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

*Mon, 7 Jun 1999 17:35:41 +0300 (EET DST)*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Stephen P. King: "[time 393] Constructing space-times"**Previous message:**Matti Pitkanen: "[time 391] Re: [time 388] Re: [time 384] Re: [time 380] Re: [time 376] What are observers"

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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