[time 475] Parallel translation, etc..

Matti Pitkanen (matpitka@pcu.helsinki.fi)
Sat, 24 Jul 1999 10:29:59 +0300 (EET DST)

I chopped the email into pieces.

On Fri, 23 Jul 1999, Stephen P. King wrote:

> Dear Matti,
> I believe that we have ideas that are very very similar, it is
> that we are using complementary ways of talking about them! I am very
> grateful for the patience that you show to me! I am sure that I am very
> aggravating and difficult! ;-)
> I cut this responce in half as it is long and my browser software
> contain all of it simultaneously.

Perhaps we should try to keep responses always by chopping them. I try to
remember this in future.

> Matti Pitkanen wrote:
> > > [MP]
> > > > Parallel transport and connections emerge both at the level
> > > > of spacetime geometry and configuration space geometry.
> > >
> > > > a) At spacetime level all geometric structures are induced
> > > > from those of imbedding space. Although imbedding space
> > > > geometry is nondynamical, spacetime geometry is dynamical.
> > > > This is metaphorically dynamics of shadows: object is u
> > > > nchanged but shadow varies when object moves.
> [SPK]
> > > This "parallel transport" notion is what Weyl made a careful
> > > of in his work and why he reasoned that Riemannian geometry needed
to be
> > > generalized! But, Weyl did not consider the possibility that
> > > space-times are needed to model multiple observers. He assumed, as
> > > most, that there is only one M^4 and its related fields/connections
> > > all...
> >
> > 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
> > for connection.
> Could we discuss the notion of connections separately? To me it is
> 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?

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.

> > > Parallel transport is the conceptual tool to model motions of
> > > "shadows", but fact is that that which is projecting the shadows is
> > > a "ONE" in the ontological sense, it is the "MANY" possible finite
> > > subsets of U. They project onto each other...
> >
> > Is there slight misformulation here.
> > In case of 'shadows' it is the induction of parallel transport which
> > occurs. Parallel transport in imbedding space is projected to
> > Induction and parallel transport are two different things. Parallel
> > transport is the great construction principle of GRT and gauge
> > and in fact all recent QFT:s. Induction is purely TGD:eish addition
> > to the conceptual arsenal used.
> Ok, I see the difference; but, I would like to better understand
> notion of "induction of parallel transport".

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

A=A_k partial_(alpha) h^k dx^alpha

and here it is!:

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.

> > > This is a very important point that becomes obvious in a
> > > about the difference between "actual" and "potential" infinities...
> > > is ontologically a priori is the Universe, which is an "Actual"
> > > infinity, thus it is ONE. The dynamics of the shadows are
> > > infinities, thus are "MANY", thus the uncertainty formalism that
> > > has discovered and discussed in his papers connects the time of the
> > > to its measure of uncertainty.
> > > The idea that the Universe, as a "static, pre-given entity"
> > > non-dynamical is part of my thinking and that of Hitoshi and,
> > > appearently yours.
> [MP]
> > But what one means by Universe! I do not mean by Universe imbedding
> > Rather, Universe is one possible spacetime surface classically.
> > configuration space spinor field quantum mechanically. And Universe
> > in latter sense is replaced by a new one in any quantum jump. It can
> > quite well occur that the sequence of quantum jump produces almost
> > fixed points: at least parts of universe can become almost fixed
> > this is what self organization by quantum jump produces.
> > But basically there is endless evolution: the prime characterizing
> > topology of universe increases all the time.

> Here is the most dramatic difference in our thinkings. I am saying
> 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.

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.

Where I speak of quantum superposition you want to introduce almost
disjoint spacetimes as spacetimes of observers.

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.

>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.

> > > Michael C. Mackey also addresses this notion in his
> > > book Time's Arrow with his "God Theorem"... The key notion that we
> > > seem to agree on is that the "parts" can have dynamical behavior
> > > not at equilibrium) while the "whole" is static. The notion of time
> > > 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
> > > 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!
> No. There can be no ultimate self associated with the entire
> if we are talking about the Totality of Existence. It is static, it does
> not evolve, it merely exists.

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.

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



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