[time 493] Re:[time 486] Re: [time 479] Parallel translation,etc... part V

Matti Pitkanen (matpitka@pcu.helsinki.fi)
Sun, 25 Jul 1999 11:50:37 +0300 (EET DST)

Dear Stephen,
I begin to be exhausted....

Dear Matti,

Matti Pitkanen wrote:
>
> Here begins my reply to your second email about parallel
> translations etc...
>
> On Fri, 23 Jul 1999, Stephen P. King wrote:
snip
[SPK]
> > OK, on this point let me be clear: The Universe "just exists"! It is
> > the experiences that are computed by the interactions (observations)
of
> > the finite subsets (LSs) of the Universe.
[MP]
> I can agree here in the following sense. Each quantum jump
> is quantum computation in the manner I explained in
> part IV of my reply to earlier posting. Infinitely large
> quantum computer performing infitely long quantum computation
> and halting: Psi_i--Z UPsi_i-->Psi_f.
>
> Actually the sequences of computations of this kind represent thoughts
> which generate cascades of selves. Infnite number of
> simultaneous experiences by the hierarchy of dynamical selves
> forming abstractions about the experiences of their subselves.
>
> NP-computability is replaced by quantum computability in this
> sense in TGD framework.

Yes, but can you see that physics needs a model wherein
n-observers are
participating to generate each others local framings ("reality",
Schommers calls them "pictures of reality")? The fact that so many
observers are involved, is, I believe why we must use statistics very
carefully. Peter's Secondary Observer Conjecture is an expression of
this.

[MP] I think selfs and hierarchy of the, provides this model.
Subselves can also represent objects of external world. Houses, people,
anything. Even notes and instruments in a piece of music:
enormous mimicry occurring all the time.

snip
[SPK]
> > Perhaps I am not understanding how the "definite sector D_p of
the
> > configuration space" is selected. I say that the act of observation is
> > the act of selection, in the sense that a given transition A -> B on
the
> > material configuration involved in the particular observation is
allowed
> > iff the information content of B implies the information content of A.
> > This is the essence of an infomorphism!
[MP]
> Configuration space decomposes into sectors D_p allowing effective
> p-adic topology with prime p. The localization of state
> to D_p is forced by Generalized Unitarity.

I need a link to a definition of "generalized unitarity"!

[MP] This is somewhat technical concept.

a) The point is that S-matrix is in some sense a 'sum',
or rather collection, of S-matrices
S[p_i-->p_f:mn], where p_i and p_f label the effective
p-adic topologies of initial and final sectors.
This matrix is p_f-adic valued since transition takes to sector p_f
and inner product involves p_f-adic valued integral over final
sector D_pf. Collection of sub-S-matrices in belonging to different
number fields is involved: how one could generalize unitarity
to this kind of situation. This is the problem!

b) Unitarity conditions say that inner product of two rows for
S-matrix for transitions p_i-->p_f is ZERO or ONE
for any p_f and p_i. I do not bother to write the formulas
explicitely. 1 and 0 appear on the left hand side of conditions.
The problem is that different number fields are involved!

c) The solution of problem is that 1 and 0 appearing on the left
hand side of unitarity conditions are in well defined sense COMMON
to reals and all p-adics: 1 and 0, the neutral
elements of multiplication and sum, are the only elements of
reals mapped to them selves under canonical identification always.
Binary numbers are common to reals and p-adics, one can say.
Thus unitarity conditions make sense. Logic has certain
kind of universality in it!

d) Note that unitarity conditions state that total sum of p-adic
probabilities from sector p_i to p_f not equal to p_i vanishes: this
does not certainly make sense in real context but makes sense
p-adically! I will not go to the physical interpretation here.

This is not yet enough.

a) One cannot calculate probability for quantum jump
in which final state is superposition of states in several
sectors p_f: the resulting number would be sum of
p-adic numbers in different p-adic number fields: something
totally nonsensical. This forces localization to definite
sector D_pf in quantum jump. This implies evolution: p_f increases
in the long run since the number of primes larger than given
prime is larger than the number smaller than p_f.

b) The quite recent realization was that more than localization
in D_pf is involved: the mere requirement that quantum jump
is quantum measurement local at the level of configuration
space requires localization in continuous zero modes.
Localization into D_pf follows as a consequence.

> The requirement that quantum jump corresponds to measurement
> which is local at the level of configuration space sharpens
> localization hypothesis: localization occurs to definite
> values of zero modes characterizing the classical features of
> spacetime.

I believe that the "volume" involved in the localization process
is
related to the \epsilon of accuracy of bisimulational predictability
(perhaps also to the "log-likelyhood" of the prediction). I see the
Planck constant as an expression of this, it is a "mean" of the
observers that are participating in our little corner of the Universe!

[MP] One cannot exclude the possibility that localization
in zero mode is not exact: pinary cutoff could determinne
the volume involved. I am however rather convinced that exact
localization occurs: huge simplification in the calculational
formalism results and one can indeed calculate everything.

> One could say that selection is involved at the level of zero modes.

Yes!

> Our volition probably corresponds to selection between degenerate
> absolute minima of Kahler action characterized by *discrete* zero
> modes: no experience selection in continous zero modes
> since it is not possible to experience what it is to select
> between continuum of alternatives.

Yes, thus we do not experience the 'objective' continuity of
spectra
that the W^n has! Weyl has right!

> I do not assume quantum jump A-->B means that information content
> of B implies information content of A. This is too strong requirent.
> p-Adic evolution however implies that information gains tend to
increase.

Yes, I weaken the strict binary implication (Chu_2 = {0,1} =>
Chu_[0,1]) to the fuzzy "similarity" of B's content to that of A. The
"more similar" the contents are to each other, the closer to binary
isomorphism they become! It is interesting to note that the constant Phi
found though out Nature is smallest scaling that maintains the
invariance with respect to arbitrary scale transformations!

> Note also that there are infinitely many types of informations
> and each is characterized by its own information measure:
> not only single abstract information.

Yes, I think that each LS has for its own one of these measures! I
also
believe that they are all related to Fisher Information! (pg. 32 of
Physics from Fisher Information)

[MP] As I said earlier I tend to regard Fisher information as too
technical: the motivation for it is that it leads to Laplace
type equations but already d'Alembert type equations involves
algebraic trickery forced by Minkowski signature of metric.

Somehow Frieden's information concept involves the method of getting
information too much: I am right? I was almost saying 'In my case
information measures for quantum histories must be completely observer
independent, nothing about method getting information'.

This is not the case! One maps the reality to the p-adicity of the
local observer with his own p-adic p, calculates p-adic information gain
and maps it to reals and gets the upper bound p*log(p)!

There are measures for information can be about spacetime geometry,
information about spacetime spinor fields, local information
about configuration space spinor fields etc.... Conscious information gain
is difference for these 'dead' informations for initial and final
states of quantum jump.

[MP]
> > One can even consider the primordial chaos obtained formally by
putting
> > p=1: effective topology is roughest possible: distance between two
> > points is zero or one. I have essentially you picture but
> > at the level of effective experienced topology.
> > **
[SPK]
> > This is modeled by the Chu_{0,1} that Pratt discusses in ratmech.ps!
I
> > am using the generalization that would be modeled by a Chu_[0,1], it
> > allows the modeling of \epsilon accuracy involved in dissipative
> > transitions of the configurations, e.g. memory fades, spectra shift to
> > red, entanglements decohere, etc. I wish I had a better grasp of the
> > formal language needed to communicate this precisely, but I guess that
> > that it why I am a philosopher, not a physicist. :-)
[MP]
> I have the feeling that Chu spaces might enter TGD in description of
> cognition as cascades of selves created with selves.

Me too! :-) To be specific, in the way that that causality is
defined...

[MP] Causality problem disappears in quantum jump picture. Quantum
jumps indeed define Brownian motion type process in zero modes
and there is no requirement of propagation with light velocity .
Note that light velocity is property of metric. But line element
vanishes in zero modes!

[SPK]
> > > Is there any relationship between "zero modes" and "null geodesics"?
> > > Could there exist an infinity of almost disjoint hull hypersurfaces?
[MP]
> > All curves in zero modes are null geodesics formally. I do not
> > believe that this is however a useful concept. The space of zero
> > modes is infinite-dimensional. And each surface in this space
> > as formally vanishing metric. Or better to say: no metric at all.
> > Only symplectic structure making integration possible.

[SPK]
> > Interesting! Is it true that we could find every possible set of null
> > geodesic somewhere "in" this space of zero modes? Could you elaborate
> > about the role of this "symplectic structure"?

[MP]
> All curves would be null geodesics since line element vanishes:
> thefore the concept of geodesics becomes useless.

Yes, but does it vanish identically for all LSs, or just all but a
finite number of them? This allows for the use of interaction (or
connection matrices) that are inverse (?) functions of Hamming distance
(http://www-dept.cs.ucl.ac.uk/staff/S.Bhatti/D51-notes/node30.html) or
"dissimilarity", I think. Thus we avoid Olber's paradox and explain why
only a finite number of observers can interact at any (subjective)
moment! This gives each observer an asymptotically decreasing causal
influence on any other when considering Real values, and upon mapping to
p-adics, we get cut-offs. This gives a p-adic definition to a black-hole
event horizon! Wow!

> Symplectic structure is defined by antisymmetric tensor J^kl.
> One can define J_kl by the condition
>
> J^kr J_rl = -delta^k_l, Kronecker delta.
>
> Note that metric is not needed: it emerges
> only when one wants to define J^k_l by index raising operation
> applied to J_kl.
>
> J^kl defines Poisson bracket in the space of functions define
> on manifold
>
> {f,g}= J^{kl}f_lg_k
>
> _l means partial derivative. Poisson bracket is antisymmetric and makes
> function algebra infinite-dimensional Lie-algebra. Jacobi identities are
> indeed satisfied. Functions generate canonical transformations
generated
> by the vector fields
>
> X^k = J^{kl}f_l, f arbitrary function.

I am not a mathematician :-( Words, please?

[MP] This is somewhat technical. It is very difficult to
express the concept of canonical transformation without formulas.
In two-dimensional case canonical transformations are
*area preserving* transformations. This is something simple
and comprehensible. But in higher dimensions situation is not
so simple.

> Canonical transformations preserve all 2n dimensional integrals
> defined as integrals of J^J...^(n J:s ) over 2n-dimensional
> submanifolds.

[MP]Areas, 4-volumes, ....2n-volumes defined by J and its tensor
powers are preserved. This is the 'wordy' characterization of
canonical transformations.

>
> Canonical transformations are the isomorphisms of canonical
> formalism of classical mechanics. The are crucial also for
> canonical quantization. Quantization in standard QM means
> the replacement of function algebra with the algebra
> of Hilbert space operators and the replacement of Poisson
> bracket with operator commutator AB-BA.
>

Little by little! :-)

Best,
MP

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