On Wed, Aug 01, 2007 at 11:31:51AM +0200, Bruno Marchal wrote:
> >
> > No, I mean all information known by the observer (including, but not
> > exclusively information know by the observer about erself).
>
>
> OK, but then adding "about the universe" is confusing at this stage.
> You interpret the quantum state as describing knowledge. (And then I am
> not sure I follow what you mean by quantum state: you are supposing the
> quantum hyp. here, aren't you (or perhaps your linearity hyp. only?
> Again where would that linearity come from?).
>
Sorry, I realised I hadn't responded to this before. Things have got
away from me, including recovering from a harddisk crash.
I am using universe somewhat colloquially here, to help intuition. But
sometime it doesn't help.
What we have are observer moments, which somehow contain all
knowledge, or are circumscribed by all knowledge that an observer has
at an instant of time.
The use of the word "universe" was meant to make some connection back
to discussions of "many universes", or "many worlds", but to be
precise we are just talking about observer moments which are in a
sense primitive.
>
> >
> >>
> >>
> >>> this led me to identify the observer moment
> >>> and the quantum state vector.
> >>
> >>
> >> ... and the partial relative quantum state vector corresponding to the
> >> observer. OK, but at this stage this would be cheating. We can not yet
> >> explain why the quantum histories wins over the comp/number relations.
> >>
> >
> > Well I have my own reasons, considering knowledge acquisition as an
> > evolutionary process. But I disagree about it being cheating, because
> > I don't a priori assume quantum states are elements of a Hilbert
> > space. That is a derived property.
>
>
> So, how do you define quantum state?
>
I don't define quantum state. I use the word state as a synonym of
observer moment, again as a means of contact with quantum
terminology. I make the statement "identify observer moment with
quantum state" as a shorthand for the following argument.
Assume that the state (or observer moment) undergoes evolution (I'm
refraining from qualifiying this with Darwinian) in that:
1) subsequent OMs (obviously a successor relationship is a
prerequisite here - something I call the TIME postulate) are
related closely to the previous OM, ie they inherit.
2) There is variation between successor OMs - ie the "many worlds"
idea.
3) That a particular successor OM x_i of OM y is what is observed
("anthropically selected"), with a probability P(x_i|y). The
probability function P(x|y) satisfies the Kolmogorov probability
axioms. This also implies that OMs must satisy set axioms. I also
call this third assumption the "PROJECTION postulate".
There is a final assumption. The initial OMs are drawn from the set of
all OMs according to some sort of measure, which happens to be complex. Since
measures can be more general than complex measures, I'm not entirely
sure why the measure should be restricted to being complex.
And that is it. From this idea (that OMs evolve), the following three
postulates of QM follow by a mechanistic proof
1. States are elements of a Hilbert space over a complex field
2. States evolve unitarily (ie according to a Schroedinger equation)
i\hbar d\psi/dt = H\psi
between measurements
3. The probability function P(x_i|y) satisfies the Born rule
P(x_i|y) = |<x_i|y>|^2 / <x_i|x_i><y|y>
Now some people have complained about how one can derive quantum
probabilities from the Kolmogorov axioms. It seems
counterintuitive. But this part is the most rigorous. The argument has
been put for the last seven years, and a number of very smart people
have looked at it without finding a flaw. Of course that doesn't mean
there isn't a flaw, but it would have to be quite subtle.
>
> >
> >>
> >>
> >>> This is not incompatible with with your
> >>> notion of the OM being a Sigma1 sentence, but it places severe
> >>> restrictions on the form of the quantum state vector.
> >>
> >>
> >> The OM are the Sigma1 sentences, when they are considered as third
> >> person constructs.
> >
> > Third person is that which is accessible to all observers.
>
>
> ? (This correspond more to the first person plural notion as I have
> defined it in most of my papers: observers appeared in the fourth and
> fifth hypostases, and perhaps already a part of it appears in the third
> one; but there are no observer in the second or first hypostases).
>
> cf:
> 1 p (truth, 0-person)
> 2 Bp (provable, 3-person)
> 3 Bp & p (knowable, 1-person)
> 4 Bp & Dp (observable, measurable; 1-plural-person)
> 5 Bp & Dp & p (sensationalisable, feelable, personally
> observable/measurable, 1 person again)
>
Thinking about it, I'm not sure our x-person terminology is completely
compatible. And it comes down to the problems I've had even in
understanding (or grokking, more to the point) the Theatetus
definition of knowledge. I can understand it from a purely
intellectual point of view ("henceforth, what we mean by `knowable' is
Bp&p"), but I have never grokked it.
--
----------------------------------------------------------------------------
A/Prof Russell Standish Phone 0425 253119 (mobile)
Mathematics
UNSW SYDNEY 2052 hpcoder.domain.name.hidden
Australia http://www.hpcoders.com.au
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Received on Mon Aug 13 2007 - 22:07:39 PDT