[time 32] Re: My comment of the trinity of nondeterminisms in TGD, I.

Matti Pitkanen (matpitka@pcu.helsinki.fi)
Fri, 19 Mar 1999 07:34:52 +0200 (EET)

Before answering to you questions: it would help me if I could get
addreses of web page containing basic vocabulary of LS theory.

On Thu, 18 Mar 1999, Stephen P. King wrote:

> Dear Matti and Friends,
>
> I will now attempt to comment on Matti's ideas in order to see
> if we can tease out the commonalties and differences [mutual subsethood!
> :)]
> with Hitoshi's formalisms, as I understand them; so I will take all the
> heat for any errors found. :)
>
> Matti Pitkanen wrote:
> >
> > Holy trinity of non-determinisms in TGD
> >
> > In TGD there are three different non-determinisms.
> >
> > 1) The non-determinism related to the quantum jump between quantum
> > histories.
>
> A one-to-one mapping can't be, in general, defined between
> objects, using the term loosely :), that represent quantum histories. I
> believe
> that LS can be defined as such objects, with the caveat that LS do not
> have a restriction on the possible mappings between them, so the former
> idea stands. There may be a topological property of "multiple
> connections" leading to some form of non-commutative relation that may
> account for some of this. (cf. Alain Connes' work:
> http://www.math.ohio-state.edu/connes/)
>
Loosely speaking: quantum histories correspond to solution of Schrodinger
equation of of any deterministic field equations. One must however
generalize the concept of determinism because of the nondeterminism
of Kahler action discussed below.

> > 2) The classical non-determinism associated with the absolute
> > minimization of the Kaehler action.
>
> I still do not have a clear enough mental image of the Kaehler
> action to comment on this. :(

Kahler action is Maxwell action for CP2 Kahler form projected to
spacetime surface. CP2 Kahler form vanishes for extremely large variety
of 4-surfaces: it is enought that the projection of surface to CP2
belongs to so called Lagrange manifold (in general two-dimensional).
In canonical coordinates for CP_2 (symplectic manifold) Lagrange
manifolds have form

P^i= Nabla_if(Q^1,,Q^2), f arbitrary function.

Thus the restriction of Kahler action to M^4xY^2, Y^2 any Lagrange
manifold is purely vacuum theory!! This means huge vacuum degeneracy.
Vacuum extermals are obviously nondeterministic.
The small deformations of vacua yielding nonvacuum extremals are
expected to inherit part of the nondeterminism of vacuum extermals.

Kahler action is used to define Kahler function K defining the Kahler
geometry of the configuration space of 3-surfaces.
Kahler function K(Y^3) with Y^3 at light cone boundary is defined
as the absolute minimum of Kahler action for spacetime surfaces 'spanned'
by Y^3 (note the analogy with soap films). By nondeterminism
the absolute minima are expected to be highly degenarate (several
soapfilms spanning the same frame). Of course, there is no
mathematical proof for this: this is just physicists intuition about
obvious.

>
> > 3) The p-adic non-determinism implied by the existence of p-adic pseudo
> > constants which are functions of p-adic coordinate constant below some,
> > arbitrary small but finite scale.
>
> This appears to speak of truncations or cut-offs in the
> interaction terms of the Hamiltonians defined using a p-adic coordinate
> system (see
> for instance: http://www.unipissing.ca/topology/p/a/a/c/10.htm,
> http://www.elec.uq.edu.au/~annis/doubt_misc/janet4.html
> http://www-i2.informatik.rwth-aachen.de/Forschung/MCS/Mailing_List_archive/con_hyperarchive_1995/0194.html,
> etc.) which makes sense since we understand that any meaningful
> measurement will be a finite quantity and will take into account the
> former states of the LSs involved to guarantee, as it were, the logical
> consistency of the measurement.

Solutions of p-adic differential equations contain 'integration constants'
as do ordinary diff equations. Integration constant is function having
vanishing derivative. In p-adic context integration constants are
functions depending on finite number of positive pinary digits:
taking any function f(x) and replacing x by its pinary cutoff x_c(n):
x= SUM(m)x_mp^m --> SUM(m<n)c(m)p^m=x_c(n)
and you get a function having vanishing p-adic derivative and acting as
p-adic integration constant. One can say that any piecewise constant
function is p-adic integration constant. When this constant depends on
time you get nondeterminism.

Truncations of interactions are extremely natural in p-adic context.
For large p:s these truncations lead to extrely powerful approximation
scheme since series is in powers of p (think about p=2^127-1 associated
with electron!).

>
> > It took relatively long time to realize that there must be
> > very close connection between these three non-determinisms and that
> > this connection provides a precise criterion making it possible to
> > assign definite p-adic prime to a given spacetime region.
>
> Question: Could we then think of a "given spacetime region" as
> bounded, closed, or finite? Would the distribution of the primes and a
> (bijective) mapping between primes within in R and the set (or class or
> hyperset, etc.) of all possible spacetime region follow some statistical
> form?
>
Answer is 'mostly yes' to the first question. Regions have finite spatial
size but can have infinite temporal duration. I do not quite understand
the second question. I do not consider all possible spacetime regions.
I believe that the shape and size of region is determined by the dynamics
(actually the connection between nondeterminisms provides a criterion for
determining the size and shape of region corresponding to given p!).
Typically a region with given p represents a given spacetime sheet,
elementary particle, hadron, nucleus,atom,.... and have typical size given

> > This connection also gives strong quantitative grasp on the behaviour
> > of cognitive spacetime sheets: even our own behaviour should reflect
> > the characteristic features of p-adic non-determinism for some values
> > of p!
>
> Could we think of cognitive "spacetime sheets" as a grouping of
> all of the "thoughts" or representations of the information content of
> such > "thoughts" possible given an arbitrary finite configuration of
> matter/energy?

Perhaps I should answer Yes. Cognitive spacetime sheets provide cognitive
reprsentations for 'material world' by their interactions with them.
Essential element is the nondetermnism of cognitive spacetime sheets.

> Do the statistics of the distribution of primes within the Reals
> important? How do we model this?

physically interesting primes correspond to subset of primes.

a) p modulo 4=3 requirement is needed if one wants to introduce sqrt(-1)
as algebraic extension and do standard QM.

b) p=about 2^k, k prime or power
thermodynamics for Super Virasoro representations. This hypothesis
follows from what I call elementary particle blackhole analogy by
generalizing the Hawkings entropy formula to p-adic context and applying
it to CP2 type extremals representing elementary particle. Horizon is
no the region in which Minkowski signature of exterior metric changes
to Euclidian signature of CP2 type extremal.
If one requires that the radius of elementary particle horizon
is p-adic length scale itself one indeed obtains p=about 2^k, k prime or
power of prime.

c) Thus it seems that the distribution of k:s is important in principle
but probably not in practice. In physically intersting length scales k is
very small. Up to cosmological length scales k stays below 300!

>
> > 1. Quantum non-determinism <--->classical nondeterminism of Kaehler
> > action
> >
> > Only few months ago it became clear that classical nondeterminism and
> > quantum nondeterminism correspond to each other very closely. The quantum
> > jumps between quantum histories typically *select one branch from a
> > superposition of branches of a multifurcation of classical history
> > determined by the absolute minimization of Kaehler action* [timesc].
>
> This selection process, can it be modeled using something like a
> tournament of n-player games?

The crux of the matter is that single quantum jump cannot be modelled!
It is something totally irreducible, the moment of consciousness! QM
however predicts probabilities for various outcomes and if one can model
the statistical predictions of QM in the manner your propose, one can also
model TGD:eish quantum jumps.

>
> > [Here of course, quantum superposition of infinite number of classical
> > spacetime surfaces in the sense of quantum parallel classical worlds,
> > is in question].
>
> In Hitoshi's Local Time theory we do not consider LS to have any
> "classical features" internally; these only arise when we consider the
> the "affections from other particles outside of the local systems" (

I am not sure whether the meaning of 'classical' is same to us.
In TGD the only non-classical feature of quantum theory is quantum jump.
In TGD quantum states correspond to modes of *classical*
configuration space spinor fields. Oscillator operators are typically
thought as something quantum mechanical but in infinite-dimensional
geometry they are competely classical objects.
Furthemore, 'classical' is not result of approximation: classical
physics is genuine part of TGD based quantum theory and emerges
at the level of configuration space geometry: definition of Kahler
function associates definite classical spacetime surface to a given
3-surface.

> There is a passage in "From Here to Infinity" by Ian Stewart pg.
> 128-131. that I believe is relevant to this discussion!
>
> "Fake 4-space
>
> One absolutely basic question is the uniqueness of the smooth structure
> on a given smooth [Euclidean] manifold. In other words, is there only
> one consistent way to define 'smoothness' on a given space? Originally,
> everyone pretty much assumed the answer would be 'yes'; but Milnor's
> exotic 7-sphere (mentioned in chapter 1) showed that it is 'no'. Indeed
> there are exactly 28 distinct smooth structures on a 7-sphere. However,
> it was still hoped that for good old *Euclidian* space of dimension n
> there should be a unique structure, namely the standard one used in
> calculus. In fact this was known to be true for all values of n except
> 4. But at much the same time that Freedman was showing that 4
> dimensional space is in most respects pretty much like five or higher
> dimensions, Simon Donaldson, in 1983, made the dramatic discovery of a
> non-standard smooth structure on 4-space.
> His proof is difficult and indirect, making essential use of
> some new ideas cooked up between them by topologist and mathematical
> physicist
> called *Yang-Mills gauge fields* (see chapter 10). These were invented
> to study the quantum theory of elementary particles, and were not
> intended to shed light on abstruse questions in pure topology! On the
> other hand, the topological ideas involved were not invented for the
> analysis of fundamental particles either. It is quite miraculous that
> such disparate viewpoints should become united in such a tight and
> beautiful way. Ironically, many of Freedman's ideas are essential to the
> proof; and the starting-point is a curious group-theoretical
> coincidence. The *Lie* group of all rotations of n-dimensional space is
> simple, except when n=4. In that case it breaks into two copies of the
> rotational group in 3-space. ...
> At any rate, we now know that 4-dimensional space is quite
> unusual, compared to any other dimension. In fact, not only is the
> smooth
> structure non-unique: it is *extremely* non-unique. Milnor's 7-sphere
> has only a finite number of smooth structures. But there is an
> *uncountable infinity* [isomorphic to R^n ?] of distinct smooth
> structures on 4-space. However, things are not quite as bad as they
> currently seem. Among these is one 'universal' structure which in a
> sense contains all the others. ..."
>
> leaping ahead in the book:
>
> "...Bizaca starts with a four-dimensional ball, whose surface is
> a three-dimensional sphere. This is a king of 'curved three-space'...
> Inside the three-ball he draws lots of wiggly loops, which look just
> like knots and links in ordinary three-space. Then he 'glues' certain
> kinds of handle[s] on to the ball so that, in effect, their edges run
> along these links. ... A systematic way to work with these links, called
> Kirby calculus, was introduced by Kirby in 1978. The main point is that
> you can define lots of four-dimensional spaces by drawing the
> Bizaca's link is constructed so that it corresponds to a *fake*
> Euclidean four-space. The link is built from infinitely many closed
> loops, structured into levels as shown in the figure (the link to the
> right of the dotted line is the same as that to the left.) At level one
> there is one 'kinky handle', the thing that looks like a little
> lifebelt. At levels two, three and four there are two kinky handles. On
> level five there are two hundred kinky handles; on level six the number
> of kinky handles is the utterly huge number 2 x 10^10^10^10 where the
> arrows indicates forming a power. ... On higher levels the number of
> kinky handles is even vaster: on level n there is 2 x 10^ ... ^10 of
> them, where the number of 10s is 8n -44.
> Got that? Now exercise your imagination to 'glue on' an infinite
> system of handles, one for each loop in Bizaca's link - and so he
> proves, is a
> fake Euclidean 4-space. The prescription may seem curious, but what
> makes it all the more intriguing. Don't worry too much about what it
> *means*: just revel in its strangeness. Its importance is that it
> provides an explicit construction instead of mere existence. So that we
> have - so to speak - a handle on what fake four spaces look like."
>
> There is much more related information and a good bibliography
> in Stewart's book that I do not have space or time to type here. :( In
> short I will say that it just might be that your "jumps between quantum
> between LS iff we can model the internal Euclidean L^2(R^3n,C^k) space
> of quantum mechanical particles consistently using the *Lie* group of
> all rotations of 4-dimensional space, where "it breaks into two copies
> of the rotational group in 3-space."
> I need to see what the relationships between these two "copies"
> have! The space of the "total Universe" may be well represented by the
> "one
> 'universal' structure which in a sense contains all the others. ..." :)
>
> > This realization led to the understanding of the
> > psychological time and its local arrow [timesc].
> > The argument goes roughly as follows.
> >
> > a) One can distinguish between volitional and cognitive nondeterminism:
> > volitional nondeterminism corresponds to multifurcations of classical
> > time development having large, long lasting and macroscopic effect in the
> > scales of conscious observer. For cognitive nondeterminism
> > the effect of multifurcations is small, short lasting and always
> > localized to a finite spacetime volume (cognitive spacetime
> > sheets having finite time duration).
>
> I see this as distinguishing between "cosmos" sized interactions
> such as involved in Galaxy and Quasar formation and stability and
> "benchtop"
> sized situations such as discussed in section 7 and 8 in
> http://www.kitada.com/time_I.html and section 5 in
> The connection between "volitional" and "cognitive" nondeterminism
> is not obvious at this point in Local Systems theory, frankly we have
> not
> tried to address it yet but, as it seems, perhaps your work spans that
> crevasse! :)
> Another point that, I think is relevant, is the size of the
> configurations spaces involved. If we were to think of them as some sort
> of memory, which appears to be necessary in any model of consciousness,
> the "amount" of information and its statistics would come into play!
>
> > b) Conscious experiences associated with the choices selecting between
> > brances of volitional multifurcations generate psychological time and its
> > local arrow. The value of psychological time for a given quantum jump
> > corresponds roughly to the moment of multifurcation. The fact that
> > spacetime surface belongs to the Cartesian product of *future light cone
> > M^4_+* (this is essential!) and CP_2 implies that volitional
> > multifurcations have time ordered hierarhical structure: multifurcations
> > having long lasting effect affect future rather than past. There is no
> > strict time arrow associated with cognitive quantum jumps: our thoughts
> > contain contributions from past and possibly also from future.
>
> I do not see any non-trivial disaggrement of this idea of your
> with that of LS theory! Hitoshi has stated: "The times are defined only
> for
> local systems (H_nl, \*H_nl). The total universe \theta has no time
> associated. The local times arise through the *affections* from other
> particles outside the local systems (definitions 1-3). The uncertainty
> principle holds only within these local systems as the uncertainty of
> the local times. Quantum mechanic[al] ... [behaviour] is confined within
> each local system in this sense. The quantum-mechanical phenomena
> between two local systems appear[s] only when they are combined [in the
> context of a given experimental 'act' as in D. Finkelstein's work] as a
> single local system. in the local system, the interaction and forces
> propagate with infinite velocity or, in other words, they are
> *unobservable*.
> Each local system [is]... the observer of other [local] systems.
> In this situation, the local systems are not correlated in general.
> Therefore, ther is no reason to exclude classical mechanics in
> describing the *observable* relative behaviour of the observed systems
> with respect to the observer [representable by an LS]."
> We also see from Cramer's work
> (http://weber.u.washington.edu/~jcramer/theory.html) that the Wave
> function includes both "past" and "futute" contributions simultaneously;
> and since the LS is, in this case a Wave function + its propagation
> space, it is symmetrical with respect to temporal orientation "on the
> inside". It is how LSs interact that seems to "break" this symmetry,
> since the interactions have a finite "propagation speed", that of light
> or less, that may dovetail into your thinking!
> The analysis of how the interactions of finite object with
> internal infinite speed "motions" construct a cosmos with finite signal
> propagation deserves a post of its own! Suffice it to say, that we get
> into computational complexity and some properties of graph and measure
> theory.
>
> > c) With some additional input (theory of infinite primes) one ends up with
> > a mathematical generalization of von Neumann's intuition about
> > brain as an ultimate state function reducer. The allowed quantum jumps
> > can only reduce quantum entanglement between cognitive
> > and material regions of spacetime. Cognitive regions
> > have vanishing total energy and other quantum numbers and typically
> > correspond to spacetime sheets having finite time duration
> > whereas material regions carry nonvanishing classical charges and have
> > necessarily infinite duration by conservation laws. Thus one can say
> > that Matter-Mind duality is realized at the level of spacetime geometry.
> > Of course, Mind refers now to cognitive representations, not
> > consciousness.
>
> I have been considering Matter-Mind duality for quite some time and was
> about to give it up until I read V. Pratt's wonderful paper:
> http://boole.stanford.edu/chuguide.html#ratmech
> It is a difficult paper, but after some study of Chu spaces and the
> work of Barwise and Wegner have come to appresiate the depth of subtle
> beauty of them. Again, they are for a differnt posting... :) Needless to
> say, "consciousness" is a dynamic process, not representable by any
> static structure; thus your statement "Mind refers now to cognitive
> representations", is one I totally agree with! :)
> I again must insist that if the are looking at our cosmos with the
> duality paradigm, we must take into acount in our models of phenomena
> both the physics of matter and the logics of information inference. As
> Pratt at al explain, it is the n-ary relation between them that is the
> "link" that Descartes sought but never found...
>
> I will post my responce and analysis of the remaining section in an
> upcomming post.
>
> Onward to the unknown,
>
> Stephen Paul King
>

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