[time 455] Re: [time 454] Re: [time 450] Re: Fibering, manifolds and sums

Stephen P. King (stephenk1@home.com)
Sat, 17 Jul 1999 19:05:44 -0400

Dear Matti,

        We are communicating well!

Matti Pitkanen wrote:
> On Sat, 17 Jul 1999, Stephen P. King wrote:
> > Dear Matti,
> >
> > Matti Pitkanen wrote:
> > snip
> > > Important constraint for the fiber space is that it should explain
> > > as much as possible facts unexplained by standard physics.
> > > In Hitoshi's/your approach R^6/W^6 explains how local quantum mechanical systems
> > > combine with global general relativistic spacetime. Fiber abstracts the
> > > concept of local nonrelativistic quantum system. What troubles me
> > > in this approach is that every point of X^4 contains local system. Somehow
> > > only some fibers are 'active'. This same feature troubles me also
> > > in Bohm's theory. Only some classical orbits are 'activated' in the
> > > hydrodynamical flow defined by Schrodinger amplitude and correspond to
> > > classical particles.
> > Umm, some misunderstanding... I am proposing a Weyl geometry W for the
> > manifold into which the LSs are projected; Hitoshi uses a Riemannian
> > manifold X.
> [MP]
> Of course! You have general relativity
> with Riemann geometry replaced by Weyl geometry.

        Close... The local dynamics of GR are given by the local compositions
of LS projections into W. I am considering GR to be a model of the
"glocal" structure of extremal geodesics, but as I see it, we move from
the assumption of a single Riemannian manifold X to a supermanifold W
that has as submanifolds almost disjoint X_i. Each has a slightly
different inner product, maybe? This is an attempt to model how each
observer perceives itself to be in the center of a finite universe that
has a Robertson-Walker like metric.

> > The R^6 refers to the 3N + 3N Euclidean space of positions
> > and momenta that the LS propagator "lives" in. The breaking of scale
> > invariance of the W manifold, I think, is due to the way that the LS's
> > projections partition it into local logical consistent subspaces. This
> > has to do with the epsilon bound on the accuracy that LSs can predict
> > each other's behavior.
> > This is in the spirit that each LS has an associated M^4 that is
> > constructed from W by the act of projection. This projection is an
> > identification or mapping between the internal configurations of the LS
> > and a finite subset of W.
> There is probably some computationalistic motivation for this
> map.

        Yes! The computational aspect is the way that the dynamics of the
physical (material) aspects act as the "hardware" whose tendency toward
local equilibrium generates the solution of the Lagrangian of the
"next" state, the "software" is the information content. I say
poetically: "Guided by the future, powered by the past, I become"

> Does the map to finite subset of W mean that different
> particles in LS are mapped to different points of W, their positions?

        Yes, almost, the map to finite subset of W is a model of the "unity of
consciousness" of the LS. The quantum entanglements of the subsets
(quantum particles) of a given LS acts to unify the mapping. The
positions that are being mapped are the centers of mass of other LSs
that are being projected into W themselves. I wish I could draw a
picture of this here! The problem is that the image I have is not
reducible to 2 dimensions! :-(
        The notions of positions is a relative notion in my thinking. It is
perhaps good to think of a graph were the centers-of-mass of other LSs
are the nodes and the edges are the relative overlaps in W. In this way
we define the space aspects of the observation in terms of extensions
and the time aspects in terms of durations, given the particular
internal behavior of the LS propagator.
        Thus, if the LS is in a bound state, it has not time associated and all
extensions are null, e.g. LSs that are at internal equilibrium have no
space or time associated. The consequences of this when considering how
other LSs, in interaction with such, is interesting! It looks like the
situation where a worm hole, connecting separated space-time regions in
the classical GR picture, is pinched closed.

> > The way that the Weyl scale invariance
> > connects the spectra of particles and their histories, is important
> > since the structure of the LS's M^4 requires that no logical
> > inconsistensies are present. Each LS would have a M^4 that has a causal
> > structure that is, from the point of view of the LS in question,
> > logically consistent. The objection that is used against Weyl, I am
> > turning around! I say that observers do not see spectral smearing
> > because they can't see all possible histories of particle motions!
> > LS that have different quantum histories will be able to interact only
> > within the bounds of the intersection of their histories, e.g. those
> > part wherein they agree. This is a very relativistic notion since it
> > shows that the observations of LS depend on their associated quantum
> > histories. The divergence of the frequency of particles generates the
> > appearence that they are moving away from each other!
> >
> This looks like fiber bundle structure: different LS:s as regions
> of fiber bundle related by transition functions. One could not
> define uniquely single LS but would have some minimum number of
> LS:s, patches of the bundle. OK? But the mapping of LS to
> several points of W breaks this picture.

        The key notion is how the "connections" are defined! LS have no a
priori connections between them, this allows connections, which define
the forces between LS, to be dependent on the local histories of the
LSs, e.g. the information content of the LS's quantum history constrains
the mapping. The fiber thus look more like a tree, N to 1 map, than a
string, 1 to 1 map.

> Generalized fiber spaces with projection mapping fiber to
> several points? Could the number in image depend on base point and
> could one allow the image be empty set? In this manner one
> would avoid the counter argument about fiber space
> realization of LS. When image is empty there is no LS.

        Yes to all! :-) Also, we can reason that an empty LS maps to the whole
W, as the null or empty set. A singleton LS, e.g. an LS having a single
quantum particle maps to all singleton points of W.

> > > In my approach CP_2 geometrizes elementary particle numbers and classical
> > > gauge fields in spirit very much to that of Kaluza-Klein theories.
> > > Local system is now spacetime sheet. Cartesian product x in your
> > > and Hitoshi's approach is replaced by topological sum # of 3-surfaces
> > > representing local system and its complement (drill holes D^3 in
> > > LS and complement and connect resulting boundaries S^2 by a tube S^2
> > > xD^1).
> > Umm, your topological sum is more akin to the way that LSs are composed
> > from other LSs, not the way they fiber the base manifold X or W. The
> > worm holes S^2xD^1 that you propose connect the boundaries S^2, I see,
> > as a way of defining field lines in TGD. I remember the discussion in
> > MTW's Gravitation... :-) I think that this is a very fruitful notion! Do
> > you think of the relationship between the LS and its complement as
> > synonymous to the relationship between subject and object in an
> > observation?
> More or less but quite not as I realized just now.
> I try to explain.
> a) The decomposition of spacetime surface
> to cognitive spacetime sheets having *finite time duration* (I stress
> this since this is crucial) and material spacetime sheets has turned out
> instrumental for the model for self and binding.

        Umm... I still have trouble with this wording, but I can see how you
use it, I think. I see "cognitive space-time sheets" and "material
space-time sheets" as Stone-Birkhoff duals of each other, as Pratt
defines the latter. I see the ideas implicit in the concept of a
"space-time" as applicable to both information and matter, each has
extentional and durational attributes. The identification between the
two is modeled by the "informorphism": material event A causes material
event B if and only if the information content of B implies the
information content of A, thus A -> B iff A* <= B*; "<-" is material
causation and "<=" is logical entailment or implication.
        Other key differences are in the way the "physics" of the two work.
Material "bodies" are incompressible and rotatable, information "minds"
are compressible and irrotatable. Bart Kosko explains the latter
properties in his book Fuzzy Engineering...
> b) In quantum jumps only the entanglement between cognitive and material
> spacetime sheets can be reduced. This generalizes von Neumann's
> intuitions.

        This idea, to me, speaks to a generalization of Pratt's notions. The
transition or quantum jump from event A to event B is a process of
selecting the particular points of W that are dual to the information
content of the LS. Umm, I have been trying to work out an idea where the
points on the surface bounding an LS encode a n-ary bit of information,
but this is still very crude. I am using the ideas of Wheeler about the
information content of a Black-Hole's event horizon.
> c) Selves are pairs of material-cognitive
> spacetime sheets unentangled with the other
> selves and containing as nested subsystems lower level selfs:
> Russian dolls inside Russian dolls. Self is
> synonymous to observer.

        I agree with this. I believe that the notion of a homunculus existing
inside our head with a mini-humunculus inside of its head, etc. is
generated by this p-adic hierarchy of LS selves! :-)
> Thus LS would correspond to a pair of matterlike and mindlike subsystems
> rather than single spacetime sheet.

        Yes! I think of these in terms of Pratt's BODY and MIND, as defined in
ratmech.ps... An LS is a CHU space.
> By the way, this picture works: I just worked out
> a beatiful model for what happens when we wake-up or fall
> asleep, or get older and eventually die. The basic phenemenology is
> reproduced beatifully and one can even say definite things about what
> happens in death. The Buddhist view about gradual evolution of
> self to higher and higher levels of subjective existence seems to be the
> only reasonable conclusion. Amazing!
        Umm, I am skeptical of the notion of the literal transmigration of
souls, since it assumes that the awareness of self is continuous a
priori and the idea that the self is a "rigid object" that "move in
space-time". This contradicts the basic notion of quantum jumps. I am
claiming that the notion of rigid bodies is at best an illusion!
        The unity of "flow of consciousness" is given by the unitary evolution
of entanglements of the material particles that make up a mortal body.
When a body dies, the entanglements decohere and so does the unity of
consciousness that is associated. This idea looks very bleak, I know!
But I found another mechanism that models most paranormal phenomena as
well as the notion of transmigration. We need to consider that quantum
entanglements "connect" subsets of LSs that can have very different
space-times associated, thus we can are resonances in the information
content. Carl Jung's discussion of "synchronisities" is relevant!
> > > The problem why the universe of conscious experience looks classical while
> > > quantum universe is nonclassical, has bothered also me. For long time I
> > > thought that the association X^3--> X^4(X^3) forced by
> > > 4-dimensional General Coordinate Invariance might be all that is needed
> > > to understand this but I was wrong. The hypothesis that quantum jumps
> > > correspond to quantum measurents, which are local at the level of
> > > configuration space of 3-surfaces implies
> > > localization of configuration space spinor fields in zero modes: this
> > > means that moment of cs makes the world essentially classical.
> > I, unfortunately, do not follow all of what you are saying here. :(
> > Could you elaborate on what "localization of configuration space spinor
> > fields in zero modes" means?
> a) Configuration space of 3-surfaces has fiber space structure.
> Fiber corresponds to cm degrees of freedom and vibrational degrees
> of freedom of 3-surface: 'vibrational' motions do not change
> the macroscopic shape and size nor classical Kahler field of
> 3-surface.

        What is "cm"? Have you read about the resonances of a fractal surface?
I have a book that discusses this. It is possible that the boundary of
an LS is a fractal in most cases! The notion of tree fibers points to

> Metric of configuration space is nontrivial in
> fiber degrees of freedom and quantum fluctuations occur in
> these degrees of freedom. These are the degrees of freedom of
> ordinary QFT.

        How is the metric generalize to our tree-like fibers? The "leaves" of
the tree-fibers are LSs and the truck roots into a non-zero region of 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 difference
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 my
picture into words better...
> b) Base corresponds to zero modes in which configuration
> space line element vanishes: there is however symplectic form
> so that integral can be defined. Zero modes characterize
> shape size and classical Kahler field (which often reduces
> to classical em field) of 3-surface. These degrees of
> freedom are new and result from generalization of elementary
> particle concept. They are classical degrees of freedom:
> all that which we see around us and in terms of them we
> formulate classical physics.

        Umm, this seems to follow the "classical" notions. We need the
generalizations were there is not absolute space-time! The illusion of
classical emerges as a weighted averaging over many such space-times in
> c) In quantum jump the state must go to an unentangled
> state. The state can be expressed formally as quantum superposition
> of 3-surfaces

> SUM(X^3) C_nN (X^3)|n>|N>

> where summation symbolizes sum over 3-surfaces and
> n and N denote labels for the quantum states in fiber:
> these states correspond to quantum states of
> ordinary QFT. SUM(X^3) is not present in QFT since
> X^3 is not dynamical.

        Yes. The dynamics involve the changes of the X^3s which I see as finite
partitions of W. The topology changes abruptly!
> c) Locality of NMP in configuration space requires that
> quantum jump must reduce the entanglement in local manner.
> This suggests that localization to single X^3 must occur:
> otherwise one simply cannot get unentangled product state.
> This would mean that each quantum jump leads to state
> localized in single 3-surface and spacetime.

        Yes, but I say that this occurs for each LS... The overlap of fibers in
W creates the appearance of a single space-time with a Robertson-Walker
metric, e.g. with a Hubble expansion related to the local "age" of

> This would be extremely classical but
> cannot be true since various symmetries do not commute
> with fiber degrees of freedom and resulting
> state could not be eigenstate of say momentum.

        Could we deal with this? What if we allowed for multiple states and
think of eigenstates as not strictly orthogonal for all fibers. Remember
that the ordinary mathematics of QFT assumes binary complete knowledge
and it is obvious that such is an idealization...
> d) Symmetries however act
> in fiber and since they are gauge symmetries one
> must require that entanglement coefficients C_nN depend
> *only on zero modes* but not on fiber degrees of freedom.
> This means that localization in zero modes is all that
> is needed. And this indeed makes the universe of
> conscious experience classical!
        I keep reading this and it makes no sense to me as a whole. :-( I only
understand pieces...
> In each quantum jump the state UPsi_i generated
> form Psi_i is reduced to state Psi_f localized completely
> in zero modes. Superposition of 3-surfaces which
> differ from each other only in vibrational and rotational (in particular
> color-rotational) degrees of freedom and
> are macroscopically identical, is created.

        I think I understand! :-)

> Informational time development U makes universe classical
> but moment of consciousness makes it classical again.
> When Djinn comes out of the bottle universe becomes
> nonclassical: when it returns to bottle the universe
> becomes classical again

        Umm, is this saying that during the transition from classical
space-time to another, the universe is quantum mechanical? This allows
space-times with different global topologies to be connected? 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
> > It seems to me that GCI notion is built on
> > the notion that all observers live in one and the same space-time, thus
> > the need to transform a single set of physical laws, like a rigid
> > structure, so that all would obey it. If we instead consider that
> > observers can only interact to the degree that their own set of physical
> > laws agree, we can avoid the problems that GCI has with QT!
> [MP]
> Yes. I see the idea.



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