# [time 468] Re: [time 464] Parallel transport, etc....

Stephen P. King (stephenk1@home.com)
Thu, 22 Jul 1999 16:16:47 -0400

Dear Matti,

I pasted your response to my last query at the end...

Matti Pitkanen wrote:
>
> Below continuation to the answer of earlier message.
>
> On Tue, 20 Jul 1999, Stephen P. King wrote:

> > > [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.
[SPK]
> > Yes. Can we talk about the "connections" involved? Particularly, I
> > would like to understand how "parallel transport" is modeled in your
> > theory.
[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.

This "parallel transport" notion is what Weyl made a careful analysis
of in his work and why he reasoned that Riemannian geometry needed to be
generalized! But, Weyl did not consider the possibility that multiple
space-times are needed to model multiple observers. He assumed, as do
most, that there is only one M^4 and its related fields/connections for
all...
Parallel transport is the conceptual tool to model motions of the
"shadows", but fact is that that which is projecting the shadows is not
a "ONE" in the ontological sense, it is the "MANY" possible finite
subsets of U. They project onto each other...
This is a very important point that becomes obvious in a discussion
about the difference between "actual" and "potential" infinities... What
is ontologically a priori is the Universe, which is an "Actual"
infinity, thus it is ONE. The dynamics of the shadows are "potential"
infinities, thus are "MANY", thus the uncertainty formalism that Hitoshi
has discovered and discussed in his papers connects the time of the LS
to its measure of uncertainty.
The idea that the Universe, as a "static, pre-given entity" is
non-dynamical is part of my thinking and that of Hitoshi and,
appearently yours. 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!

> 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.

The argument about "a priori orderings" that I have been trying to
communicate since the beginning of the Time List relates to this! The
Oneness aspect of the Universe is forever out of reach since it is
Everything simultaneously. It would have at least a multiple connected
topology - everything mapping isomorphically to everything else... I am
proposing that the subsets of U are simply-connected manifolds, up to
isomorphism in the classical limit of hbar = zero.
Here is a crude sketch of my idea about the applicability of Weyl's
geometry:
"The necessary and sufficient condition that a Weyl geometry may be
reduced to a Riemannian geometry is that a vector keep its original
length after transplantation along an arbitrary closed trajectory.
Indeed, the condition of such a length preservation is ...

(15.40) \surface integral_C of dl/l = \surface integral_C \PHI_\alpha
dx^\alpha = 0

[forgive my terrible ascii]

and it is well known that \PHI_\alpha|\beta - PHI_\beta|\alpha
\equivalent 0 is the necessary and sufficient condition for the
integrability requirement of (15.41) in simply connected regions."
[from Introduction to General Relativity by Adler, Bazin and Schiffer,
1975. pg. 496]
These "simply-connected regions" would have properties that are
represented by Maxwell field equations. These define the patterns of
causal connections that M^4's represent. They have the appearance of
being unique from the point of view of an observer O because they are
the framing within which an observation by O is possible. If we consider
for the sake of illustration the ability to observe this projection
process "from a divine point of view" we would see that each O (an LS)
has its own M^4 constructed piece by piece by mappings between its
internal propagator's (quantum mechanical) dynamic's configuration space
and points in the manifold W.
What is difficult to grasp is that the "points" that make up W are the
possible exteriors as centers of mass ("cm"). In other words, the
"exteriors" of LS are ontologically different from the "interiors"! The
former are facets of the "MANY" and thus appear "classical" while the
latter are finite facets of the "ONE" and are quantum mechanical! The
ONE and the MANY are co-inductive aspects of the Universe, they define
each other, but the "meaning" of such definition is only possible for CE
reasons as being "in time" and since each LS, as a unique facet of U,
has its own time and gauge scale, it constructs its own local consistent
"reality".

> 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!

Umm, I fail to see how it is necessary and sufficient to require that
spinoral properties be aspects of the ONE. I see them as constructed
dynamically in the act of observation in that they are aspect of the
framing. Thus I do not see them as a priori givens! Is it not enough to
show that the "shadow" is a construction of the mapping between LSs? Do
we really have to suppose that the "statue" exists "out there" prior to
the sculptor's work?! It "exists" only in the sense of being a
possibility! The metric or related properties are given by the act of
observation which is determined by the local logical entailment aspect
of the informorphism that is the interaction between LSs.
To be sure, I need to discuss the "compactness" notion that you
mentioned before with regards to the W(r)...

> 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.

Umm, I an missing something! :-( I say that each LS observes a
classical like world, thus can be approximated and modeled by classical
math, but we need to consider co-induction, not just classical
induction... Perhaps I am misunderstanding you!

> 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
> totally.

Sure, the particular symmetries "fix the metric", but I argue that this
fixing is a computational act, not an a priori synthetic! This is
metaphorically similar to the situation of a new-born baby; it has the
potential to become many different adults, what determines the actual
adult it becomes depends on both its interactions with the world
constrained by the consistency requirements of the world determined by
others that are interacting wherein. The World is not a priori given in
an ontological sense, it is co-indictively defined by the interactions
of its members, it is the intersection of its member's possibilities
that define it!
The way that M^4s are constructed "piece-by-piece" is well illustrated
in Penrose's discussion of quasi-crystals and it highlighted in by
Forbidden Symmetry paper:
http://members.home.net/stephenk1/Outlaw/forbiden.html

[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 W:s.
[SPK]
> > 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.
[MP]
> 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.

But how do we derive particular actual properties by merely evoking the
fact that they exist? Existence is "outside of time", it is tenseless.
The dynamics that define sharp properties only occur "in time", in fact,
I propose that in a subtle sense, they ARE time, but since each LS has
its own unique dynamics (defined by its internal QM propagator), each
has their own time and thus their own length scale or gauge. Thus I say
that each LS constructs its space-time M^4 framings by interaction with
other LSs.
The Cauchy problem holds in the static case since the Uncertainty
principle prohibits the a priori definite "sharp" properties! It is for
this reason that Einstein was very opposed to QM! His statement "GOD
does not play with dice" [to determine the properties of the world]
speaks directly to this!

> 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 say that observations are co-inductively related posets of points (as
an abstraction since true infinitesimal points can not be distiguised
from each other as they can not encode any information!!!), not
individual points per say. To call them "space-time framings" speaks to
the fact that we always make observations in terms of a M^4 frame... The
quantum jumps are more epiphenomena that objective, but the restriction
that observers can only communicate effectively about M^4 framings that
do not logically contradict each other follows from CE! The
schizophrenic is an example of an observer that is attempting to
communicate about a M^4 framing that is logically inconsistent with
another's!

> 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.

I see the evolution of information in terms of the evolution of the LSs
as they interact. Thus the act of the Universe 'experiencing itself' is
an ongoing process. It is what "concurrent computation" is all about is
a fundamental sense! Again, this is why I find Peter's work so useful!
:-)

> 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 hope to discuss this when you receive the copy of Schommers' paper...

> 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.

Could you give me the reference of this paper? I can order a copy from
the library! :-) The loss of "general coordinate invariance" does not
bother me at all, in fact, I say that it is an illusion that results
from the fact that LSs can not communicate if they have different
histories. I say: LSs can interact only to the degree that their
histories are not inconsistent with each other. If the think of the LS's
histories in terms of M^4s, we see that to be consistent is for there be
isomorphisms between at least one point and that the "Hamming distance"
between points in the the Powersets of the LSs observations is
equivalent to the degree of mutual consistency or "agreement" between
the LSs.

[SPK]
> > 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 the
> > 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?
[MP]
> 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.

Yes, but here the error is in the tacit (subconscious) assumption that
the psychological time of one person is one and the same of that of
another! The clocking by the QM propagator of the LS defines the
individual time of the LS, thus representing psychological time very
well!
The "everything has not already happened" notion is indeed related to
psychological time, but in the sense stated above. "Everything" much
included all possible "actual" experiences, and obviously, these involve
NP-complete computational issues! This later notion is at the heart of
my argument.

> I regard this concept as badly wrong. In TGD framework subjective time
> corresponds to quantum jumps and there can be *no first quantum jump*.

I am not communicating my notion since your are not aware of the
NP-completeness problem! Karl Svozil's papers point the way! We agree
that "there can be *no first quantum jump*"! The ideas of co-induction
and related issues involved are in Peter's papers...

> This requirement plus p-adic evolution
> 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.

universe" is given in my thinking in the sense that the overall
concurrent interactions of LSs are modelable in this way. It would be a
representation of the accumulated "experience of the Universe", but with
the caveat that it is information that can not be gotten in finite time
or finite energy with arbitrary accuracy. The operator formalisms that
Schommers talks about, I believe, is useful to us in thinking about
this!

> > > [SPK]
> > > This notion is very different from Hitoshi's idea, but perhaps the
> > > difference 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 jumps
> > > "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!
[MP]
> I have the feeling that this is not a matter of perspective. Our
> basic philosophies are different.

In a way, yes. You are a Platonist, and I something different. I think
that Plato's Idea Reality *is* the Universe in-itself, but as Kant
argued well, is not knowable in-itself. I see it as Existence itself!
All experiences, measurements, observations, qualia or what ever, are
not given directly by their mere existence (as all exist in the
ontological sense), but have finite properties given by the interactions
between the finite subsets or "facets" or LSs of the Universe. We can
only observe shadow, consciousness is not capable of knowling the
"in-itself"!

> > > [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 classical
> > > spacetimes since spacetime is made quantum dynamical. Localization in
> > > zero modes however effectively makes dynamics by quantum jumps to hopping
> > > in zero modes: like Brownian motion.
[SPK]
> > 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
> 3-geometries).

How could this be? The very definition of such requires the a priori
computation of the causal structure of such a space-time! Newton et al,
assumed that God did this "in the beginning" and thus they give us a
"clock-work" paradigm of the world. I am speaking to a paradigm that is
more like a network of CGI (computer graphic interface) that is
constructed by the very act of interaction, thus the "means" of
communicating between LSs is given simultaneously with the "ability" to
communicate! There is no "absolute" ex nihilo beginning and no absolute
pre-defined causal ordering and thus no pre-given M^4!
I am trying to model this by using a generalization of Weyl's ideas...

[SPK]
> > I would like to discuss the basic notion that observations are the way
> > 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?
[MP]
> 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.

Is there any relationship between "zero modes" and "null geodesics"?
Could there exist an infinity of almost disjoint hull hypersurfaces?

> 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.

Does this implicitly assume a unique set of measuring "tools" for all
possible observers? If so, how? I am very aware that the usual
explanation involves the notion that numbers "exist" a priori, but,
really, how can it be affirmed that something is knowable just by
postulating its existence? CE does not imply necessity of knowledge or
experienciability! In order to allow for the uniqueness of the
individual self, it is necessary and sufficient that there is at least
one attribute that can be unique to such. The mere CE postulate is not
enough, we need a dynamical difference. When one posits a pre-given
metric for all and assumes that all observer exist *in* the same space,
it is the postulation of a prison for the self from which there is no
escape.

> This brings in also psychological time as center of mass coordinate of
> cognitive spacetime sheet, which is zero mode too.

This seems to follow from Hitoshi's theory...

> 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
> sense.

Sure, I can agree with that. Each person's sensory experiences can be
explained a unique integration of many almost disjoint quantum
measurements, thus I think of posets... But what is troubling me is the
notion that the localization is somehow independent of what it is that
makes the poset different for each person. I think that if we start out
with the notion that each measurement is informative only in comparison
to some aspect of the measuring device, here a person, we could perhaps
avoid the trap of classically. The idea of a pre-existing classical
world or space-time in which all observers are "embedded" is the error,
but it is very hard to escape it! It must be done. We need a radical

Matti Pitkanen wrote:
>
> On Thu, 22 Jul 1999, Stephen P. King wrote:
>
> > Dear Matti,
> >
> > Just a quits question that comes to mind as I read you post on parallel
> > transport: What is the embedding space? What are its properties? Is it a
> > "static" entity? (You relate it to Plato's Ideal world) Most off all, is
> > it similar to the "Totality of Existence" in-itself?
>
> Sorry for not making clear what I meant with the concept.
> Embedding space refers to the 8-dimensional M^4_+ xCP_2, future lightgone
> times complex projective space of 4 real dimensions. Isometry group
> is Poincare times SU(3), color group, the symmetry group of strong
> interactions (quarks, gluons). CP_2 spinor connection has as its holonomy
> group (group of parallel translations around closed curve in given point)
> electroweak gauge group.

Again, I fail to see how you propose to claim that a unique M^4_+ xCp_2
*is* the TOTALITY OF EXISTENCE! These are definite actual properties and
as such can only be given within the context of particular experiences!
The Universe in itself can only *exist* by CE reasoning, the
construction of M^4_+ xCP_2 is a subjective projection; it is phenomena,
not noumena, e.g. I believe that the particular patterns of behavior
that is called "physics" is derived from observation, it is not priori
to it. This is the case presented by Frieden!
Is is the "behavioral form" that the "shadows" take that is given by
the interactions between the finite facets of the Universe. All we can
observe is the shadows and the "wall" upon which they think that they
are "projected" is just the Powerset of the set or equivalence class of
Again we are merely debating the semantics of our perspective model of
our worlds, and thus it is the usefulness of our notions that matters in
the long run... :-)

> Embedding space is stati, pregiven entity.
> Imbedding space provides only geometric frames for existence. It is
> only finite-dimensional, etc..
> Totatility of existence is the space of all configuration space
> spinor fields=quantum histories.

I agree with this only in the sense that all possible quantum histories
"exists", but the "ordering" of them, that makes consciousness "flow" is
not a priori! This is the basis of the computation argument!

> Best,
> MP

Onward!

Stephen

PS I am very tired mentally so my arguments may very well be just a lot
of noise! :-)

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