On Mon, Sep 24, 2007 at 09:52:12AM -0700, Youness Ayaita wrote:
>
> I really enjoyed reading your paper "Why Occam's Razor?" and I'd never
> pretend to understand your derivation of quantum mechanics better than
> you do. But maybe, I have another perspective on it (or even an
> addition). Explaining this will reveal why I think that the Born rule
> supports materialism/realism against idealism.
>
> >From a physicist's point of view, your derivation is complete and
> doesn't require any addition. We get the postulates as they can be
> found in every introductory textbook on quantum mechanics. But someone
> starting from the idea of the Everything ensemble won't be completely
> satisfied. You introduce an unspecified probability distribution P_psi
> which is essential in the definition of the inner product. In physics,
> the Hilbert space of physical states can be different for every
> system; a physical theory must specify the inner product of each
> Hilbert space from which we can reconstruct the distribution P_psi by
> applying the Born rule. Though, if we start from a theory of
> everything, we want a fundamental explanation for the specific
> distribution.
Given the topic of Born's rule is now white hot (the New Scientist
article, eg), its good to dig into some of these topics. P_psi(psi_A) is
defined as the probability of observing A as the outcome of some
experiment, given one is in the state (or OM) psi to start with. OK,
somewhat more conventional notation would write this as P(psi_A|psi),
but I wrote it that way as it connects better with the projection
operators (the curly P's).
The point is that the measure of OMs do not come into the Born rule
calculation at all! They are renormalised away in taking the
conditional probability. Now I think that requires that the measure be
at least a division algebra, but since most people assume the measure
is positive this is hardly a restriction.
>
> The materialist approach (of the Everything ensemble) would say that
> P_psi(psi_a) is given by the measure of psi_a divided by the measure
> of psi. Here, the measure of psi(_a) is meant to be proportional to
> the 'number' of 'worlds' forming psi(_a).
This requires a positive measure, which is not always the case. In
fact what I demonstrate in my proof (assuming no fundamental error in
it of course, as only a handful of people have checked it) is that
assuming a complex measure, the Born rule is what you'd expect for the
observation probabilities. This entails that the above simple idea
(which you call materialism, but I suspect that is terminological
abuse) is just wrong. Restricting the measure to a positive one does
not give the right answer.
--
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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 Sep 24 2007 - 22:08:21 PDT