# [time 489] Re: [time 475] Parallel translation, etc..Part I

Matti Pitkanen (matpitka@pcu.helsinki.fi)
Sun, 25 Jul 1999 08:35:08 +0300 (EET DST)

On Sat, 24 Jul 1999, Stephen P. King wrote:

> 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.

[Matti] This was based on the conservative idea that you somehow
generalize manifold concept and the idea that you get concrete
representation of manifold by mapping it to surface. Almost disjoinness is
then a fixed purely geometric notion. For instances two spheres in E^3 can
intersect along curve (generically as surfaces), or in discrete point
(very rare occasion as surfaces).

>
> 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.
This might be roughly what I said above.

>
> 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)
>

This sounds much more general: I do not know whether you start from
manifold concept. By the way, epsilon accuracy obviously corresponds to
pinary cutoff in TGD framework: reality is mapped to p-adicities
of selfs with resolution defined by pinary cutoffs.

> 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...

I must admit that my mathematics is quite too conservative to say
anything useful about the problem. I do not know whether fuzzy
mathematicians have been able to define concepts like fuzzy manifolds.

[Matti] Recall that the problem of mapping observers to each other
rises from the assumption that there are no realities there: in TGD
the existence of quantum histories leads to the mapping of
reality to p-adicitiies and pinary cutoff replaces epsilon accuracy.

> 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!

[Matti] I try to describe parallel translation of vector
in sphere. Take vector in itial point of a curve. Translate the
vector along curve in such a manner that the angle between
the vector and curve is not changed during the process.
When you go through a closed curve you find that the direction of
vector is not the original direction: sphere is curved space.

More generally: parallel translation means transport of vector
spinor etc. such that this object suffers a tranformation
represented by the operation of gauge group when it is
transported along infinitesimal portion of curve.

Transformation is infinitesimally:

delta V = A_mu dx^mu o V ,

where A_mu is matrix acting on vector and called connection. Combining
these infinitseimal transformation one obtains net transformation
of vector when one goes through closed curve.

>
> > 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?]
>
No. In Kaluza Klein approach imbedding space would be dynamical
and its quantum version (quantized metric, etc) it would be the
quantum reality. In TGD imbedding space is pregiven passive object
fixed uniquely by symmetry requirents guaranteing the existence
of the space of 3-surfaces.

Note however that the dynamical nature of Kaluza Klein approach
destroys the basic idea behind: namely that the symmetries of
the fiber of higher dimensional spacetime explain particle symmetries.
Symmetric fiber corresponds to extremely special solution of
Einstein-YM equations.

> > 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?

Wormholes are purely geometric concept: the 2-dimensional illustrations
on my homepage might help. You have seen Einstein-Rosen bridge
leading from one spacetime to second spacetime represented as
infinite planes. Make these planes finite and put Einstein-Rosen bridges
near their boundaries.

Subsystem is key notion in definition of strong form of NMP dynamics of
subjective existence. Spacetime sheets indeed define subsystem concept
basically: the definition reduces to purely geometric definition in which
subsystems essentialy correspond to spacetime sheets, cognitive with
finite time duration or material with infinite time duration.
This is of course consistent with how we intuitively define subsystem:
geometrically, as subset of spacetime.

Many-sheeted spacetime is basic prediction of TGD. In particular,
finite size of the sheet: recall that my body is finite spacetime sheet.
At my skin my personal spacetime ends and outside my body there
is second larger spacetime sheet.

>
> > >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... :-)

OK. We must discuss about this. This is highly nontrivial notion
differentiating between QFT defined in fixed spacetime and TGD defined
in the space of 3-surfaces<--> in the space of allowed spacetime surfaces
defined by absolute minimization of Kahler action.

[snip]

> [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!

Single step in the drifting is indeed interpretatble quantum computation
in quantum TGD.
>
> > 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

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