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

*Thu, 3 Jun 1999 16:30:42 +0300 (EET DST)*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Matti Pitkanen: "[time 383] What is information?"**Previous message:**Matti Pitkanen: "[time 381] Re: [time 378] Re: [time 376] What are observers"**In reply to:**Stephen P. King: "[time 378] Re: [time 376] What are observers"

Previous discussions about information concept inspired

the attempt to associate *a well defined information to configuration

space spinor field as a property of quantum history.*

Situation does not seem so hopeless as I have believed!

In accordance with intuitive expectations the

information contains *infinite part, which does not however depend

on the state*! Therefore it is possible to compare the information

contents of different quantum histories! The information

measure relies on Shannon formula and idea of selection. Entanglement

plays now no role. Definition works also in ordinary quantum mechanics.

The argument goes as follows.

Concept of configuration space spinor field

In absence of nondeterminism of Kaehler action configuration space spinor

field would be completely determined once its values

on the lightcone boundary are fixed. Nondeterminism however

implies that given 3-surface on the lightcone boundary corresponds

to several absolute minima. This forces the generalization of

the concept of 3-surface. The space of 3-surfaces on the lightcone

boundary is like manysheeted like Riemann

surface with various sheets corresponding to various absolute

minima X^4(Y^3) fixed by choosing some minimal number

of 3-surfaces from particular absolute minima: these

association sequences provide geometric representation

for thoughts. What is essential that everything

reduces to lightcone boundary since inner product for configuration

space spinor fields can be expressed as *integral over the space

of the 3-surfaces Y^3 belonging to lightcone boundary xCP_2

plus summation over the degenerate branches of X^4(X^3)*, this

space will be referred to as *reduced configuration space*.

How to measure the information associated with

configuration space spinor field?

The idea of selection and Shannon entropy works also here.

a) The probability that 3-surface X^3 in volume element dV of

reduced configuration space is selected is

dP = R*dV

where R is 'modulus squared' for the configuration space spinor

field at Y^3, which is essentially the norm of state in fermionic

Fock space.

b) The information associated with configuration

space spinor field is just the negative of Shannon entropy. Using

division into volume elements dV

I= -SUM(X^3) dP log(dP) = -SUM(X^3) R*log(R)*dV -SUM(X^3) R*dV log(dV)

= -INT R*log(R)dV - log(dV).

The first part gives well defined integral over reduce configuration

space. Second term is infinite but does not depend on state!! This

infinite term tells that the information contained in state is infinite,

which is not at all surprising. One can however forget this

infinite since it *is information differences which matter* so that

one can define:

I== -INT R*log(R).

This kind of formula of course applies also in case of ordinary quantum

mechanics. Perhaps one should call I as *available information*, perhaps

information available to conscious experience

c) The degeneracy of absolute minima brings in

summation over branches but this is only minor complication and

can be included in the definition of integral over reduced

configuration space.

Connection with the concept of cognitive resources

One can decompose configuration space spinor field as

Psi = exp(-K/2) f,

where K is Kaehler function and f is fermionic Fock state

depending on X^3. This makes it possible to express

information in the form

I== <K> -<log|f|^2>,

where the first term is expectation value for the

Kaehler function. |f|^2 is Fock state norm squared.

What is remarkable that first term is a direct generalization

of the purely classical hypothesis that

*Kaehler function gives information type measure for the cognitive

resources of the 3-surface measured by the number of degenerate absolute

minima proportional to exp(K_cr), where K_cr is Kaehler function at

quantum criticality.*

This suggests that 'ontogeny repeats phylogeny' principle is at work also

here in the sense that

*vacuum expectation for the classical measure for cognitive resources

equals to the quantal information of the vacuum state (apart from

infinite state dependent term).*

p-Adicization implies some modifications (restriction to definite

sector of configuration space and replacement of the logarithm with

its p-adic countepart Log_p(R), which is integer valued and

determined by the p-adic norm of R.

Best,

Matti Pitkanen

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