Matti Pitkanen (matpitka@pcu.helsinki.fi)
Mon, 7 Jun 1999 17:35:41 +0300 (EET DST)
Here is the latest version of the TGD based
definition of information, information gain
in conscious experience and information flow.
***************************
1. Introduction
The concepts of information, information gain of
conscious experience and of information flow
would seem to have a natural place
in any theory of consciousness. It has however turned out
that these concept are not at all easily definable.
a) One can associate to the density matrix of subsystem entanglement
entropy. The interpretation as a measure for the lack of self knowledge
suggests itself. Entanglement entropy is reduced to zero in quantum
jump leading to an eigenstate of density matrix and the suggestive
interpretation is as information gain. This interpretation
makes sense in the sense that, since quantum jump involves a conscious
choice between final states of the quantum jump, conscious
information is created when the choice is made.
This information does not seem to have anything to
do with what we usually regard as information (say information
obtained by reading this page). For instance, entanglement entropy
depends on the general tensor product decomposition of the quantum state
only: there is no reference to the geometric
representation of the quantum state
as configuration space spinor field. Hence it seems that entanglement
entropy cannot measure the information about the geometry of
configuration space and of spacetime surfaces.
Entanglement entropy has also an interpretation as
a measure for the 'catchiness' of potential conscious experience
so that strong NMP would state that only the most 'catchy' conscious
experiences are actually experienced.
b) The hypothesis that the number N_d of the degenerate
absolute minima of K\"ahler action is proportional to
the exponent for the negative of the K\"ahler function
(N_d \propto exp(-K_{cr}))
leads to a very attractive realization of quantum criticality.
At quantum criticality N_d and vacuum functional exp(-K) compensate
each other so that TGD is analogous to string model at critical
temperature (for which degeneracy g(E) of energy eigenstates
behaves as exp(E/T_H) and compensates Boltzmann weight
at Hagedorn temperature T_H). In TGD inspired theory of consciousness
N_d(X^3) measures the number of possible thoughts associated with
a given 3-surface and realized as spacetime surfaces going through X^3.
If TGD were a strictly deterministic theory, there would be no
no one to realize it. The logarithm of N_d is clearly an entropy type
measure for the cognitive resources of X^3 and as such kind of
intelligence quotient rather than a measure of information content.
It is obvious that configuration space spinor fields contain the available
information. Configuration space
spinor fields have indeed interpretation as both objective
realities and Platonic Ideas, the latter being
suggested strongly by the possibility to
interpret fermionic Fock state basis as a Boolean algebra of statements
about statements. The basic statements are most naturally statements
about spacetime geometry since fermionic oscillator
operators are determined by the second quantized free quantum field theory
for the induced spinors on spacetime surface X^4(X^3).
One should be able to associate a well defined measure of
information to configuration space spinor field.
This measure should also make it possible to associate
a well defined information gain to quantum jump as the difference of
the informations associated with the initial and final quantum histories.
One however expects that the information contained by entire
universe is infinite and that one can associate definite
information gain to quantum jump only provided the infinite
contribution to the information is independent of quantum state.
In the following a specific proposal for the information measure
is discussed.
a) The definition relies on Shannon information
and the defining formula is same as in kinetic theory of gases
based on probability distributions for single particle states.
Entanglement plays now no special role and no decomposition
into subsystem and complement is involved.
Definition works also in ordinary single particle wave mechanics
but has no obvious generalization to quantum field theory context.
b) Not surprisingly, the proposed measure of
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 and information
gain associated with conscious experience is well defined and finite!
c) The definitions generalize to p-adic context and
information has probabilistic interpretation, which could
explain the generalization of Hawking-Bekenstein formula inspired
by the elementary particle black hole analogy.
d) It seems also possible to define the concept of information
flow: this is essentially due the fact that the
proper time coordinate for lightcone is Lorentz invariant.
2. Information carried by configuration space
spinor field and information gain associated with quantum jump
Configuration space spinor field is determined once its values
on the lightcone boundary are fixed. Nondeterminism
implies that given 3-surface Y^3 on the lightcone boundary corresponds
to several absolute minima X^4(Y^3). 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 delta M^4_+xCP_2
plus summation over the degenerate branches of X^4(Y^3).
2.1 Information associated with configuration space
spinor field
The definition of information to be discussed is used also in
kinetic theory and relies on the idea of selection defined by
configuration space spinor field and on Shannon entropy.
a) The probability that 3-surface Y^3 in volume element dV of
configuration space is selected is
dP = R(Y^3)*dV ,
where R is 'modulus squared' for the configuration space spinor
field at Y^3, which is essentially the norm of the state of fermionic
Fock space.
b) The information associated with a configuration
space spinor field is defined as the negative of Shannon entropy. Using
division into volume elements dV
I= -SUM_{Y^3} dP *log(dP)
= - SUM_{Y^3} R * log(R)* dV - SUM_{Y^3} R*dV* log(dV)
---> - INT R*log(R)DX^3
- log(dV)_| dV-->0} .
The first part gives, at least formally, a well defined integral
over the configuration space. Second term is infinite.
That the information contained by quantum history is infinite,
is not at all surprising. Rather remarkably,
the infinite term does not depend on state! Therefore one can
forget the infinite contribution since it is information differences
which matter and one can define information as
I== -INT R*log(R)*DX^3 .
This kind of formula of course applies also in ordinary
single particle wave mechanics. One should perhaps call I as {\it
available information}.
c) The degeneracy of the absolute minima brings in
summation over branches X^4(X^3) but this is only a minor complication and
can is included in the definition of integral.
2.2 The information gain associated with moment of consciousness}
Each quantum jump is preceded by the action of 'time development' operator
U_a (a--> \infty is lightcone proper time)
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 the configuration space. This obviously means
the reduction of the information and the interpretation is that
the difference
Delta I = I(U_a|Psi_i>) -I(|Psi_f>)
of the informations associated with the time evolved
initial state U_a|Psi_i> 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) term *automatically disappears*
from Delta I! This is quite sensible: it is conscious information
gain, which matters and this must be well defined and finite (at least
formally). It is important to notice that U_a|Psi_i>
rather than Psi_i appears in the formula: if this were not the
case, the information gain would not be positive in general.
Thus the presence of U_a is absolutely essential for
intelligent universe as it is also essential for p-adic evolution.
2.3 Objections
There are some problems to worry about.
a) One can argue that I is actually entropy rather than
information. On the other hand, the larger the value of 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 is important is
that the definition of conscious information gain is unique.
b) One can also worry about General Coordinate Invariance.
In case of a nonrelativistic Schroedinger equation the information
is Galilei invariant. In case of QFT Lorentz invariance
is lost since probability density behaves like a component
of a four-vector. In Lorentz transformed system the troublesome
volume element dV^3 would be
multiplied by a proportionality factor changing the value of the
infinite contribution to information. In quantum TGD situation
R is both Lorentz invariant and General Coordinate Invariant
so that no problems are encountered.
3. Properties of the information and information gain
3.1 Connection with the concept of cognitive resources
One can decompose configuration space spinor field as
Psi = exp(K/2) f ,
where K is K\"ahler function of configuration space. This makes it
possible to express information in the form
I= -<K> -<log|f|^2> ,
where the first term is expectation value for the negative of the
K\"ahler function.
What is remarkable that first term is a direct generalization
of the purely classical hypothesis that K\"ahler function gives entropy
type measure for the cognitive resources of the 3-surface
measured by the number N_d of the degenerate absolute minima
assumed to be proportional to exp(-K_{cr}), where K_{cr} is
K\"ahler 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).
3.2 Decomposition of information to contribution related
to reflective and proto levels of consciousness
Information measure decomposes into several
parts. K\"ahler function represents
vacuum contribution to information, f can in turn decompose
to a product of zero mode functional and fermionic part
giving an additional bosonic contribution to the information.
The purely fermionic Fock space part of the f can be interpreted as
the information related to the reflective, 'consciousness about
consciousness', level of consciousness whereas
bosonic contribution has interpretation as the information related to
the proto level of consciousness. The fermionic part of
f describes all fermions associated with the 3-surface representing
universe: note that in TGD framework elementary bosons are
regarded as fermion antifermion bound states in length scale
of CP_2 so that all matter in the form of elementary particles
corresponds to the reflective contribution to
information. If f decomposes into a product of unentangled
states then also information reduces to a sum of informations associated
with these subsystems.
3.3 Entanglement and information
Intuitively it is clear that unentangled states contain minimum
information.
>From the inequality
<x>^<x> <= <x^x>
it indeed follows that information for f representing
entangled state of two subsystems is in general larger
than the average information for the unentangled states appearing
in the superposition. Same is true for entanglement
in coordinate degrees of freedom. The smallest information
is associated with separable states which are products of
functions depending on various configuration space coordinates
(standard example are provided by eigenstates of hydrogen atom
and momentum eigenstates). This means that the measurement
of subsystem density matrix occurring in quantum jump tends
to maximize information gain.
3.4 Dispersion creates information
I 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 (R= 1/Delta V),
the information I = log(Delta V) becomes negative
and infinite whereas at the limit, when configuration space spinor
field is totally delocalized (R= 1/V), I= log(V)
becomes positive and infinite if the volume is infinite.
The interpretation is obvious. Completely localized configuration
space spinor field does not carry (potential information) whereas
delocalized field carries a lot of information.
For instance, the maximum information carried by phase angle in case
of an angular momentum eigenstate
exp(im*phi) is I= log(2*pi).
3.5 What happens in case of wave mechanics and QFT?
In ordinary quantum mechanics the definition of information
as entanglement entropy for the density matrix
of entire universe does not work since
entropy is constant of motion ad vanishes for pure states.
The proposed definition however works also in case
of wave mechanics.
What is remarkable is that the dispersion
associated with the Schroedinger time evolution
in general increases the information
(potential information gain of quantum jump). Only for
energy eigenstates information is constant of motion.
It is also obvious that the information for
harmonic oscillator states/states
of hydrogen atom increases, when the energy increases
since states become increasingly delocalized. Thus information
and energy measure also the complexity of the quantum state.
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 is different since
it is not possible to interpret time evolution
as evolution in any kind of configuration
space (the required assignment of the space
of quantum states to single point of 3-space
does not make sense). Problems are also caused by the fact
that probability density is not scalar quantity anymore but
time component of a 4-vector.
In quantum 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'.
4. p-Adicization
4.1 p-Adic information concept
The logarithm of R is problematic in real context
and one can quite well wonder whether the integral
over configuration space is well defined.
p-Adicization implies the restriction to definite
sector of configuration space and the 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)*DX^3
=<n> = SUM_n p_n *n
expressing information as p-adic expectation value of Log_p(R)=n.
p_n is the probability that Log_p(R) equals to n.
If n is finite integer it can be regarded as p-adic pseudoconstant.
n can be however infinite.
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 different
values of p give their contribution to the information
gain so that this is indeed the only sensical possibility.
4.2 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/T_p} is the counterpart of Boltzmann weight exp(-E/T),
1/T_p is integer valued inverse
of the p-adic temperature and g_n is the degeneracy
of the state having 'energy' Log_p(R)= n.
In p-adic thermodynamics used to calculate 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 generalization of Hawking-Bekenstein formula: the generalized
formula states the proportionality
of the particle mass squared and p-adic entropy of
the particle: S propto M^2.
The entropy in question
could be entanglement entropy of the particle in p-adic thermodynamical
equilibgrium or the proposed information
measure for the quantum state describing elementary particle:
the connection between geometry and information theory
suggests that the information associated with configuration
space spinor field is in question. These interpretations
need not be mutually exclusive since entanglement increases the
value if I.
Elementary particle blackhole analogy
in turn leads to 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.
5. Can one define the concept of information flow?
One is accustomed to speak about communication as information flow.
Therefore one could wonder whether it is possible to define the concept
of information current somehow in quantum TGD framework.
U_a, a-->infty is indeed defined as a time evolution operator associated
with Virasoro generator L_0 playing the role of Hamiltonian.
Hence it should be possible to formally associate with
the time evolution U_a a conserved probability current having time
component I^a plus spatial components in the degrees of
freedom characterized by the coordinates of the reduced configuration
space. This assigment would be completely analogous to that performed for
the ordinary Schroedinger equation and the Lorentz invariance of
the lightcone proper time coordinate a would make this assignment
possible.
In p-adic context n= Log_pR is pseudo constant
for finite values of the integer n and this would mean
that information current would be conserved locally in p-adic sense.
This would *not* imply the conservation of information even
in the case that n is pseudo constant everywhere.
If this indeed works, one could assign
with a given time evolution U_a transfer of information in
the reduced configuration space of 3-surfaces. The zero modes
characterizing the nondeterminism of the K\"ahler action
contain information about the moments for multifurcations in the
time development of the spacetime surface and this
gives hopes of approximate reduction of this
information flow to an effective information flow
occurring at the level of 'quantum average effective spacetime'.
With Best,
Matti Pitkanen
http://www.physics.helsinki.fi/~matpitka/
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