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

*Wed, 16 Jun 1999 08:13:38 +0300 (EET DST)*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Matti Pitkanen: "[time 406] Dissipation and quantum jumps between quantum histories"**Previous message:**Matti Pitkanen: "[time 404] On the Problem of Information Flow between LSs"**Next in thread:**Stephen P. King: "[time 407] Re: [time 405] How to define information measures for conscious experience?"

Dear Stephen,

Below is qmind message summarizing the recent situation in problem of

defining measures for information content of conscious experience. There

is also a new chapter in TGD inspired theory of consciousness.

By the way, quantum jump is jump to a state with vanishing entanglement.

In earlier posting You proposed that entanglement could perhaps vanish

only to some accuracy epsilon. I disagreed saying something

like 'philosophy with accuracy epsilon is not attractive idea'.

I was wrong.

Common sense indeed suggests that you are correct.

There are several arguments.

For instance, conscious experiences bind to single experience if there

is arbitrary but nonvanishing small entanglement present. It is difficult

to understand why conscious experiences would become separate experiences

precisely when entangelment is zero and integrate to single experience

for arbitrary small entanglement. Rather, one would expect

some critical entanglement entropy below which integration

does not occur.

I found that this is the case!

The point is that real states are mapped to their p-adic counterparts

and if real entanglement entropy is smaller than the pinary resolution

(recall pinary cutoff) then real entanglement with entanglement

entropy below pinary cutoff is mapped to zero p-adic entanglement!

p-Adic entanglement could be even defined as entanglement with

the unique pinary cutoff! Pinary cutoff defines the resolution

of conscious experience also.

Best,

MP

***********************************************************************

The work of Frieden [Frieden] and the very stimulating discussions

with Stephen King [SKing] inspired serious consideration of

the problem of defining measures for the information content of conscious

experience. This resulted in a new chapter 'Information

theoretical aspects of TGD inspired theory of consciousness'

at

http://www.physics.helsinki.fi/~matpitka/cbook.html#Ch9.

Abstract

The basic task is to associate information measures with conscious

experience and possibly also with quantum histories. Also it

would be nice if one could assign information currents

with the 'informational' time development U_a, a--> infty associated

with quantum jump (analogous to the time development operator of

ordinary QM). This would make it possible to construct quantum

model of conscious communication.

This indeed turns out that this possible. Crucial role in the

construction is played by the fact that in quantum TGD physical

states can be regarded as classical spinor fields in infinite-dimensional

configuration space. The resulting formulas apply also in

nonrelativistic QM but not in quantum field theory

and the existence of unique Lorentz invariant time coordinate

is crucial ingredient of the approach.

The key ideas are following:

a) Information content of conscious experience can be defined as the

*difference of the informations associated with initial and initial

quantum histories*, which are well defined geometric objects and to which

classical information theory applies.

b) The requirement that *information gain of conscious experience reduces

to sum of information gains associated with irreducible sub-experiences*

implies that information measures are *local* at configuration space

level.

c) The *real-to-p-adics correspondence* crucial for the understanding

of the p-adic aspects of quantum TGD relies on phase preserving

canonical identification with pinary cutoff. Pinary cutoff

makes it possible to assign unique information measures with various

geometrical objects and one can interpret the *characteristic

coarse graining of conscious experience as resulting from

pinary cutoff*.

It is found that the definitions allow a beautiful

information theoretic interpretation for the evolution by quantum jumps:

one can speak of 'holy trinity' of time developments: classical

time development at the level of 3-surfaces, time development by

quantum jumps and 'informational' time development defined

by the unitarity time development operator U_a, $a-->infty

(a is lightcone proper time).

******

1.Information content of conscious experience as difference

of informations associated with initial and final quantum

histories

The basic objection against assigning information measures to conscious

experience is that it is impossible to characterize

conscious experience by a bit sequence and hence also impossible

to assign information measure to it, say, as the length of the bit

sequence or the length of the minimal program producing the bit sequence.

One can however circumvent this problem! Conscious

information can be defined as the difference of the informations

associated with the initial and final quantum histories

U_a*Psi_i and Psi_f, where U_a, a-->infty

could be called 'informational time development' operator

analogous to time development operator in ordinary QM:

Delta I = I(U_a*Psi_i>) -I(Psi_f>) .

Therefore quantum jump can be said to transform

part of the information of the initial state to conscious information.

The problem of defining information measure for conscious experience

reduces to that of associating information measure(s) to quantum

histories! Since quantum histories correspond to well defined geometric

objects, the methods of the existing classical information theory are

available. Second consequence is rather paradoxal: the larger the

entropy, the larger the potential information gain in the quantum jump.

Hence entropy can be regarded as potential information and it is

only a semantic question whether to speak about entropy or information.

What matters is the information gain of conscious experience and this is

uniquely defined.

******

2. Strong NMP as a heuristic guide

Strong NMP (see the chapter 'Strong form of NMP' of [cbook])

gives strong constraints on the detailed definition of

the information measures. Strong NMP predicts the decomposition of the

conscious experience into separate sub-experiences corresponding to

irreducible subsystems characterized by finite values of

the p-adic prime whereas entire universe (as 3-surface)

is characterized by infinite p-adic prime. Hence arbitrary information

gain should reduce into a sum of the information gains associated with

separate conscious experiences. Furthermore, these information gains

should be real counterparts of the p_i-adic information gains,

where the finite primes p_i label separate 'sub-universes':

only in this manner information gain is finite (log(p)/log(2)

conversion factor multiplying real counterpart of information gain

diverges if p is infinite). This also guarantees

upper bound for the information gains of p_i-adic subsystem.

Hence the decomposition of the spacetime surface to regions

characterized by finite p-adic primes p_i

seems to be more than a convenient approximation

associated with QFT limit of TGD.

*********

3. Each information type defines its own information measure

Information is always about something and information

measure depends on what this something

is. Therefore one can define large number of information measures.

a) The first thing to come in mind is to define real information

as integral over the configuration space for the information density

i= R*log(R) ,

where

R=exp(K)*|f|^2

is the Fock space norm of the configuration

space spinor field at the surface Y^3 in the reduced configuration

space consisting of 3-surfaces on lightcone boundary. Information

density i measures basically information about the position of

the Universe ('configuration space fermion')

in configuration space.

Positional information clearly measures how

much quantum state deviates from classicality. For

living systems, which are very information rich, the deviation

is expected to be especially large.

Note that i does not give information about

spacetime surface. 'Informational time development'

operator U_a, a --> infty is expressible as the exponential

of the Virasoro generator L_0 acting essentially as

spinor Laplacian of the configuration space and therefore causes

dispersion. Hence the action of U_a increases I.

Since quantum jump always involves localization into

a definite sector of the configuration space, the positional

information gain associated with the quantum jump is always positive.

The classicality of quantum state increases dramatically in quantum

jump.

b) The precise definition of the real counterpart of

the positional information involves

decomposition to a sum of the contributions associated with

irreducible (minimal unentangled) sub-Universes characterized by

finite p-adic primes p_i giving rise to the decomposition

f= PROD(i) f_i

of the fermionic part of the configuration space

spinor field: this means that one can write

the real counterpart of information as

I^R= sum_i (I_i)^R .

Calculation of I_i involves the mapping

of the configuration space spinor field to its p_i-adic

counterpart using the phase preserving canonical identification

with pinary cutoff and the inclusion of

the corresponding tensor factor f_i of the

configuration space spinor f to get

I_i = INT R * Log_{p_i}(|f_i|^2),

where Log_p function is integer valued p-adic logarithm depending

on the p-adic norm of its argument only. INT denotes

integration over the configuration space. (I_i)^R is defined

using canonical identification

x = SUM(n)x_np^n --> x_R= SUM(n) x_n p^(-n)

and including the conversion factor converting

information into bits:

(I_i)^R = (I_i)_R * log(p_i)/log(2) .

(I_i)^R has interpretation as conscious information gain

associated with the the i:th conscious experience.

Similar decomposition should hold true for all information

consciously available for finite subsystems.

*********

4. Locality of the information measure guarantees

the decomposition into sum of informations associated with

sub-experiences

The decomposition of information to a sum of informations

associated with irreducible subsystems requires that conscious

information measure is local at the level of configuration space.

This means that information is expressible as integral of

information density

i= R*X

over configuration space. R is probability density associated with

configuration space spinor field. Local information measure X(Y^3) is

function of 3-surface Y^3 characterizing local information about

configuration space geometry, information about spacetime surface X^4(Y^3)

and about configuration space spinor at Y^3. Locality makes it possible

to decompose the information into contributions associated with separate

conscious sub-experiences and also allows to associate information current

to the 'informational time development' U_a, a --> infty.

Any orthonormal state basis of configuration space or its sector

D_p can be defined as eigenstates

of some set of mutually commuting observables and the Fourier

coefficients of the configuration space spinor field in this basis

define probabilities p_n appearing in the p-adic counterpart

I= SUM(n) p_n*Log_p(p_n)

of the Shannon formula defining a measure

for the potential information contained in the state about these

observables. Here Log_p(p_n) is integer valued p-adic logarithm.

Locality requirement however allows only the 'position

eigensate basis' and the only freedom remains in choosing configuration

space spinor basis at each point Y^3 of the configuration space.

Unless the definition of the configuration space spinor

basis contains implicit information about spacetime geometry,

these information measures give information about configuration

space spinor but not about spacetime geometry.

As a special case one finds interpretation for entanglement

entropy as the information relating to the state basis,

for which configuration space spinors are eigenstates

of the density matrix for the quantum jumping subsystem.

Entanglement entropy gives information about spacetime surface since the

definition of the subsystem concept reduces to the level of spacetime

surface and information about spacetime geometry is therefore present

implicitely.

********

5. Real-to-padic correspondence allows to define

unique information measures

The simplest manner to define information is as the number

of bits in a series of binary digits.

The construction of the p-adic counterparts of various geometric

structures (spacetime surfaces, spacetime spinors, configuration space

spinor fields) relies on the phase preserving canonical identification

with minimal pinary cutoff (see the chapter 'Mathematical concepts and

ideas' of [padTGD]). Minimal pinary cutoff

means finite number N of pinary digits and finite

information. The obvious question is whether this

pinary cutoff might have information theoretic meaning and whether

it could even describe the coarse graining characterizing conscious

experiences. Minimal pinary cutoff allows to construct very general

unique measures for the information about various aspects

of the spacetime geometry, local configuration spacetime geometry

and configuration space spinors. The fact that the

real counterpart of p-adic N is finite even when N is infinite as real

integer, has a nice interpretation: finiteness reflects the

the finite intelligence of a subsystem characterized

by a finite p-adic prime.

For instance, in case of spacetime surface only the discrete

pinary cutoff of the canonical image can be continued to a

smooth p-adic spacetime surface satisfying the p-adic counterparts

of the field equations implied by the absolute

minimization of the Kaehler action.

Minimality requirement means that the completion to a

smooth surface resembles the canonical image

as closely as possible. Since finite number

of pinary digits of the imbedding space coordinates

are used to characterize the points of the spacetime surface,

the information associated with the value of a given imbedding

space coordinate at given spacetime point

is finite and given by the canonical image of the number N of

pinary digits in the pinary expansion of the coordinate

if log(p)/log(2) is used as a unit of information. Obviously

this information is bounded by p*log(p)/2.

For example, one can calculate the p-adic information contained by

a given 3-surface Y^3 on lightcone boundary as sum of the informations

associated with discrete spacetime points. As a real

number this information measure would is infinite but its

canonical image defines finite real valued information measure.

The real counterparts for the expectation value of the p-adic

information measure in given quantum state is also well

defined and information gain (or loss!) in quantum jump is also

well defined. One can also associate information measures

to induced gauge fields and induced metric.

One can also calculate the information content of a given region

of spacetime surface and this makes it in principle to make

information map of average spacetime surface telling at which

regions information content for given sub-experience is maximal:

the time value around which maximum occurs can be defined

as the *value of psychological time*. Obviously

information content is at maximum in regions containing

cognitive spacetime sheets. Information measures

allow also rigorous discussion of the *quantum correlates of

sensory qualia*.

*******

6. Information current

The 'informational' time development U_a, defined by the exponentiation

of the Virasoro generator L_0 acting as configuration space spinor

Laplacian, allows to identify probability current having probability

density as time component J^a and infinite number of spatial components

J^k in configuration space. This current is conserved. One can

also assign information current

I^i = J^iX

to any local information measure defined as J^a X, where X characterizes

particular information measure. These information currents are not

in general conserved locally and even when this is the case, information

itself is not conserved in general. Information current concept makes

it possible to model the *communication occurring in a given quantum

jump*.

The existence of information current relies crucially on

the fact that imbedding space is Cartesian product of future

lightcone with CP_2: it is lightcone cosmology, which allows

to identify Lorentz invariant time coordinate a as the proper time

of the future lightcone. Second crucial element is that

configuration space spinor field describes Universe as classical

'fermion'. Also nonrelativistic QM allows similar local information

current whereas relativistic quantum field theory does not allow

the definition of either information density or information current

at spacetime level.

********

7. 'Holy trinity' of time developments

The construction of information measures led to realization

of 'holy trinity' of time developments. There is classical time

development defined by the Kaehler action, time development

by quantum jumps and 'informational' time development

by U_a in the space of quantum histories just like

there is holy trinity of existences: matter ('res extensa') as geometry,

subjective existence as quantum jump and objective

existence as quantum histories/objective realities/ideas.

Rather remarkably, informational time development can be also

interpreted as a quantum computation in a very general sense

of the word so that the idea about Universe as conscious Quantum

Computer makes sense.

********

References

\bibitem[King]

http://members-central.home.net/stephenk1/Outlaw/Outlaw.html

[Frieden]

R. Mathews, I is the law,

Popular article about Roy Frieden's work in New Scientist 30,

Jan 1999.

\bibitem[TGD]

M. Pitkanen (1995)

Topological Geometrodynamics

Internal Report HU-TFT-IR-95-4 (Helsinki University).

http://www.helsinki.fi/~matpitka/.

[padTGD]

M. Pitkanen (1995)

Topological Geometrodynamics and p-Adic Numbers.

Internal Report HU-TFT-IR-95-5

(Helsinki University).

http://blues.helsinki.fi/~matpitka/.

[cbook]

M. Pitkanen (1998)

TGD inspired theory of consciousness with applications to biosystems.

http://blues.helsinki.fi/~matpitka/cbook.html.

With Best,

Matti Pitkanen

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