Matti Pitkanen (matpitka@pcu.helsinki.fi)
Fri, 4 Jun 1999 08:22:13 +0300 (EET DST)
Here previous posting with some additions.
***************************************************
The 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.
This would make possible to associate *genuine information
gain to quantum jump as difference of the informations associated
with initial and final quantum histories*.
[Entanglement entropy would indeed measure the 'catchiness'
of conscious experience rather than its information content.]
It seems that this is possible!
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. Most importantly,
the information gain associated with conscious experience is well
defined since infinite terms cancel each other in difference!
The definition is based on Shannon information. Entanglement plays
now no role. Definition works also in ordinary wave mechanics
but has no obvious generalization to quantum field theory context.
The argument goes as follows.
*******
Concept of configuration space spinor field
Configuration space spinor field is 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).
*******
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
configuration space is selected is
dP = R*dV
where R is 'modulus squared' for the configuration space spinor
field at X^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).
Here INT denotes integral. The first part gives well defined integral
over 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.
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.
******
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. 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.
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).
*******
Properties of the information and information gain
The information defined in the proposed manner is not positive definite.
This follows from the dropping of the infinite background contribution
guaranteing positivity.
At the limit, when configuration space spinor field is located
to infinitely small volume the information becomes negative
and infinite whereas at the limit when configuration space spinor
field is totally delocalized, I becomes positive and infinite.
The interpretation is obvious. Completely localized configuration
space spinor field does not carry (potential information) whereas
delocalized field carries a lot of information.
Each quantum jump is preceided by the action of 'time development'
operator
U_a acting on the initial quantum history. This means dispersion in
the reduced configuration space so that information increases.
The final state results in a quantum jump involving localization to some
sector of configuration space. This obviously means
the reduction of information and the interpretation is that
the difference
Delta I = I_i-I_f
of the informations associated with the
initial and final state is the *information content of conscious
experience* (which in general decomposes into separate sub-experiences).
What is nice that the ill defined log(dV) factor automatically disappears
from I_i-I_f. This is quite sensible: it is conscious information
gain which matters and this must be well defined and finite(?).
One can of course argue that I is actually entropy rather than
information. On the other hand, the larger the I the larger
the potential information gain in quantum jump leading
to localization in configuration space. Therefore one can say
that entropy is a necessary prequisite for information gain and
could as well be regarded as (potential) information.
Only sinner can have the moment of mercy!(;-)
*******
What happens in case of wave mechanics and QFT?
The definition of information concept works also in case
of wave mechanics. What is remarkable is that the dispersion
associated with Schrodinger time evolution increases the information
(potential information gain of quantum jump). The information
associated with density matrix associated with pure state is constant.
For instance, the information for harmonic oscillator states/states
of hydrogen atom increases, when the energy increases
since states become increasingly delocalized. One can generalize
the definition also to the case of many particle wave mechanics
by replacing 3-dimensional configuration space with 3N-dimensional
configuration space.
In quantum field theory situation seems to be different since
it is not possible to interpret time evolution in any kind of
configuration
space. In TGD the situation is saved by the fact that configuration space
spinor fields are infinite-dimesional classical spinor fields so that
one can regard states of universe as states of single gigantic
classical fermion.
********
p-Adicization
The logarithm of R is problematic in real context
and one can quite well wonder whether the integral is well defined.
p-Adicization implies some modifications (restriction to definite
sector of configuration space and replacement of logarithm with
its p-adic counterpart Log_p(R), which is integer valued and
determined by the p-adic norm of R. Hence on obtains extremely simple
formula
I= Int R(X^3) n(X^3) dV =<n> = SUM(n) p_n n
expressing information as p-adic expectation value of n= Log_p(R).
p_n is the probability that Log_p(R) equals to n.
In p-adic context information gain in quantum jump must be defined
as the difference for the *real counterparts of the p-adic
information* for initial and final quantum histories. For the state
U_a |Psi_i| preceiding quantum jump p-adic sectors with all values of
p give their contribution to information so that this is indeed the
only sensical possibility.
*******
Connection with p-adic thermodynamics and
generalized Hawking formula
A good guess is that the huge complexity of the infinite-dimensional
situation implies that the probabilities p_n can be calculated from p-adic
thermodynamics and are hence of the form
p_n= g(n) p^(n/T_p) ,
where p^(n/F_p) is counterpart of Boltzmann weight exp(-E/T),
1/T_p is integer valued p-adic temperature and g_n is the degeneracy
of state having 'energy' Log_p(R)= n.
In p-adic thermodynamics determining the values of particle mass squared,
exactly similar formula for particle mass squared as analog of thermal
energy results. Elementary particle black hole analogy leads to
the hypothesis that Hawking-Bekenstein formula stating the proportionality
of particles mass squared and p-adic entropy of particle generalizes!
This in turn leads an intuitive justification for the p-adic length
scale hypothesis stating that p-adic primes near prime powers of
two are the most interesting ones physically: geometrically the hypothesis
means that the radii of elementary particle horizons are p-adic
length scales themselves. Thus it seems that generalization
of Hawking-Bekenstein formula might derive from the proposed
definition of information for quantum state.
********
To sum up, the prospects seem good! Of course, it takes week or
two to think all details through but I think that I can express
already now my gratitude for Stephen for very stimulating discussions!
Best,
Matti Pitkanen
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