Matti Pitkanen (matpitka@pcu.helsinki.fi)
Thu, 1 Apr 1999 07:42:11 +0300 (EET DST)
Below answer to your questions. It happened as it often happens: the
posting became very long. I apologize.
On Wed, 31 Mar 1999, Ben Goertzel wrote:
> Hi,
>
> >In TGD the entropy associated with density matrix of subsystem is in
> >key role: strong form of Negentropy Maximization Principle states
> >that in a given quantum state quantum jump is performed by the subsystem
> >for which the negentropy gain is maximum in quantum jump reducing
> >entanglement entropy to zero. The 'physical' interpretation is following:
> >entanglement is measure for attentiveness not yet involving consciousness.
> >Entanglement entropy measures, not the information content of
> >conscious experience, but how 'catchy' the potential conscious
> >experience is. The most catchy consciouss experience is experience.
> >Mass media people would certainly agree with this!
>
> I don't know what "entanglement entropy" is, sorry. What is the formula
> for this? --
> in the discrete case (to keep things simple)
In QM entanglement is one of the most QM:eish phenomena and plays key
role in quantum compute sciencer. Schrodinger cat provides the standard
example of entanglement. The state of cat and bottle of poison is
superposition of cat dead-bottle open and cat alive-bottle closed states.
In TGD inspired theory of consciousness these kind of superpositions
appear at macroscopic level and our volitional acts select one branch
in the superposition.
Entanglement entropy serves as a measure for entanglement.
Let entangled state be
Psi = C _Nn |N> |n>
where summation occurs over M and N
In entangled state one cannot associated pure quantum state to subsystem
and density matrix provides the description of subsystem. Density matrix
is obtained from formula
rho_mn= C_MmC_Mn^*, (sum over M)
One can clearly integrates away external world states labelled by M.
Density matrix can be diagonalized and eigenvalues have interpretation
as probabilities p(m) for occurrence of particular state m in the
entangled superposition.
The amount of entanglement is measured by entanglement entropy
S= Tr(rho*log(rho))= SUM p(m)log(pm)
and is of same form as Shannon entropy characterizing how far subsystem
is from pure state.
TGD based QM measurement theory postulates that density matrix rho is the
universal observable measured in quantum measurement and that subsystem
goes in quantum jump to eigenstate |m> of rho with probability p(m)
and thus ends up to pure state without any entanglement. In this quantum
jumps S goes to zero and negentropy gain is just S: one could perhaps say
that subsystem gains self knowledge (or rather is conscious). The
original interpretation of S was as a measure for information content of
conscious experience but the interpretation as a 'catchiness' of conscious
experience has turned out to be a more natural interpretation.
There are some interesting and very important delicacies related
to the definition of S in p-adic context but I will not go to them here.
>
> >The problem is to find also a measure for the information content
> >of conscious experience and there are quite explicit ideas also about
> >this. The modification of Roy Frieden's ideas to TGD context lead to
> >the idea that the number of degenerate absolute minima of Kahler action
> >going through given 3-surface X^3 (there are several of them by classical
> >nondeterminism) is entropy typ measure for the cognitive resources of
> >3-surface.
>
> I don't understand this. How do we get from this mathematical measure
> to "cognitive resources"??
This is a long story told in my homepage
(http://www.physics.helsinki.fi/~matpitka/cbook.html).
In TGD quantum states are replaced with quantum *histories* and moments
of consciousness correspond to quantum jumps between them. Contents of
conscious experience are assumed to localize into region where
nondeterminism of quantum jump is localized: consciousness is where the
free will is.
The problem is following: since entire quantum histories are in question,
the *contents of conscious experience should depend on the initial and
final history only*: spatial localization of contents of conscious
experience is possible but there seems to be no time localization
and hence no psychological time.
The solution of problem is provide by TGD and relates directly to General
Coordinate Invariance. 3-surface in M^4_+xCP_2 is
basic dynamical object of TGD and the so called
Kaehler action associates to every 3-surface X^3 a four-surface X^4(X^3):
this makes possible *4-dimensional* general coordinate invariance in
space of *3-surfaces* X^3 since Diff^4 can act on this 4-surface.
The unique property of Kaehler action is its *classical
nondeterminism*: there is large number of 4-surfaces X^4_i(X^3) going
through X^3 and having same absolute minimum value of Kaehler action.
For instance, one can have situation in which spacetime surfaces differ
in a finite time interval only: nothing like this occurs in standard
physics based on strictly deterministic action principles.
This classical nondeterminism makes possible quantum nondeterminism:
many particle states can get quantum entangled with various
classical branches X^4_i(X^3) of spacetime surface and if these
branches differ in a finite time interval, the nondeterminism of quantum
jumps selecting one of the branches is located in finite time interval.
Hence the contents of conscious experience give dominantly information
about physics in that time interval and psychological time emerges.
The number N_d of different absolute minima X^4_i(X^3) clearly
tells how many possible alternative choices subsystem defined by X^3
has. The larger the value of N_d, the larger the amount of free will.
Also cognition involves free will: we select between thoughts when we
direct attention to something. Thus N_d measures also cognitive
resources, the number of possible thoughts subsystem can have.
Of course, it measures also the ability of subsystem to change the world
since some bracnhes of multifurcations have long lasting effects on the
development of spacetime surface.
>
> The relation between your very interesting TGD theory and Hitoshi's very
> interesting
> global/local theory is not at all clear to me.
>
> Evidently you all think there is some kind of conceptual correspondence between
> them in spite of the different mathematical vocabularies.
I try to list some of the common features.
a) Hitoshi postulates X^4xR^6: R^6 corresponds to phase space of
classical QM and he wants a marriage between QM and General Relativity.
This is what I also want. One can say that Hitoshi separates QM and
General Relativity to Carteisian product to achieve this.
b) I want to get rid of the energy problem of General Relativity
(isometries of spacetime are lost when spacetime becomes curved and it is
difficult to find satifactory definition of energy since Noether theorem
is not at use). I postulate that spacetime is surface in M^4_+xS
X^4 in M^4_+xCP_2
and assume that Poincare symmetries act on M^4_+ (future lightcone)
rather than spacetime so that Poincare is broken only cosmologically.
S=CP_2 is fixed by the requirement that standard model gauge group SU3 xU2
(color group +electroweak group) results from the geometry of S. Also
geometrization of elementary particle quantum numbers and classical gauge
fields results in this manner. TGD can be also regarded as a
generalization of string model: strings are replaced with 3-dimensional
surfaces and the miraculous properties of the 4-dimensional future
lightcone guarantee that Super Virasoro and Kac Moody algebra
structures quintessential to string models generalize.
c) The concept of local system has as its TGD analog spacetime sheet of
finite size. The idea of local system is however realized quite
differently in TGD. Hitoshi introduces clock at every point (I
apologives if I have not understood correctly!). In TGD approach
spacetime sheets representing elementary
particles, nuclei, atoms,...ourselves,.... , galaxies,... are
local systems realized as spacetime sheets which have contact to larger
spacetime sheets via extremely tiny wormholes. This is like taking
extremely thin 3-dimensional parallel slightly curve membranes or glass
plates of various sizes and gluing them together by these tiny wormhole
contacts. Each plate represents its own sub-universe, local system with
respect to larger glass plates. LS property is relative property.
Every spacetime sheet which is 'General Relativistic' object as such
is LS relative to larger spacetime sheets.
>
> Maybe it would be useful for you two to articulate what you think the
> points of commonality
> and points of difference between the two approaches are. If you have been
> over this ground
> before in other forums please excuse me for being so presumptuous!
>
> I think that if we arrive at a sufficiently abstract and foundational
> perspective we will be able
> to see exactly where the two approaches coincide, and then where they
> diverge in the process
> of making additional mathematical assumptions to turn philosophy into science.
>
> Matti, does your theory fit into the general framework of
>
> -- one set of laws for parts
> -- one set of laws for wholes
> -- a bridging principle explaining how whole-laws and part-laws interrelate
TGD inspired theory of consciousness relies on subsystem-complement
separation. Quantum states=quantum histories =objective
realities (=Platonic ideas somewhat surprisingly) correspond to all
possible wholes: they are unconscious. Quantum jumps between
quantum histories give rise to moments of consciousness creating
the experiences of separation. There is no unique objective reality/whole
as in materialistic world view since quantum jump replaces the cosmology
with a new one: as conscious beings also we are (mini)Gods(;-).
A different aspect to whole/part distinction is related to the
manysheeted spacetime concept. Different spacetime sheets correspond
to different branches of physics: at nuclear spacetime sheets nuclear
physics applies and at atomic spacetime sheets atomic physics is
satisfactory description. The reason why these physics are practically
separate is that interactions between different spacetime sheets are
weak. Many-sheeted spacetime makes possible to explain
generation of structures: in standard quantum field theory basic objects
are point like particles and all geometric structures are ad hoc concepts
introduced as macroscopic idealizations for modelling purposes. In TGD
size and shape of 3-surfaces are completely new degrees of freedom
bringing into physics matter in the sense of res extensa at fundamental
level.
>
> ??
>
> This seems to be the philosophical structure of Hitoshi's theory...
> If your theory could somehow be cast into this form this would give us a
> way to proceed
> in producing a "conceptual diff" of the two theories...
>
> As I said before, I think that getting the ideas right is the key here and
> that mathematical
> tricks are not going to be the answer. There are too many mathematical
> tricks out there,
> the mathematical universe is full of elegance, but our universe only
> implements a limited
> assortment of the really nice things in the mathematical universe...
>
I do not believe in mathematical tricks (although I have tried them
occasionally(;-)). My basic philosophy has been to construct quantum TGD
using only the basic classical spinor-geometry generalized to
infinite-dimensional context.
Matti
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