzzy logic...
snip
[SPK]
> > Ok, I see the difference; but, I would like to better understand your
> > notion of "induction of parallel transport".
[MP]
> Geometrically induction is extremely obvious: regard curve of
> surface as curve of imbedding space and perform parallel transport
> in imbedding space.
>
> Technically this reduces to projection. Take U(1) connection represented
> as one-form, covariant vector field in imbedding space.
>
> A= A_k dh^k
>
> where A_k are components of connection and dh^k are coordinate
> differentials giving a basic for one-forms. Restric A to
> surface
>
> A=A_k partial_(alpha) h^k dx^alpha
>
> and here it is!:
Oh, so close! I need more. I have to build a picture in my head to
understand. Could you use words instead of symbols. I know this is very
tedious, but please, it would help me so much!
> Components of connection on surface are just projections of A_k:
> A_alpha= A_k partial_alpha h^k.
>
> This method generalizes trivially to connection with structure group
> G in which case A_k are Lie algebra valued. Also line element
> for metric is restricted in the same manner. In particular,
> vielbein connection in its various representations (spinor
> connection) is induced in this manner.
>
> >From Riemann connection which has 3 tensor indices instead
> of one, situation is quite not this. One must first project
> metric to spacetime surface
>
> ds^2 = gkl dh^k dh^l = g_kl partial_alpha h^k partial_beta h^l dx^alpha
> dx^beta = g_alphabeta dx^alpha dx^beta
>
> and calculate Riemann connection from induced metric.
I get an intuition that I agree with this, but I don't have written
words for this. :-(
snip
[SPK]
> > Here is the most dramatic difference in our thinkings. I am saying that
> > the Universe can not possible be "one possible spacetime surface"
> > classically or otherwise, this is inconsistent, since the existence of
> > such requires, at a miminum, that the information content of such to be
> > knowable by an arbitrary entity.
[MP]
> Actually I am saying just the same thing but from different view point.
> Universes are represented by quantum superspositions of classically
> equivalent spacetime surfaces which are dynamical, determined by absolute
> minimization of Kahler action.
>
> Imbedding space is pregiven but it is NOT universe, it is
> only the fundamental framework of the geometry: imbedding space
> geometry contains very little information as such already because
> of its extremely high symmetries. Dynamical spacetime surfaces
> are carriers of geometric and topological information.
Is the "inbedding space" \subset of my "MANY"? [inbedding =
embedding?]
> Where I speak of quantum superposition you want to introduce almost
> disjoint spacetimes as spacetimes of observers.
I think we are really talking about the same thing... But until I get
meaning of the math into pictures in my head, I can't be sure...
> To add confusion note however that also I introduce the many sheeted
> spacetime: spacetime sheets are almost disjoint: only tiny wormholes
> connect them. Your many spacetimes aspect is in well defined
> sense present also in my thinking: there is single spacetime
> surfaces decomposing to almost disjoint spacetime sheets.
> The spacetimes of classical observers.
OK, these "wormholes" connect finite regions or identify pinary points?
Why do we need the "spacetimes of classical observers"? what function
does it serve? It is illusion, MAYA! Do you use it like Bohr used it in
the Copenhagen Interpretation of QM?
> >This is why the classicists, such as
> > Newton and Laplace, relied on Gods or other "supernatural" entities to
> > observe such and thus make it actual. Existence and actuality are NOT
> > the same. The Universe in it-self can only exist, it can not be a
> > "space-time" in-itself. The experiences of finite LSs of it, are given
> > in terms of space-times, yes, but to identify a space-time with the
> > Universe is not helpful!
> >
> Well, I agree here completely. As I already explained, quantum
> superposition of classically equivalent manysheeted spacetimes is the
> universe in my approach.
I think that this notion needs to be refined. But I am just a mindless
philosopher... :-)
> > > > Michael C. Mackey also addresses this notion in his
> > > > book Time's Arrow with his "God Theorem"... The key notion that we
> all
> > > > seem to agree on is that the "parts" can have dynamical behavior
> (are
> > > > not at equilibrium) while the "whole" is static. The notion of time
> is
> > > > given in the manner that the parts, by interacting among themselves,
> > > > evolve toward equilibrium, e.g. evolve toward isomorphism with the
> > > > whole: the "Union with All" of mystics. But the key notion is that
> this
> > > > process literally takes forever to accomplish!
> > > >
> > > p-Adic evolution indeed never ends. Even God as the self of entire
> > > universe is evolving all the time!
[SPK]
> > No. There can be no ultimate self associated with the entire universe,
> > if we are talking about the Totality of Existence. It is static, it does
> > not evolve, it merely exists.
[MP]
> I agree if I interprete the Totality of Existence as the space
> of all configuration space spinor fields=quantum histories. In
> this space subjective time development is studying by hopping around
> and gradually drifting to more and more interesting corners
> of this space where spacetime surfaces itself contain more and
> more cognitive spacetime sheets and possess p-adic topology
> with ever increasing p.
Yes, this "drifting" is what I call "computation"! By computing "what
is most likely to happen next given what I know now" the self
projects/teleports it-self forward!
> Most differences in our opinions result from different interpretations
> for the notions of Universe, Totality of Existence, and so on.
>
> And from different notion of existence. I talk about
> material (geometric), subjective and objective (quantum histories,ideas)
> existences. By the way, do you have similar classification of
> existences? This might help me to get more precise view about your
> thinking.
I will in the forthcoming responses. :-)
> Best,
>
> MP
Onward!
Stephen
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