Matti Pitkanen (matpitka@pcu.helsinki.fi)
Thu, 1 Apr 1999 08:38:03 +0300 (EET DST)
On Wed, 31 Mar 1999, Stephen P. King wrote:
> Dear Ben & Matti,
>
> 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)
>
> Could "entanglement entropy" be similar to "mutual entropy"? I can
> think of the "catchyness" of an idea (or to use Richard Dawkin's term
> meme) as an example is interesting. Could we consider there to be a
> "susceptibility" to a bit of information that is the complement of the
> "catchyness"? Could we quantitate this in terms of the configurations
> that interactive local systems (LSs) have "in common". I think of this
> as how individuals have similar ideas and can thus understand each other
> based upon these similar bits of information. We can think of bits as
> either particular configurations of matter, as in the case of ink on
> paper, or using information theory, as the bit string that maximally
> describes the computation that defines the bit.
I explained entanglement entropy in separate posting so that I will not
go to this. I think the essence of 'catchiness' is
that it is purely quantum mechanical concept (since it measures the amount
of entanglement): quantum entanglement has no classical model
(Schrodinger cat illustrates this well). In quantum computationalism
one could could perhaps introduce this concept. When quantum computation
halts, state function collapse occurs and there is a definite entanglement
entropy reduction associated with this collapse.
It seems that one cannot associate any bnary interpretation to catchiness:
entanglement entropy gain just measures the reduction of lack of knowledge
when a choice is made and is thus purely 'psychological' concept.
Basic point is that there is competition between subsystems/potential
conscious experiences: only the most catchy experience is actually
experienced and corresponding subsystem has a moment of free will.
In real context this competion would be among all subsystems of the
universe and this leads to nonsense result: only single subsystem at given
moment of consciousness would be conscious. In p-adic context strong NMP
becomes a local principle: each moment of consciousness involves
large (infinite) number of conscious subsystems with separate experiences.
>
> > >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 type measure for the cognitive resources of
> > >3-surface.
> >
> > I don't understand this. How do we get from this mathematical measure
> > to "cognitive resources"??
>
> Matti, I am still stumbling over what "Kahler action" is. :( Can you
> think of me as the "man from Mars" and explain it to me in baby steps?
> :)
I try!
a) CP2 has Kahler structure: real unit is represented by metric
g_kl and imaginary unit is represented as an antisymmetric tensor J_kl
satisfying J_k^rJ_rl= -g_kl. J_kl is like Maxwell field: it is closed
form (the other half of Maxwell eequations) and satisfies vacuum Maxwell
equations.
b) One can induced metric and any tensor quantity of imbedding space
H=M^4_+xCP_2 to spacetime surface by projecting it (I would be happy
to give some reference about submanifold geometry, Eisenhart's old book
provides an enjoyable representation but I do not have bibliodata now).
J_munu = J_kl partial_mu h^k partial_nu h^l
(sums with respect to indices are performed)
c) Maxwell action is defined by action density
B^2 -E^2
where B and E are electric and magnetic fields. In covariant notation
this reads
F_munuF^munu
d) Kahler action is Maxwell action for induced Kahler field J_munu
J_munuJ^munu sqrt(g).
g is the determinant associated with the induced metric and index raising
is performed using induced metric.
e) The primary dynamical variables are not the components of vector
potential A_mu now but CP2 and M4 coordinates. This is an important
difference. The field equations are not the ordinary vacuum Maxwell
equations stating that currents and charge densities vanish and
only superpositions of planewaves are possible. Only at the limit
of flat induced metric Maxwell equations are a good approximation.
The field equations have hydrodynamic character: they can be expressed as
conservation laws for 4-momentum and color charges (CP_2 has SU(3)_c as
its isometry group and this gives rise to conservation laws).
I do not type them here them since they are really complicated.
New physical effects are predicted: for instance, vacuum
charge densities (no elementary particles carrying the charges) are
possible unlike.
f) Vacuum degeneracy is the characteristic feature of the Kahler action
and it has turned out that entire quantum TGD and TGD inspired theory of
consciousness relies on this feature. Also ordinary Maxwell action action
allows vacuum degeneracy in the
sense that any vector potential, which is gradient of scalar function,
is vacuum extremal. Now however gauge invariance says that all
gauge related Maxwell fields are physically equivalent so that
degeneracy does not represent anything physical.
In TGD situation is somewhat different.
Any 4-surfaces X^4 whose projection to CP_2 projection belongs to
2-dimensional Lagrange sub-manifold Y^2 defined in canonical coordinates
P_i,Q_i of CP_2 as
P_i = nabla_i f(Q1,Q2), f *arbitrary*(!!) function
Vacuum four-surfaces can have arbitrary topology and they develop
nondeterministically.
Canonical transformations of CP_2 generate
new vacuum surfaces. Any function of CP_2 coordinates serves as a
Hamiltonian of canonical transformation. So that entire function space
labels this further vacuum degeneracy.
Canonical transformations of CP_2 leave Kahler form invariant
and act therefore as U(1) gauge transformations: this also clear from the
fact that single function generates both canonical transformation and U1
gauge transformation.
There is however an important difference between TGD and Maxwell's ED: CP2
canonical transformations *do not act as symmetries of nonvacuum
extremals*. Therefore these symmetries are not gauge symmetries. This
implies spin glass analogy: huge number of almost symmetry related
spacetime surfaces whose physics differs only because induced metric
describing classical gravitation is slightly different.
Classical gravitation breaks U1 gauge invariance. This scenario in
fact realizes Penrose's intuitions to some extend.
Note: this breaking of U(1) gauge invariance does not break the U1 gauge
invariance of electromagnetism: electromagnetic U1 gauge invariance is
still a symmetry and is related to the gauge group of the vielbein
connection of CP_2.
>
> > The relation between your very interesting TGD theory and Hitoshi's very
> > interesting global/local theory is not at all clear to me.
>
> To put in my 2 cents :), we hope that a discussion with a wide variety
> of people with differing backgrounds and specializations but with the
> common goal of a good model of quantum gravity will accomplish more that
> individuals working independently. ;)
>
> > Evidently you all think there is some kind of conceptual correspondence between
> > them in spite of the different mathematical vocabularies.
> >
> > 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 we are in the process of doing just that. :)
>
> > 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
> >
> > ??
>
> If I could interject, Are either of you familiar with Bart Kosko's
> work? He has a formalism that proposes to "bridge" the whole-part
> interrelationship. It is the concept of "fuzzy subsethood" See: Neural
> Networks and Fuzzy Systems, Prentice-Hall, 1991 (ISBN 0-13-611435-0). I
> hope to elaborate on his ideas in the near future. :)
>
I looked at his homepage but I am not familiar with this work.
> > 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...
> >
> > ben
>
> I think of mathematics as the result of humanity's effort to make sense
> of the world that we share. We can work out grand schemes in our
> attempts to "explain everything" and only succeed in making what to
> others appears to be strange and cryptic markings in the sand. It is the
> ideas, not the linguistic, math or English, representations that are
> important. I find that the fact that "our universe only implements a
> limited assortment of the really nice things in the mathematical
> universe" is an example of how Local Systems can only implement (within
> their finite configurations) a finite subset of the Universe, which
> Hitoshi and I consider to be, in itself, Infinite, the Totality of
> Existence.
>
> Onward to the Unknown,
>
> Stephen
>
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