# [time 304] Re: [time 303] Mapping p-adic spacetime to its real counterpart

Stephen P. King (stephenk1@home.com)
Tue, 11 May 1999 10:00:46 -0400

Dear Matti,

Matti Pitkanen wrote:
[SPK]
> To say the truth, I am in awe of your knowledge of the math
> involved! I am happy that critique is useful. :) I have a few silly questions...
>
> Matti Pitkanen wrote:
> >
> > I have been pondering more and more seriously the problem of mapping real
> > spacetime surface and imbedding space to their p-adic counterparts.
> > I proposed already earlier a possible solution of problem but it
> > was not satisfactory: thanks for Stephen for critical comments.
> > The basic problem that canonical identification mapping real coordinates
> > to their p-adic counterparts is not manifestly General Coordinate
> > Invariant concept and one should be able to identify preferred
> > coordinates of imbedding space, where canonical identification applies
> > in some form, in order to achieve GCI.
[SPK]
> Is it possible that there exist a class of pairs {M_i, I_p}, where
> M represents the spacetime [hyper?]surface and I_p represents the p-adic
> imbedding space, such that there is an asymptotic hierarchy of
> inclusions of the {M_i, I_p} that, at the limit of +/- \inf, there is
> isomorphism between M_i and I_p and a dismorphisms for any i, p > \inf.?
> I hope this makes sense! ;)
>
> [MP]
> I am not sure whether you regard M_i as real or p-adic manifold and
> I could not quite understand what you mean by asymptotic hierarchy of
> inclusions. Certainly this kind of sequence might exists but since
> I do not know about motivations for the existence of sequence I cannot
> imagine any concrete example.

I was trying to think of M_i as a real (analytic) manifold. By the
"asymptotic hierarchy of inclusions" I was thinking of how the p-adic
manifolds would contain as subsets other p-adic manifolds. I find the
means to map the reals into this nesting structure to cause problems
with the usual topology assumptions of the real manifold. I will try to
be more explicit: When we map a subset of R to another subset of R the
usual assumption seems to involve the Borel property, which, frankly, I
don't understand well :(. So I will use a simple example from elementary
algebra. We think of an open set of points from R^1 as a line segment
without the endpoints, conventional topology seems to be built from R^n
versions of there. Ok, if we were to take an arbitrary sized open set
and try to map it to a subset of a p-adic manifold, how would we know
that it "fit" exactly and uniquely? more on this below...

[MP]
> What I am considering is basically the problem of mapping real manifold
> M_R to its p-adic counterpart M_p (in present case M is the 8-dimensional
> space H where spacetimes are 4-surfaces). The idea is that this mapping
> induces the mapping of submanifolds of M_R (spacetime surfaces)
> to submanifolds of M_p somehow.

That is the relationship (explicitly!) between the 4-surface subsets(?)
of M_R?

[MP]
> > First some general comments on frames of reference question
> > and then a brief description of how the concept of preferred
> > frame appears as a purely technical concept in the formulation
> > of quantum TGD.
> >
> > General Coordinate Invariance
> >
> > The Principle of General Coordinate Invariance states that the
> > laws of physics cannot depend on frame of reference. A slightly
> > different formulation says that diffeomorphisms of spacetime do
> > not represent genuine physical degrees of freedom. You do not get
> > new physical configuration by mapping various tensor quantities
> > describing physical fields to their diffeomorphs.
> > This means that GCI as a symmetry
> > is like gauge invariance: there are not conserved quantum numbers
> > associated with infinitesimal general coordinate transformations.

Is it absolutely necessary to assume that quantum number conservation
is "external" to possible local observations? Would it not suffice to
show that for any sequence of observations is constructed such that in
any given transition in the sequence the total number of quantum numbers
is constant. This is a subtle but important difference. It shifts us
from a priori postulates to local logical consistency derivations.
Weyl's original gauge theory made this explicit, but the lack of
imagination of other physicists at the time prevented proper
understanding. I have read and re-read Pauli and Einstein's critique of
"spectral indefiniteness" and Weyl's response of "unobservability". I
tell you that Weyl is correct! If we consider that each observer has its
own unique set of measuring tools, instead of some a priori egalitarian
handout from above, each observer would perceive a discrete spectra
-*relative to their local gauge*!
Thus the "smearing" of spectra from a so-called unique discrete set to
a continuous one *would only be observable* to some ideal observer that
has the gauges of all others as explicit subsets of its own local gauge.

[SPK]
> The use of infinitesimal (under any circumstance!) is suspicious
> to me, since it tacitly assumes zero error (no uncertainty) observations.
>
> [MP]
> Infinitesimal is convenient physics shorthand: there is
> rigorous group theory behind this all. One defines
> conserved charges as variational derivatives of action with respect
> to the action of various abelian subgroups generated by Lie-algebra
> elements of symmetry group. For instance, angular momentum in
> specific direction corresponds to the action of rotation around
> the direction of angular momentum in that direction.

But is it not true that the Lie groups, for example, assume
infinitesimal changes x -> x within points \subset S^1? or what ever
analytic manifold that is used...
What does "specific direction" mean? Do we assume a unique basis of
independent vectors for all possible measurements of angular momentum? I
am reminded of the thought: "In space you can't tell up from down..." Is
this "specific direction" assuming a preferred frame for all?

> [Stephen]
> I believe that these are *very* special cases, and need to be better
> understood! One thing that infinitesimals do is that they create the
> illusion that all observations use the exact same measuring 'rod'. This
> may not be the case. I have been trying to explore Weyl's gauge theory
> to understand how we can do physics when each observer has its own
> unique measuring rod and clock, as opposed to assuming that an absolute
> standard is imposed "from above"!
>
> [MP]
> Noether theorem makes the symmetry thinking of physicists mathematically
> rigorous. The concept of Noether charge as completely independent
> of any measurement theory: it is purely group theoretical concept:
> infinitesimals are only bad linguistic habit of physicists (or who
> knows!?). This group theoretical aspect becomes decisive in quantum
> mechanics which to large extend reduces to a representation theory for
> symmetry groups.

This rigor can only come at a high price of ideality! Umm, I do need to
understand what is being written about Noeter charge, could you give me
an elaboration on it or a good explicit (with examples) reference? :)

> My personal belief that the concept of measurement comes into play only
> at the level of quantum measurement theory, which is the poorly
> understood part of quantum mechanics and involves
> the concept of quantum jump and basically consciousness. On the other
> hand, Riemannian geometry is classical theory of length and angle
> measurement, and I take it as God given and go on to postulate that entire
> QM with quantum jump excluded is just infinite-dimensional Riemannian
> geometry. I could be wrong!!
> Somehow I however believe that Riemannian geometry is something final.

Should we not consider a geometry with a maximum amount of possible
properties to be the most "primitive"? This is why I think Weyl's
geometry is more primitive that Riemann's as the former contains the
latter as a special case! The point you make: "Riemannian geometry is
classical theory of length and angle
measurement, and I take it as God given..." is what bothers me. :( I am
reminded of Lee Smolin's discussion of "ideal elements" in his paper
"Space and Time in the Quantum Universe" found in "COnceptual Problems
of Quantum Gravity" A. Ashtekar & J. Stachel eds. Birkhauser QC 178.C63
1991, pg. 228-288
Smolin discusses the Principle of Sufficient Reason of Leibnitz: "...in
a complete theory of the universe, every question of the form: 'why is
the world this way rather than that way?' must have an answer." (pg.224
ibid.) He goes on to show that this implies that any measurement (or
answer to such questions!) must be done relative to some subset (my
paraphrasing) of the whole and not some "external unobservable", such as
Newton's God Clock and Ruler and Ordering. These infinitesimals are an
explicit example of Newton's attempt to eliminate the finite subjective
observer and impose an absolute apartheid upon the Universe. Suffice it
to say, he is wrong! The subjective view is *all* that is accessible to
any, any observer! We must not forget, ever!

[MP]
> I have make little quantitative progress... :(

YOu are not alone, The above mentioned book is full of despondent
physicists!

[MP]
> > This is in fact leads to the basic conceptual problem of General
> > Relativity: one does not have any GCI definition of energy and momenta
> > since Noether theorem gives identically vanishing conserved diffeo
> > charges.
[SPK]
> One thing that I have always wondered about Noeter's theorems,
> which relate conservations to symmetries, is that the symmetries are
> always considered using "time" as a parameter; but it is a "time" that I
> would call exactly "periodic". Ben discusses a spiral/fractal time in
> http://goertzel.org/ben/timepap.html that IMHO is possibly more
> realistic. We must remember that the Noeter theorems are phrased in
> classical thinking, and as such are ideal.
>
> [MP]
> What Noether theorem says that variation of action for a volume of
> space vanishes and by equations of motion the variation reduces to
> a surface integral. What happens under suitable
> additional assumption is that the surface integral reduces
> to contributions from two time=constant surfaces and these contributions
> must cancel: this says that classical charges at these t=constant
> surfaces are identical: charge is conserved. Metric signature
> does not matter: only the idea that there is cylinder like region such
> that charges do not flow out from the sides of the cylinder.

"Infinitesimal variation of action"? Action = ? "Integration" with what
measure? I think that your "additional assumptions" are correct! It
localizes and normalizes the measure to "between" synchronous observers
only. We can define space-like hypersurfaces relative to these two
time=constant surfaces. But notice what happens if we parametrize the
norm to the phase difference between observer "motions"! As observers
become asynchronous in their motions, their norms diverge from each
other until they can no longer perceive each other! This is the essence
of the bisimulation model of interaction (communication).

[MP]
> Since Noether theorem relies on group theory, one can generalize
> the concept of Noether charge straightforwardly to quantum theory context.
> Formulas are the same: now Noether charges become Hermitian operators
> whose real eigenvalues correspond to quantum mechanical charges, quantum
> numbers such as spin and momentum.

Yes, but the problems of the diffeomorphisms remains for QGR! The inner
product goes from unique to smeared over all possible. It is interesting
that in p-adic math, the metric is glocal, since it is defined in terms
of a prime number, and since there are \inf of them, we have \inf
different metrics, but they are disjoint! (I think ;) ) "Ultrametric
balls either have each other as subsets or are mutually disjoint."

[MP]
> > Most importantly: it does not make sense to speak about 'active
> > diffeomorphisms'. One can however speak of
> > isometries of spacetime as symmetries: in this case the action to
> > fields is different: one can say that fields are replaced
> > with general coordinate transformed counterparts but *coordinate system
> > is not changed*. This transformation creates genuinely new field
> > configuration and in case of isometries of spacetime. This new field
> > configuration solves the field equations.
[SPK]
> One question, how would an observer *know* that their fields have
> changed if their tools of measurement change also? As we consider a
> transformation of fields, we must understand that the observer is *not*
> independent of the transformations, as would the classical "external
> observer".
> [MP]
> You are certainly right. I see however this question as a problem of
> quantum measurement theory or basically problem of consciousness theory.
> The classical theory of fields (absolute minimization of Kahler action in
> TGD) as well as 'Schrodinger equation' are completely observer dependent:
> there are no observers or observations in classical world nor in single
> quantum mechanical time evolution/quantum history (so I believe). Quantum
> histories are dead: life and observations are in quantum jumps between
> these quantum evolutions. To answer your questions is indeed a great
> challenge but a challenge to consciousness theory, which we should
> construct first(;-).

That is what we are discussing! ;) Whether we assume QM or GR to be the
primitive and the other contingent is a matter of taste. Either method
works, maybe. Hitoshi's model casts QM as the primitive and GR as
contingent, you seem to reverse this. But the final predictions should
be equivalent. :) I am reminded of how Schroedinger and Heisenberg's
formalisms for QM show this same kind of duality! :)
Your point that "Quantum histories are dead", to me illustrates how,
from the GR view, Quantum histories" are static bound states spread
across space-like hypersurfaces, and have no absolute "extension" in the
time direction! I agree! Your point that "life and observations are in
quantum jumps between these quantum evolutions" I also agrees with! How
we map features on one such surface to another is our main question, as
we consider the model of QGR from your perspective. So, I believe that,
understanding the logic of consciousness is equivalent to the logic of
mapping the features of these surfaces to each other!

[MP]
> With the risk of repeating myself: there is no observer in the sense
> of continuous stream of consciousness residing in some corner
> of spacetime or floating above the Hilbert space.
> Observer exists only in the quantum jumps, moments of consciousness.
> Therefore one cannot say that there is any observer subject to these
> transformations: the problem disappears.

I agree completely! :) But, how one "quantum jumps" can encode/decode
information that is bisimilar to that of another is, IMHO, the key to
consciousness.

> > Of course, in practice one must almost always solve field equations in
> > some frame of reference typically fixed to high degree by symmetry
> > considerations. This does not mean breaking of GCI but only finding
> > the coordinates in which things look simple.
> > For Robertson-Walker cosmology standard coordinates (t,r, theta, phi)
> > are special in the sense that t= constant snapshots
> > correspond to the orbits of Lorentz group SO(3,1)
> > acting as isometries of this cosmology. t= constant snapshots
> > are coset spaces SO(3,1)/SO(3) originally
> > discovered by Lobatchewski and identical
> > with proper time constant hyperboloids of future lightcone of
> > Minkowski space.
[SPK]
> I must confess that I do not fully understand the meaning of the
> group symbols, but I am beginning to! :) Thank you Matti! ;)
[MP]
> > RW coordinates are *NOT UNIQUE*. For subcritical cosmology,
> > any Lorentz transformation generates new equally good
> > RW coordinates with different origin interpretable as position of
> > comoving observed! The cosmic time t is Lorentz invariant under Lorentz
> > transformations and is not changed.
[SPK]
> I think that Hitoshi's use of the RW metric to talk about the
> expansion of an observer's space-time applies here! We must remember that
> each observer, at each moment, is using a time origin unique to the
> individual LS, which is the observer. Thus, when we think of modeling
> the observations of co-moving observers, we are inferring from our own
> time origin point.
> [MP]
> You can interpret different Lorentz transformation related frames
> as associated with various comoving 'observers', yes. There however still
> remains rotational degeneracy of the frame (rotation group SO(3)): this
> is the problem from my point of view in the sequel.

A friend of mine (mentioned below) is working on this! From what I can
remember of our discussions, the problem results from the mutual
observations of motions given more that 2 observers. In the 2 observer
case, we have a nice reflection like symmetry, e.g. I see you move XOR
you see me move. (This is an example of my thought on subject object
symmetry, HItoshi)
But when we introduce a third observer, we break the symmetry, since it
may be synchronously co-moving with either of the original pair and the
possible observations of the 4-vectors are different! I am strongly
encouraging him to write this up for us, but he is very busy with his
family. He, like me, is not working in a university.

> [Stephen]
> A local friend and I have been exploring the implications of
> Lorentz
> transformations, and have arrived at the conclusion that such are
> restricted to the possible inferences of a single observer and can not
> be assumed to well-model the actual observations of other LSs. In other
> words, the Lorentz invariance of possible observations is a group that
> each observer has, and there is no necessary isomorphism between the
> SO(3,1) of one LS's observations and another's. All that is required for
> consistency is that there is the possibility of mutual entropy in the
> information that can be encoded in the SO(3,1) of each.
>
> [MP]
> This would be very nearly equivalent with the viewpoint of General
> Relativity. SO(3,1) is only the group of tangent space rotations
> preserving tangent space inner product and physically corresponds to
> approximate Lorentz invariance of the spacetime locally. This symmetry
> is however gauge symmetry and does not give rise to conserved charges such
> as angular momentum and is hence problematic.
>
> In TGD the big idea is that Lorentz invariance is actual global symmetry
> of the imbedding spaceH=M^4_+xCP_2, rather than spacetime itself.
> Poincare invariance is broken only by the presence of lightcone
> boundaries. Immediate prediction is standard subcritical RW cosmology.
>
> I realize that we have quite different view about observer concept. You
> assume that observer is modellable mathematically at fundamental level
> whereas I throw the observer out and leave only conscious observations
> associated with quantum jumps replacing physical time evolution
> with a new one so that also the hypothesis about single objective
> reality is thrown out. With so much thrown out also many problems
> disappear(;-). I hope that this what I am doing is not like solving
> the problem of consciousness by saying that there is no consciousness.

In Hitoshi's model, we say that there is not Time! :) These model are
complementary! We can use either the idea of consciousness XOR time as a
primitive. I think I am saying this right, please correct me other wise.
:)

> [Stephen]
> Thus we say that that rock is at such and such a position iff each
> observer involved has information encoded is a similar enough manner. I
> hope to have some more quantitative formulation of this soon! :)
>
[MP]
> > Mapping problem and preferred frames
> >
> > In TGD framework the problem of preferred frame (in purely technical
> > sense, not physically) has been one of the longstanding problems in the
> > attempts to understand the relationship between real and p-adic quantum
> > TGD (which are actually different aspects of one and same quantum TGD).
> > The problem boils down to the following mapping problem:
> >
> > ***How is real spacetime/imbedding space/configuration space of
> > 3-surfaces/space of configuration space spinor spinor fields mapped to
> > its p-adic counterpart?***
> >
> > Some form of canonical identification between real and p-adic imbedding
> > space coordinates must somehow mediate this mapping but how does it occur
> > precisely. Canonical identification mapping reals to p-adics
> > is given by
> >
> > x_R = SUM_n x_np^n --> SUM x_n p^(-n)= x_p .
[SPK]
> Is it true that the mapping can have duplication (overlap and
> underlap)? I am thinking that the equation above does not rule out the
> possibility of mapping the "same" pattern in the reals onto more than
> one p-adic set.
> [MP]
> Canonical identification from reals to p-adics is two-valued for
> reals having finite pinary digits. The reason is that in this case
> the pinary expansion of real is not unique (good example: 1= .9999999..
> which generalizes to expansions in powers of prime p instead of 10
> easily). One can however use systematically the expansion with
> finite pinary digits: this is also numerically the only possibility.

This is a point that we need to discuss separately! Some would say that
1 = .99999... to some degree equal to (1-.9999...)!

> I should mention also a second problem related to canonical
> identification: how to map negative real numbers to their p-adic
> counterparts. Just now I believe that the requirement (-x)_p = -x_p so
> that negative of real number is mapped to the negative of its p-adic
> image, is the correct option. It follows from the mapping of complex
> numbers to their p-adic counterparts described in following by restricting
> the map to real axis.

Umm, this is interesting! :) We are working out the local relative
identity, by showing the inverse... Umm, very cool!

> > The problem is that canonical identification is *NOT MANIFESTLY GCI*
> > since it must be defined in PREFERRED COORDINATES!
> > For instance, if one goes to new coordinates the p-adic image of
> > the new spacetime surface is not identical with the original one.
> > GC transformations do NOT commute with canonical identification map!
> >
> > How to find preferred coordinates for imbedding space?
> >
> > In order to achieve GCI, one must be able to find some PREFERRED
> > COORDINATES for imbedding space H=M^4_+xCP_2, in which canonical
> > identification map is performed. If the preferred coordinates are
> > unique, everything is ok. If NOT, then the coordinate transformations
> > between preferred coordinate systems must COMMUTE with the canonical
> > identification map.
> >
> > a) If imbedding space were not nondynamical, no preferred coordinates
> > would exist and p-adic quantum TGD would break GCI. This would be
> > the end of p-adic TGD.
> >
> > b) Fortunately, imbedding space H(=M^4_+xCP_2) is NOT dynamical
> > but fixed by symmetry considerations and by the requirement that
> > configuration space geometry exists mathematically. The coordinates,
> > which transform *linearly under maximal subgroup* of SO(3,1)xSU(3)
> > (Lorentz group cross color group), form a family of preferred
> > coordinates. The preferred coordinates are just linear Minkowski
> > coordinates and complex coordinates of CP2 transforming linearly under
> > some subgroup U(2) of SU(3).
[SPK]
> I hate to say so, but your requirement that the imbedding space is
> nondynamical is problematic. Unless we can show that it is a fixed
> "portrait" of a higher dimensional dynamical space, we would not be able
> to have thermodynamics on it. I may be miss understanding your
> [MP]
> The basic point is that dynamics for imbedding space CREATES the problems!
>
> Only the dynamics of spacetime surfaces is needed classically.
> The dynamics of surface motion makes induced spinor connection and metric
> dynamical despite the fact that imbedding space metric, etc. is
> nondynamical.

I think that I understand now, given the above discussion! Thanks. ;)

> The classical dynamics of TGD is dynamics of shadows: for instance, the
> metric of imbedding space is rigid object and induced metric is its
> dynamical shadow on 3-surface moving and changing its form.

Shadows on Plato's Cave Wall? ;)

> At quantum level the dynamics is at the level of configuration space
> of 3-surfaces but basically reduces to the dynamica of 3-surfaces plus
> that for induced spinor fields.
>
> The existence of configuration space of 3-surfaces as a Kahler
> manifold with spinor structure relies on the symmetries of the imbedding
> space metric: for generic imbedding space without any symmetries
> the entire construction would fail. The reason is that the symmetries
> of imbedding space are 'lifted' to much larger symmetries of
> configuration space and these infinite-dimensional symmetries
> guarantee that Riemann connection exists (this was observed
> already by Freed when he constructed the geometry of loop spaces
> in his thesis). Consistency implies existence philosophy works here.

I am a bit lost here, but from I can understand this is very good! :)

> Quantum theory for dynamical imbedding space would also lead to horribly
> nonrenormalizable theory: for instance, low energy limit of string models
> is nonrenormalizable theory since it is defined in
> 10-dimensional spacetime. Also physics would come out wrong: for
> instance, TWO gravitons would be predicted since the
> dynamical metric of the imbedding space would also give rise to graviton
> besides the graviton predicted by quantized dynamics of 3-surfaces.

Question: In the TGD model, how would we make observational
measurements of gravity waves and/or gravitons?

> > c) There are however QUITE TOO MANY coordinate choices
> > in this family parametrized by SO(3,1)xSU(3)
> > and SO(3,1)xSU(3) *cannot commute* with canonical
> > identification. One must be able to specify preferred
> > coordinates more uniquely. This is possible.
> >
> > d) Given 3-surface Y^3 on lightcone boundary representing initial
> > state of a particular spacetime surface has well defined classical
> > momentum P^k and angular momentum vector w^k as well as
> > classical color charges Q_a: these charges are Noether
> > charges associated with the absolute minima \$X^4(Y^3) of Kahler
> > action.
[SPK]
> Why do we not identify coordinate choices with observers, there
> are MANY of each... Is the cardinality of the class of each identical?
> Each observer has its own Y^3!
>
> [MP]
> Interesting idea, I have sometimes pondered the idea that
> consciousness brings in breaking of GCI by introducing coordinates.
> I considered this also a possible solution of problems with GCI
> but since GCI has been behind all real progress in TGD gave up
> this idea.

This is the key point of the Local (QM) Systems model.... all classical
type physics is contigent, e.g. perpective based...

[MP]
> >From TGD point of view each Y^3 (X^4(Y^3) represents entire
> classical world: quantum history is superposition of these.
[MP]
> My idea is to fix preferred (perhaps even unique)
> imbedding space coordinates. This choice depends on Y^3:
> if the choice of H-coordinates is sufficiently unique, the
> mapping of X^4(Y^3) to its p-adic counterpart induced by the
> mapping H_R-->H_p becomes unique and one has the highly desired
> GCI in the sense that possible allowed coordinate changes
> commute with identification map.

I am looking into the computational complexity of such imbedding space
coordinates. From what I have leamed so far, ([time 302]) "The
equivalence is not polynomial time equivalent or even decidable for
4-manifolds in general." This has very important implications!

> > d1) One can require that the preferred Minkowski
> > coordinates correspond to *rest frame* of Y^3 and that
> > spacelike angular momentum vector w^k defines the direction
> > of one of the coordinate axes, say z-axs. Hence
> > coordinate system is specified only up to planar rotations
> > around the z-axis forming group SO(2).

I think that these Y^3 frames are "what an observer perceives while"
thinking from the LS perpective. Each observer, having their own Y^3
frame with its configuration support with which to encode algorithmic
information (computer version of Caucht data), can make a computation
within the transformation of the planar rotations. I am thinking of the
SO(2) as giving the lenght or duration of the computation available to
the observer. Any behaviour simulatable within this computation is "what
can be percieved" by that observer!
The important point is that we can discontinuously "jump", as you show,
from one set of perceptions (a "quantum history") to another. Umm, this
make a lot of sense! :)

> > d2) In a similar manner one can show that
> > the complex coordinates of CP_2 are speficified only up to a
> > color rotation in Cartan subgroup of U(1)xU(1) of SU(3) representing
> > color rotations generated by color hypercharge and color isospin.
> > Here very special properties of SU(3) are crucial: SU(3)
> > allows completely symmetric structure constants d_abc so that one
> > can from from the vector Q_a of classical color charges second
> > vector R_a = d_a^bcQ_bQ_c commuting with Q_a as an element of SU(3)
> > Lie-algebra. Q_a and R_a as Lie-algebra elements span the unique
> > Cartan Lie-algebra U(1) xU(1, which generates the allowed color
> > rotations.
> > What is important is that this gives an additional item to the list of
> > arguments stating that CP_2 is unique choice for imbedding space.
[SPK]
> You lost me here. :( I'll re-read until it makes sense. One
> question, do you think that only one group of each type "exists" in the
> Platonic/ontological sense? If so, would it still make sense to think of
> many "perspectives" of the One, like the many possible shadows on
> Plato's cave wall?
>
> [MP]
> This is somewhat technical: the goal is to show that one can
> find preferred coordinates unique modulo the phase rotations
> in subgroup of SO(3,1)xSU(3), which is SO(2)xU(1)xU(1) and
> corresponds to spin and color isospin and color hyper charge.
> >From this GCI follows with suitable definition of identification map.
>
> I believe that imbedding space and its isometry group is indeed unique
> in Platonic sense. I would guess that the infinite-dimensional symmetry
> group of configuration space isometries and corresponding veryvielbein-
> group probably contain however most finite-dimensional groups as
> sugroups. Note also that vielbein group which is infinite-dimensional
> unitary group contais representations of all finite-dimensional
> Lie groups.

If I understand this correctly, you are talking about the "X" in
Hitoshi's model that is orthogonal to the R^6N space of QM! We need to
look at this more closely. :)

[MK]
> > Comment: Already at this stage one notice precise analogy with
> > quantum measurement theory. SO(2) belongs to and U(1)xU(1) is
> > the group spanned by maximal commuting set of observables associated
> > with isometries of H!
[SPK]
> How many "mutually commutative" sets of observables could exist if
> we allow for each to be "almost" convex?
>
> [MP]
> You are probably speaking of G= SO(3,1)xSU(3). Observables correspond
> group theoretically to Lie-algebra of G. Mutually commuting observables
> correspond to Cartan subalgebra generating maximal abelian subgroup of G.
> For G under consideration Cartan subgroup is H= SO(1,1)xSO(2) xU(1)xU(1).
> Lorenz boosts in given direction and rotations around that direction
> plus rotations generated by color isospin and hypercharge. Any
> Cartan subalgebra gives a set of mutually commuting observables.
> The space of them is the coset space G/H, H Cartan subgroup.

Ok, but what is the cardinality?

> BTW, for SU(3) this is SU(3)/U(1)xU(1), which a mathematician Barbara
> Shipman found to be related in mysterious manner to the mathematical model
> of the dance of honeybee and guessed that quarks must somehow be involved!

;)

> TGD inspired explanation of the appearence of this space is on my homepage
> and relies crucially on the fact that TGD predicts classical color
> fields in all scales (although gluons are confined) and to to the fact
> that macrosopic 3-surfaces have besides ordinary rotational degrees
> of freedom also color rotational degrees of freedom: color rotating
> 3-surfaces has color charges just like rotating body has angular momentum.

I need to buy your book... ;)

> > e) The problem is that one can specify the preferred coordinates
> > only up to a rotations in SO(2)xU(1)xU(1). GCI requires that these
> > rotations COMMUTE with canonical identification. This can be indeed
> > achieved by a proper definition of the canonical identification map!!
> > What is done is to notice that the rotations in question correspond
> > to *phase multiplications*, when one uses complex coordinates for CP_2
> > and for the plane E^2 orthogonal to momentum and spin vector w^k.
> > One must require that the phase exp(iphi)
> > of a given complex coordinate z is mapped AS SUCH such to its p-adic
> > counterparts: no canonical identification is involved. Geometrically
> > this means that products of real phase factors are mapped to products of
> > p-adic phase factors. The modulus |z| of z is however mapped by
> > canonical identification, which is continuous map.
[SPK]
> Interesting!
[MP]
> > f) This does not make sense unless phases are complex rational number
> > (rational numbers can be regarded as 'common' to both reals and p-adics
> > as far as phases are considered) and thus correspond to Pythagorean
> > triangle possessing rational sides
> > a,b,c:
> >
> > a= 2rs, b= r^2-s^2, c= r^2+s^2, r and s integers.
> >
> > In this case one can identify the real rational phase as such with
> > its p-adic counterpart. This means angle quantization.
> >
> > g) Actually this applies also to the hyperbolic
> > phase factor exp(eta) associated with (t,z) pair of Minkowski
> coordinates
> > and in this case quantization of allowed boost velocities mathematically
> > equivalent with Pythagorean triangles happens so that
> > the group of allowed coordinate transformations extends to the
> > Cartan subgroup SO(1,1)xSO(2) of Lorentz group (boots in direction of
> > spin plus rotations orthogonal to it). Altogether this means that
> > only the coordinates sqrt(t^2-z^2) and rho= sqrt(x^2+y^2)
> > and the moduli of CP_2 complex coordinates are mapped
> > by canonical identification to their p-adic counterparts.
[SPK]
> What happens to the remainder that is not mapped?
>
> [MP]
> That mapping is possible only for the subset of H is just what I
> want since this implies that the p-adic image of spacetime surface
> induced by this map is discrete in the generic case. Finite portions
> of H are mapped to unions of discrete 4-dimensional surfaces. The
> image of entire H is dense in H_p but does not fill H_p completely.
> [The mapping is discontinuous in phase angle degrees of freedom: the
> Pythagorean rays of complex plane are mapped to randomly mixed
> rays of p-adic complex plane].

I need to think about this for a while to get constuct my mental image
of it...

> The basic problem has been that direct canonical identification
> of real spacetime surface yields continuous but non-differentiable
> p-adic surface so that p-adic counterparts of induced field quantities
> and hence Kahler action do not exist since derivatives
> appearing in them are ill defined. The task has been to somehow
> loosen the canonical identification so that image becomes discrete
> and one can complete it to smooth p-adic surface by requiring
> that p-adic surface satisfies the p-adic counterpart of field
> equations associated with Kahler action.
>
> My earlier approach was to replace the direct canonical image
> with its pinary cutoff, which is discrete set and hope that
> surface going through these points and satisfying field equations
> would exists for some maximal, or perhaps even infinite, pinary cutoff.
> There earlier posting was about the possibility of infinite
> pinary cutoff. The problem is
> that this approach does not have any geometric appeal: it is
> too numerical! And there is no connection with quantum theory.
>
> The proposed form of real to p-adics mapping however implies
> automatically that image is discrete and one could hope that
> the phenomenon of p-adic pseudo constants and classical nondeterminism
> of Kahler action could make it possible to find unique
> p-adic spacetime surface spanned by the discrete canonical image of
> real spacetime surface. Unfortunately I have no ideas about how
> this could be achieved! Frustrating!!
>
> What makes me to take this approach
> seriously is the direct connection with geometry, number theory
> and with quantum measurement theory and sudden consciousness
> theoretic understanding of why Platon believed so firmly in
> rational world(;-).

Have heart, we are making progress! :) Others can fill in the gaps in
our individual models, we must "think outside the box"! ;)

> > Conclusions
> >
> > The conclusions are following.
> >
> > a) Quantum world according to TGD has a well defined Pythagorean
> aspect.
> > Only the discrete set of Pythagorean phase angles and boost
> > velocities are mapped to their p-adic counterparts. By the way,
> > Pythagoras was a real believer: the pupil of Pythagoras, who
> discovered
> > sqrt(2) payed for his discovery with his life! Perhaps it is easier
> > to forgive or at least understand Pythagoras now(;-).
[SPK]
> Interesting, but is it necessary and sufficient to just assume
> that a single unique discrete Pythagorean phase angles *exists*, are we
> not assuming an absolute basis to make this assumption? We can show that
> such exists, but it is only asymptotically *knowable* and that knowledge
> by one LS is not necessarily knowledge by all! I think the discussion of
> the inner product problem in QGR is related... I'll try to dig something
> up about this. It is mentioned in Conceptual Problems of Quantum Gravity
> (reference in Hitoshi's papers).
>
> [MP]
> The existence about existence of discrete Pythagorean angles reduces
> to the hypothesis about existence of imbedding space H.
> The angles are defined for preferred complex coordinates, which are
> fixed by pure group theoretical considerations.
>
> It is absolutely essential
> that H is non-dynamical and has its symmetries: if H is dynamical or
> just generic 8-dimensional Riemannian manifold,
> there are no preferred coordinates and one can safely forget
> the idea about relating real and p-adic TGD. All this relies
> on consistency implies existence philosophy.

I agree!

Continued later,

Stephen

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