Stephen P. King (stephenk1@home.com)
Thu, 18 Mar 1999 13:08:21 -0500
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/)
> 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. :(
> 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.
> 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?
> 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?
Do the statistics of the distribution of primes within the Reals
important? How do we model this?
> 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?
> [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" (
Sect. 10 http://www.kitada.com/time_I.html)
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
corresponding links ...
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
histories" are representable as ultametric (p-adic valued) *links*
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
http://www.kitada.com/time_II.html.
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
This archive was generated by hypermail 2.0b3 on Sat Oct 16 1999 - 00:29:45 JST