Matti Pitkanen (matpitka@pcu.helsinki.fi)
Thu, 3 Jun 1999 16:30:42 +0300 (EET DST)
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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