Matti Pitkanen (matpitka@pcu.helsinki.fi)
Wed, 12 May 1999 09:09:27 +0300 (EET DST)
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...
[Matti]
Your example is good. p-Adics are not well ordered and to speak
of the mapping of ordered real interval to p-adic *p-adic ordered
interval* is statement, which as such does not make sense.
Canonical identification
x_R= SUM x_np^n --> SUM_n x_np^(-n)
maps reals in continuous manner to p-adics (Here x_n has values 0, ...p-1
since pinary expansion is in question). Continuity can be checked by
direct
calculation.
One can however INDUCE the ordering of reals to p-adics just by saying
that p-adic x is smaller than y if the inverse image of x is smaller than
the inverse image of y. This ordering cannot be and is not global however
since two p-adics x_1 and x_2 can be mapped to same real and in this case
one
cannot distinguish between x_1 and x_2 since the real images of x_1 and
x_2 are
identical. This is related to the nonuqinuess of pinary expansions
of real numbers (counterpart of 1=.99999..).
This induced ordering is crucial in my proposal for
the definition of the p-adic version of definite integral reducing
the definite integral to the difference of integral function at
the endpoints of the integration interval whose ordering is induced
by canonical identification. This same ordering could make
possible to define the concept of differential forms but
this would require the touch of real mathematician.
[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?
If canonical identification is applied in standard manner to all
coordinates of say 8-dimensional M_R ==H the p-adic image of X^4 in M_R
is fixed completely. But X^4_p is NOT smooth surface p-adically: this is
the
problem. I want smooth surfaces satisfying the field equations defined
by Kahler action. What follows could be also regarded as a manner to
loosen
the mapping so that the *discrete* image of X^4 could be completed to
smooth p-adic surface.
[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.
[Matti]
Quantum numbers would be external to local observations in standard
approach
to quantum measurement theory, where observers are external agents
not described by quantum physics except phenomenologically by
state function collapse concept.
In TGD situation is different. TGD introduces theory of consciousness
as part of quantum physics: observation/observer is in quantum jump
between objective realities which as such are Zombies and do not observe
anything. This framework is much more ambitious: the theory should
predict when and where the observer and observation
are in given quantum jump, which subsystems can be said to observe
/be observed, etc...
[Stephen]
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.
[MP]
I have kind of intuitive idea about Weyl's basic philosophy. For instance,
unit of length would not be universal and change with position.
This would require giving up the good old Riemannian geometry and
personally I am conservativist in this point: CP_2 size is the fundamental
length scale in my approach.
From my view point the introduction of ensemble of ideal
local observers looks phenomenological concepts: I am convinced that we
need the theory of consciousness which only talks about conscious
experiences: conscious
experience as quantum measurement is indeed attractive hypothesis
since it builds direct bridge between physics and psychology.
It might be a good idea to ponder the distinction between
what is measured and measurement itself. I would argue that
various structures of spacetime (metric, connections) describe what
is in principle measurable. Same applies to quantum numbers
characterizing quantum states and reducing to group theory.
The real measurements itself would not be describable by the model
of spacetime or physical system: quantum jump again!
Of course, I can only compare our basic assumptions (without
any claim for the correctness of my own ones: they are just assumptions!):
I do not put the observer to the world: observer and observations are
between two worlds: initial and final quantum histories of quantum jump.
This assumption would suggest that the theory of observers and local
systems should be based on quantum jump concept and use the concept of
quantum history pair: pair of two objective realities.
Perhaps at classical limit when quantum jump does not appreciabely
change the quantum state one would obtain description based
on single objective reality.
[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...
[Matti]
You are right. This goes to the basis of calculus: manifold
theory relies on calculus and which is the proper definition of
calculus. The concept of derivative can
be formulated without notion of infinitesimal and derivatives are all
that is needed.
[Stepen]
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?
[MP]
Sorry for unclear sentence. It should have read 'For instance, the
component
of angular momentum in given direction corresponds to the action of
rotation
around the axis defined by that direction'.
The 'specific direction' was bad choice of words. There is
no selection of specific frame involved yet.
> [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? :)
[Matti]
Noether's theorem from the nonrigorous point of view of physicists can
be found from most books on quantum field theory or variational
calculus.
a) What is done is to study variation of action S= int Ld^x
under symmetries: it must vanish by definition.
b) Variational principle states that under arbtrary variation of
fields the change of action reduces to total divergence and hence
to a mere boundary term. Boundary term contains contributions
from boundaries with spacelike and timelike normals, usually
at spatial and temporal infinities. The spatial boundary term
is required to vanish but no conditions on the latter term are posed.
c) The variation under symmetry for
a solution of field equations reduces to mere timelike
boundary term which no vanishes since symmetry is in question.
This expresses conservation of classical charge. Charge is the integral
over time=constant section.
> 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
[Matti]
You might be right about 'primitive'. I have not strong preconvictions
about the correctness of any philosophy: the only test is whether
philosophy
leads to working physical theory.
I have indeed made the assumption about special nature
of Riemann geometry, this is nothing but a naive belief of poor physicists
to the Godly intuition of geometers(;-) and perhaps motivated partly my
laziness!
On the other hand, the attempts
to construct quantum geometries have not been very successful: the reason
is that 'quantization' is application of ad hoc rules.
My basic philosophy is mirror image of Smolin and others:
it is simply 'Do not quantize but classically geometrize instead'.
What gives support for my approach is that one can indeed geometrize
basic structures of QFT: oscillator operator algebras reduces nicely
to Kahler geometry. Even fermionic anticommutation relations have
nice interpretation in terms of gamma matrix algebra in
infinite-dimensional
context.
[Stephen]
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!
[Matti]
Here I warmly agree: measurement involves always subsystem-complement
division
and Newtonian picture about observer is quite too simplistic.
But I go even further: I do not regard measurement done by some
continuous stream of consciousness, observer. Quantum measurement is the
moment of
consciousness and contains conscious information like
'I am doing measurement now in lab X and the reading of measurement
apparatus is this and that' or my personal version
' There is conscious experience about existence of observer doing .....'
Subjective views are indeed what exist subjectively. I however assume that
the
quantum states=quantum histories exist objectively but change quantum jump
by quantum jump so that there is no unique objective reality.
This is just assumption: the alternative, completely logical,
possibility is to assume no objective realities and finish doing physics.
[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?
[MP]
Choice of action depends on theory. For instance, in TGD
action is what I call Kahler action and integration
measure is volume element defined by induced metric and is general
coordinate invariant with respect to spacetime and imbedding space.
Infinitesimal variation is change of action under small variation
of field quantities. In TGD field quantities are replaced by imbedding
space coordinates as function of spacetime coordinates so that
there are only 4 primary field variables since GCI eliminates four
of them: this is to be compared to the number of dynamical
field variables in typical unification of basic interactions.
[Stepthen]
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).
[Matti]
This idea about synchrony and asynchrony might have counterpart in
neurophysiology and also in my model of EEG. In my approach EEG
corresponds to spatially independent collective oscillations in some
region of
brain. Two brain regions oscillating in resonance would form
a single observer: their conscious experiences bind to single
experience. When the regions oscillate as separate units, they
are two separate observers with their own experiences.
[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."
[Matti]
You are talking about metric in sense of p-adic norm, a I right?.
To build p-adic physics one is however forced to construct p-adic
version of Riemann geometry. To generalize the expression of line element,
to generalize the concept of line length, area, etc...
Also the concept of tensor, in particular, the concept of form
must be generalized. This all is nontrivial because p-adics are
not well ordered and the concept of boundary, crucial
for integration and differential geometry and theory of forms,
is not sensical in PURELY p-adic context. Canonical identification is what
makes possible integral calculus and gives hopes of lifting
these concepts to p-adic context. But all this involves all kinds
of tricky problems.
[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! :)
[Matti]
I would claim that infinite-dimensional Kahler geometry is primitive:
perhaps everything follows from the requirement that infinite-dimensional
Kahler geometry allowing spinor structure exists(;-): spacetimes as
surfaces in H, etc...would be properties of this geometry.
Dualities might be possible. I am, and most of us are victims of our
formulations unable to see that same things can be said in different
manner.
[Stephen]
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!
[Matti]
Actually I claim even more: entire deterministic time developments from
big bang to infinite future are dead, not just hypersurfaces.
[Stephen]
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.
[Matti]
Interesting question! What is communication basically? Or what is
conscious communication basically? Is it basically fusing
of communicators to single larger conscious subsystem for
few moments of consciousness? I must say that I have not
pondered what 'communication' or 'information transfer' would
mean in my conceptual framework.
> > 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.
[Matti]
Interesting ideas and might have some relationship with what I am
trying here. BTW, I am also outlaw(;-).
> [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.
:)
[Matti] Whereas I say that there are two times, subjective and
geometric(;-).
There is no end for theorizing! In any case, I am only saying
that there is no consciousNESS (consciousness as a property of physical
state): there are only moments of 'consciousness'. In Finnish there
is nice word 'tajunta', which does not contain the the idea about
consciousness as a property. Sad that English has no such word.
> [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! :)
>
[Matti]
I stop here and represent remaining comments in separate mail.
MP
This archive was generated by hypermail 2.0b3 on Sun Oct 17 1999 - 22:10:31 JST