[time 464] Parallel transport, etc....

Matti Pitkanen (matpitka@pcu.helsinki.fi)
Thu, 22 Jul 1999 08:09:16 +0300 (EET DST)

Below continuation to the answer of earlier message.

On Tue, 20 Jul 1999, Stephen P. King wrote:

> Dear Matti,
> Matti Pitkanen wrote:
> > Dear Matti,
> respond allowance of quantum superpositions of W:s in LS framework.
> >
> > Matti Pitkanen wrote:
> > >
> > > On Sat, 17 Jul 1999, Stephen P. King wrote:
> snip

> snip

> [SPK]
> > How is the metric generalize to our tree-like fibers? The
> > of the tree-fibers are LSs and the truck roots into a non-zero region
> > W, I think... The subtleties involved is that we can invert the
picture so
> > that the leaves are points in W and the root is an LS... The
> > in these views is the difference between the subjective and the
> > objective. Umm, I am confused! :-) Maybe we say that the tree is a
> > fibration of LS_i x X^3_j -> W_ij ? I need to think about how to put
> > picture into words better...
> >
> > [MP] I think that we talk about different fiber
> > spaces! I am talking about the fiber space structure of
> > the space of the infinite-dimensional space of 3-surfaces. You are
> > talking about finite dimensional space having W as base: this space
> > corresponds to M^4_+xCP_2 in my framework.
> Yes. Can we talk about the "connections" involved? Particularly, I
> would like to understand how "parallel transport" is modeled in your
> theory.

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.

b) In accordance with this parallel transport in spacetime surface
reduces to parallel transport in imbedding space. Measuring of
distances reduces to that in imbedding space. Etc...
Especially important induced quantity is CP2 Kahler form
whose projection on spacetime surface defines Maxwell field.

c) This means geometrization of classical
gravitational and classical gauge fields.
Metric, the vector formed by components of spinor connection
of imbedding space, etc are projected to corresponding
tensor quantities defined on spacetime surface.
Note that since shadow is created by projection map,
dynamics of shadows is indeed in question in precise sense!

d) Thus everything is classical mathematics: even induction
procedure which is standard mechanism of building new bundles
by mapping manifolds to base spaces of bundles but not noticed
by say string model people.

At the level of configuration space the metric is also inherited
from that of the imbedding space in well defined sense but this
is rather technical. Suffice it to say that basic idea
is to generalize the notion of the symmetric space to
infinite-dimensional context so that symmetries fix the metric

> snip
> > [MP]
> > OK. I think that we were indeed thinking about different things.
> > Configuration space is the space in question: it presumably does not
> > have any counterpart in LS framework because X^4/W is purely classical
> > so that one does not have superpositions of parallel W:s.
> > Configuration space in LS context would be the space of all possible
> The postulation of a configuration space seems to presuppose an
> ordering and a basis for observations prior to the act of the particular
> observation itself. This is very similar to the Minkowskian notion that
> there exists a single space-time manifold that just "exists". The
> serious problem that I have with this notion is that it presupposes that
> their exists a Cauchy hypersurface of positions and momenta with
> definite a priori status.

The mere existence fixes that which exist essentially uniquely
in infinite-dimensional context: this is the basic idea. One cannot
construct dynamics for the metric of infinite-dimensional configuration
space because it is fixed already by the requirement that it
exists! The existence of Cauchy hypersurface poses problems when
one assumes dynamical spacetime and tries to quantize metric.
Now imbedding space is given and these problems disappear.

The point is that I do *not* identify observations as points of
spacetime or configuration space. They are not active 'events'.
Quantum jumps between quantum states= quantum histories are events and one
cannot localize them to anywhere (one can of course, identify
these events as pairs of possible initial and final quantum histories
so that one can speak about the space of all possible experiences).

I see no problems with Heisenberg's uncertainty relations: informational
time development operator U reduces at QM limit to Schrodinger equation.
Metric, etc.. classical gauge fields are *not* quantized in TGD.
Neither spacetime coordinates are quantized. There is
absolutely no quantization, only classical geometry of
infinite-dimensional configuration space and classical spinor fields of
configuration space. Oscillator operator algebras etc are geometrized in
this approach.

One must distinguish between coordinates of spacetime which are
completely classical quantities and position and momenta of particle
in spacetime, which are operators in wave mechanics. They are
quite different things. Quantum group people have tried to
make spacetime coordinates operators but it has not led to successes:
I just saw a paper in which it was shown that divergence problem is
not solved by noncommutatitivy of the spacetime coordinates. There is also
problem with the loss of general coordinate invariance. One must assume
special coordinates and very high symmetries if one wants
special coordinates.

> This is completely contradicted by the
> Heisenberg's Uncertainty relation. The interpretation of this aspect of
> QM is the most troublesome for physicists! What I see is that the very
> notion of a configuration space has serious problems! If we are going to
> insist on using them then we much at least make their status
> statistical, so we talk about statistical distributions of properties
> given such and such conditions. This idea seems to be the best aspect of
> the Multiple World interpretation of QM.
> My problem is that I see an unanswered question being swept under
> proverbial rug: What makes the world run? We can't just postulate that a
> God or Singularity "started" the world! Why do we experience time in the
> first place? Why has everything not already happened?

God of Singularity concept is based on traditional concept of
psychological time. Also the question why everything has not already
happened is created by the same concept of psychological time.

I regard this concept as badly wrong. In TGD framework subjective time
corresponds to quantum jumps and there can be *no first quantum jump*.
This requirement plus p-adic evolution
as gradual statistical increase of p-adic prime of the universe
immediately leads to the requirement that
also infinite p-adic primes are possible and that recent universe
must correspond to infinite prime. Every moment of consciousness
decomposes to infinite number of subexperiences with values of
psychological time ranging from zero to infinity. What we really 'know' is
that local arrow of psychological
time exists: if one is satisfied with this then paradoxes disappear.
Universe becomes *4-dimensional* living being getting conscious
information about its entire 4-dimensional body in every quantum jump.
Cognitive spacetime sheets are the sensory organs of this infinitely
large 4-dimensional living system.

> snip
> > [SPK]
> > This
> > notion is very different from Hitoshi's idea, but perhaps the
> > is due to the different ways that time is treated.
> > I still see these as complementary! You see space-times as a priori
> > surfaces, subsets of the totality U that are connected by quantum
> > "in time", Hitoshi, as I understand, sees space-times as the "clocked"
> > poset of observations of LS, which are a priori quantum mechanical
> > systems existing tenselessly as subsets of the totality U.
> > Thus you are proposing space-times as a priori and Hitoshi
> > proposes quantum local systems as a priori, this is a chicken-egg
> > complementarity! We need to see that this is just a matter of
> > perspective!

I have the feeling that this is not a matter of perspective. Our
basic philosophies are different.

> >
> > [MP]
> > Hitoshi assumes fixed spacetime which is classical
> > and satisfies field equations of GRT and puts the quantum dynamics
> > to the fiber R^6. In TGD the quantum state is superposition of
> > spacetimes since spacetime is made quantum dynamical. Localization in
> > zero modes however effectively makes dynamics by quantum jumps to
> > in zero modes: like Brownian motion.
> Umm, I did not get this thought from reading Hitoshi's papers! He
> points out that GRT field equations problematic and outlines a way out.
> His theory is incomplete in this sense, again as I see it. I do not
> suppose that a fixed space-time is necessary for LS theory, it gives us
> a way to model an alternative.

I meant that there is single spacetime, not a superposition of them (or

> I would like to discuss the basic notion that observations are the
> that the Universe realizes the existence of consciousness and that
> partial orderings of observations can, allowing for the group theoretic
> properties, generate space-time framings (subjective views) that can
> overlap (have configurations that are similar) such that the appearance
> of a single finite universe results.
> Can we put the details of p-adics aside and just talk about
> space-time's ontological status? Is it necessary and sufficient to
> assume that a single unique space-time exists? If so, how?

Let me put my recent view in nutshell:

a) My view is that one assume single unique *imbedding space*,
8-dimensional space M^4_+xCP_2. I have explained the reasons for this.
Configuration space
of 3-surfaces must allow metric with Riemann connection and finite Ricci
tensor and Ricci scalar. Also spinor structur is needed. This does not
leave many possibilities. Configuration space decomposes into
union of constant curvature spaces satisfyin Einstein equations:
these spaces are labelled by zero modes.

b) *Spacetime* are dynamical and not unique and quantum states
are superpositions of classical spacetime surface. The localization
in zero modes associated with every quantum jump localizes quantum
history to the set of spacetime surfaces which are identical in
macroscopic aspects characterized by zero modes.
This brings in also psychological time as center of mass coordinate of
cognitive spacetime sheet, which is zero mode too.
Each quantum jump changes the macroscopic characteristics of these
spacetimes in quantum superposition and they seem to naturally correspond
to our sensory information: localization in zero modes is quantum
measurement of zero modes and sensory experiences seem indeed to give
information about zero modes. Perception is quantum measurement in this



> Kindest regards,
> Stephen

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