Matti Pitkanen (matpitka@pcu.helsinki.fi)
Wed, 16 Jun 1999 08:13:38 +0300 (EET DST)
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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