Matti Pitkanen (matpitka@pcu.helsinki.fi)
Wed, 17 Mar 1999 09:54:14 +0200 (EET)
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.
2) The classical non-determinism associated with the absolute
minimization of the Kaehler action.
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.
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.
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!
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].
[Here of course, quantum superposition of infinite number of classical
spacetime surfaces in the sense of quantum parallel classical worlds,
is in question].
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).
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.
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.
*************
2. Classical nondeterminism of Kaehler action <-->
p-adic nondeterminism
I realized the connection when pondering following problem:
*What principle determines the value of the p-adic prime associated with
given spacetime region?*
The answer to this question relies on the following picture [mblocks].
a) p-Adic spacetime surfaces are in rough sense images of real spacetime
surfaces obtained by canonical identification mapping real spacetime
points x, whose imbedding space coordinates h^k have finite number of
pinary digits, to their p-adic counterparts. p-Adic spacetime
surface satisfies the p-adic counterparts of the field equations
associated with the Kahler action.
b) The characteristic feature of the p-adic differential equations is
the existence of pseudoconstants: pseudoconstants are functions of pinary
cutoffs of p-adic coordinates, which are constant below some arbitrarily
small but finite length and time scales. [Pinary cutoff is essentially
equivalent with decimal cutoff mathematically]. This property makes it
possible to construct p-adic spacetime surfaces having the properties
listed in a). In particular, the requirement that p-adic surface is almost
completely determined as the p-adic image of the real spacetime surface is
not in conflict with p-adic field equations.
This construction as such does not seem to give any hint about how the
allowed p-adic prime is determined. In order to get to the core of the
problem one must consider not only single real spacetime surface but *all
spacetime surfaces which are identical outside a given spacetime region*
V^4 and give rise to the same absolute minimum value of the Kaehler
action. There are a lot of them because of the classical nondeterminism of
the Kahler action. The idea is simple: require that classical and p-adic
nondeterminisms are equivalent in the following sense:
**For the physical value of p-adic prime p associated with a given
spacetime region V^4, the p-adic images of various real spacetime
surfaces differing from each other only in V^4, must correspond to
various p-adic spacetime surfaces obtained by varying pseudo constants
in the representation of the p-adic spacetime surface.**
This requirement fixes the value of the p-adic prime
since p-adic nondeterminism is characterized by a fractal hierarchy of
p-adic length scales L(p,k)= p^kL_p and this fractal hierarchy must
characterize also the classical non-determinism of Kaehler action
in spacetime region characterized by p.
This hypothesis has far reaching consequences:
a) It gives precise quantitative grasp on the nature of the classical
nondeterminism of Kaehler action and hence also to the dynamics of
cognitive spacetime sheets.
b) Since classical nondeterminism corresponds also to quantum
nondeterminism, hypothesis implies that it should be possible to determine
the p-adic prime characterizing given spacetime region (or spacetime
sheet) by observing a large number of time developments of this system
(involving quantum jumps). The characteristic p-adic fractality, that is
the presence of time scales T(p,k)= p^k T_p, should become manifest in
the statistical properties of the time developments. For instance,
quantum jumps with especially large amplitude would tend to occur
at time scales T(p,k)= p^k T_p. T(p,k) could also provide
series of characteristic correlation times. Needless to say,
this prediction means definite departure from the
nondeterminism of ordinary quantum mechanics
and only at the limit of infinite p the predictions should
be identical. An interesting possibility is that 1/f noise is direct
manifestation of classical nondeterminism: if this is the case, it should
be possible to associate a definite value of p to 1/f noise.
***********
Appendix: How to construct the p-adic counterpart of real
spacetime surface?
The solution of the problem involves following argument providing a rough
construction recipe for the p-adic counterpart X^4_p of a real spacetime
surface X^4. Note that X^4_p belongs to p-adic imbedding space H^4_p
whereas X^4 belongs to real imbedding space H.
a) One can associate to a real spacetime region V^4 p-adic spacetime
region V^4_p by mapping certain points h of real spacetime region to
their p-adic counterparts h_p in p-adic H by canonical identification
x=SUM(n) x_np^(-n)--> SUM(n) x(n)p^n
[Note that this expansion is analogous to decimal expansion.]
applied to the various imbedding space coordinates h^k.
It seems that the use of canonical identification involves selection of
preferred coordinates in imbedding space. Since canonical identification
is well defined for non-negative real coordinatse only these coordinates
must be non-negative. For instance, the exponentials of the geodesic
coordinates of M^4_+xCP_2 are good candidates for the preferred
coordinates. Note that this procedure is General Coordinate Invariant
at the level of spacetime surface.
b) The trick is to perform pinary cutoff in n:th postive pinary digit
and taking the limit n--> infinity. This corresponds to cutting
off the pinary expansions of coordinate variables in some sufficiently
high pinary digit n. This procedure is completely analogous to decimal
cutoff. This means that one considers only the p-adic images of the
points of real spacetime surface for which the pinary expansions of the
imbedding coordinates contain no pinary digits higher than p^n:
h^k(x) = SUM(r<=n) h^k(r) p^(-r)--> SUM(r<=n) h^k(r) p^(r)
The p-adic images of these real points are two-valued
since there are two equivalent pinary expansions for these points
(1=.9999..). The numerically favoured option is to choose the p-adic
image of the finite pinary expansion as the p-adic image.
These points provide a discretization of the real and p-adic spacetime
surfaces becoming increasingly denser as n increases.
c) In this approximation the p-adic images of the real points with pinary
cutoff form a discrete set in p-adic imbedding space. The task is to
find smooth p-adic spacetime surface going through all these points
satisfying p-adic version for the field equations deriving from p-adic
Kaehler action. By taking the limit, when pinary cutoff is taken to
infinite (desimal expansion becomes infinitely long) the p-adic spacetime
surface is fixed uniquely. This is the hope at least!
d) There are good reasons to believe that this hope is fullfilled in
p-adic context: in real context this would certainly not be the case. The
reason is the p-adic non-determinism of p-adic differential equations,
which means that the integration constants are not genuine constants but
functions depending on the pinary cutoff x_c(n) of their argument and
having vanishing p-adic derivative. The cutoff can be arbitrarily high
and the only essential thing is that these pseudoconstants become constant
in some, arbitrary small, but finite scale. This roughly means that one
must replace the initial value for p-adic field equation with initial
values given in an entire spacetime lattice which is gradually made
infinitely dense. Thus the existence of pseudoconstants together with
classical nondeterminism of Kaehler action gives good hopes of finding
p-adic spacetime surface solving the Euler Lagrange equations associated
with p-adic Kahler action such that the real counterpart of this surface
in canonical identification coincides with the original 4-surface at
cutoff points. By p-adic fractality one can increase the value of
n defining p-adic cutoff without any essential change and at the limit
n-->infinity one obtains the desired p-adic spacetime surface
as a solution of p-adic Euler Lagrange equations.
****************
References:
[mblocks] The chapter 'Mathematical building blocks'
of the book 'TGD and p-Adic Numbers'
at http://www.physics. helsinki.fi/~matpitka/padtgd.html
[timesc] The chapter 'The problem of psychological time' of the book
'TGD inspired theory of consciousness with applications to biosystems'
at http://www.physics. helsinki.fi/~matpitka/cbook.html
******************************************************************
With Best,
Matti Pitkanen
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