Matti Pitkanen (matpitka@pcu.helsinki.fi)
Wed, 2 Jun 1999 08:49:18 +0300 (EET DST)
On Tue, 1 Jun 1999, Stephen P. King wrote:
> Dear Matti,
>
> Focussing into some detail..
>
> Matti Pitkanen wrote:
> snip
>
> > There is perhaps slight misunderstanding here. Each subsystem is
> > characterized by a density matrix and defines a potential measurement.
> > Strong NMP selects one these potential measurements. There is competion
> > among potential measurements/among potential conscious experiences and
> > only the most informative (I should not use this word!) measurement
> > occurs. The winner can of course decompose to a set of indepenent
> > sub-measurements (separate conscious experiences) and in general does.
>
> Can we discuss NMP more. Could we start with your most concise
> definition?
1. Strong NMP
Strong NMP says that in quantum state/history the quantum jump
(measurement of density matrix) is performed by that
subsystem, for which negentropy gain is maximal. Negentropy
gain equals to entanglement entropy of the subsystem before quantum
jump.
If there are several subsystems giving rise to same maximal negentropy
gain then one of these performs the quantum jump with probability
1/N.
2. Definition of entnaglmen entropy in p-adic context
p-Adic and real cases differ in the definition of entanglement
entropy concept. p-Adic case is the interesting and nontrivial one.
Assume that the initial quantum history decompose into prodcut
of unentangled states in sub-Universes described by tensor
product factors. Entanglement entropy is sum of the
real counterparts of p-adic entropies (obtained by canonical
identification) for the components of subsystem in various
sub-Universes.
The summation over *real counterparts of p-adic entropies* must be
used because
a) otherwise p-adic entropy for entire subsystem
would not be calculable in practice and would have no correlation
with the size of the system. It would be finite since
p-adic ultrametricity would guarantee finiteness but one could
not say anything about its real value. Strong NMP would not
predict anything.
b) in QFT limit statespace decomposes into tensor product of
different p state spaces and one would have sum of p-adic numbers
belonging to different number fields, which does not make sense.
3. Reduction to a local principle in p-adic context
p-Adic form of strong NMP reduces effectively to strong NMP in
sub-Universes since the contributions from various sub-Universes
are positive and can be maximized separately. Each irreducible
subuniverse (having no further decomposition into unentangled
states) gives rise to its own separate sub-quantum jump/conscious
experience.
>
> snip
> [SPK]
> > > :) Are you understanding how Peter uses non-well-founded sets to
> > > do this?
> >
> > I studied the paper. I have impression that getting rid of inductive
> > approach (forgive me for my loose use of terms) means getting rid of
> > initial value problem and hence of the problem of the initial state. One
> > cannot solve time development by starting from initial values at given
> > moment.
>
> We replace "absolute" initiality with the idea of finite windows that
> are functions of the observer's ability to distinguish properties. See
> Peter's discussion of expressiveness...
>
> > As a matter fact, in p-adic context the possibility of pseudoconstants
> > (piecewise constant functions have vanishing derivative) leads to just
> > this situation when one tries to solve field equations. One must fit the
> > solution to go through a set of points most naturally chosen
> > by using all points with given pinary cutoff:
> > x = SUM x_np^n --> x_N = SUM(n<N) x_np^n .
> > One cannot predict future or retrodict past from recent in p-adic
> > universe.
>
Best,
MP
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