Matti Pitkanen (matpitka@pcu.helsinki.fi)
Fri, 2 Apr 1999 14:32:28 +0300 (EET DST)
On Thu, 1 Apr 1999, Ben Goertzel wrote:
>
> >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.
>
>
> OK, I understand. Thanks.
>
> >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.
>
> This is really not far from the standard von Neumann interpretation
Actually the recent picture about quantu jump provides generalization for
von Neumann's intuitions about brain as ultimate reducer.
In TGD framework cognitive spacetime sheets, which are nearly vacuum and
have finite time duration. [Energy and other conserved quantities flow
from material spacetime sheets to cognitive sheets when they are formed
and back to material spacetime sheets when cognitive spacetime sheets
disappear.] The entanglement of cognitive spacetime sheets, 'Mind' with
spacetime sheets carrying matter, 'Matter' is reduced in allowed quantum
jumps.
>
> >> 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.
>
>
> I think this ties in with my theory that "consciousness is randomness"
>
I would not use the world randomness. After all, in quantum jump selection
between eigenstates of subsystem density matrix occurs: final state is not
at all random. Secondly, the probabilities for final states are not
in general same. And thirdly, strong form of Negentropy Maximization
principle selects unique subsystem which can perform quantum jump: the
system is the one giving maximum negentropy gain and thus having largest
entanglement.
[Entanglement corresponds to attention 'psychologically' so that the
most alert subsystem has moment of consciousness). The most alert
subsystem can of course decompose to mutually unentangled subsystems
having their own separate conscious experiences].
> >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.
>
> Perhaps the wormholes constitute a clock in some way?
> Just daydreaming ;)
I think that any periodic phenomen provides a clock: the basic
problem is to find someone to perceive the reading of the clock(:-).
In quantum jumps between quantum histories picture the nondeterminism of
Kahler action comes at rescue and makes possible conscious experiences
with time localized contents.
The oscillations of Josephson junctions formed by wormhole super
conductors indeed generate clocks if one believes that EEG is a clock.
Amusingly, simplest EEG clock corresponds to sequence of solitons of Sine
Gordon, which is mathematically nothing but gravitational pendulum
rotating. Also EEG oscillations equivalent with oscillating
gravitational pendulum are possible. In latter case EEG is equivalent
with the clock in the wall!
> > Quantum jumps between
> >quantum histories give rise to moments of consciousness creating
> >the experiences of separation.
>
> I think that this general concept is compatible with Hitoshi's theory, but
> he realizes the evolution of histories by a different formalism, and he
> defines
> the "jump leading to a moment of consciousness" as the classification of a
> collection
> of quantum particles as a local system (center of mass, category)
>
Subsystem corresponds to local system clearly.
> >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(;-).
>
>
> Agreed
>
> >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.
>
> In a discrete view your sheets become lattices, and we have multiple weakly
> coupled > lattices
>
p-Adicity leads in a natural manner to lattice like structure. You can
form from real axis 1-dimensional lattice by cutting, say decimal
expansion, from n:th decimal. In p-adic context cutting of pinary
expansion of pinary number so that O(p^n) part of p-adic number is put to
zero is analogous procedure but defines equivalence relation in p-adic
context. Hence one can define entire hierarchy of discrete coset spaces
R_p/E_n by this equivalence relation (denoted by E_n).
This hierarchy of lattices defines extremely rapidly converging
approximation procedure for physically interesting primes p (p=2^127-1 for
electron!). Various physical fields become in this approximation fields in
lattices.
What is especially nice is that p-adic counterpart of, say, Poincare group
respects these lattice structures. I told about how p-adic Poincare
group leaves finite p-adic spacetime cube invariant in some earlier
posting few weeks ago. One can quite well say that p-adics are Taylor made
for lattice approximation.
Personally I however believe that geometry is continuous at basic level.
The basic reason for this is that infinite-dimensional geometry is highly
unique: in TGD case the sole requirements that Riemann connection exists
mathematically + some other general requirements fix the entire
geometry and also imbedding space itself essentially uniquely. In TGD
framework this means unique physics also since physics is just
infinite-dimensional spinor geometry. The inability of physicists to
find divegence free QFT:s reflects also this high uniquess of
infinite-dimensional mathematics.
> Different sets of links on the same set of nodes, perhaps?
>
> if you could present a discretized version of the many sheeted theory it
> would make it more clear to
> everyone and might make correspondences with Hitoshi's and other theories
> more clear --
> just a pie-in-the-sky suggestion ;)
>
> >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.
>
>
> But it's all bits and bytes ultimately, Matti. Infinite dimensional math is
> a shortcut for some purposes,
> but also obscures things sometimes IMO. I say this as a mathematician who
> spent many years studying
> functional analysis etc.
p-Adic approach fits very naturally with bits and bytes philosophy.
For instance, even infinite-dimensional configuration space integral
reduces to a discrete sum. My own view is that
objective realities=quantum histories are continuous object but that our
consciousness is able to work with bits and bytes
only. TGD however leads naturally infinite primes and p-adic number fields
associated with infinite primes (which are actually very much like reals):
also infinite hierarchy of consciousnesses is predicted. Perhaps these
Godlike consciousness above us are not limited to play with bits and bytes
only(;-).
MP
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