On Mon, Jan 21, 2008 at 03:31:20PM +0100, Mirek Dobsicek wrote:
> 
> Hi Russel,
> 
> you are right, a "*positive* real Hilbert space" is a wrong term.
> However, the point of my comment was to express a belief that your sentence
> 
> >>> If quantum mechanics was done using a real-valued Hilbert space, you
> >>> simply don't get wavelike interference patterns.
> 
> is not correct. But of course, QM as physical framework and as derived
> from experiments goes with complex numbers.
> 
> Best,
>  Mirek

Yes, of course. And in fact given that the naive OM measure usually
chosen and discussed is a positive measure, putting this into the
"theory of observations" framework would give a theory without
interference effects and the like.

My point is still valid about the need for complex measure, but you
have added an important correction. As I point out in ToN (and Why
Occams Razor) complex measures are more general than real measures, so
Occams razor should choose these over positive measures. And this is
what is seen experimentally.

My concern, of course, is that more general measures exist than
complex measures. Hence this raises the intriguing possibility that a
more general measure (such as quaternions for instance) might give
rise to different physics than standard QM that could be
experimentally tested for. If found, these would be mind-blowing. If
not found, this raises the question of why complex measures are
preferred over quaternions, say.

The trouble is that at present, I'm all fingers and thumbs over
manipulating division rings, as I'm not familiar with them, only their
vector space cousins.

Cheers

-- 

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A/Prof Russell Standish                  Phone 0425 253119 (mobile)
Mathematics                              
UNSW SYDNEY 2052                         [EMAIL PROTECTED]
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