**Matti Pitkanen** (*matpitka@pcu.helsinki.fi*)

*Wed, 17 Mar 1999 09:54:14 +0200 (EET)*

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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

**Next message:**Stephen P. King: "[time 24] Cardinality of Split graphs as E (or V) -> oo"**Previous message:**Stephen P. King: "[time 22] Re: [time 21] RE: Reformulating Weyl's Unified Field Theory"

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