I enjoyed this post very much.  I have one question and a comment.
Q:  I didn't know that the most general field for a vector space
is the set of complex numbers;  why is this so?
Comment:  You ask why QM should be linear.  In the MWI FAQ, Price
gives a good Anthropic argument for why this should be so, based
on the fact that if it were in the least non-linear, then it 
would be possible to communicate between worlds.


Russell Standish wrote:
> 
> >
> > In Tegmark's paper,
> > in section 2G, he makes a crucial point that the fewer axioms
> > you use to define your mathematical structure, the larger is
> > the ensemble.  This provides a concrete justification for the
> > principle of Occam's Razor.  Similarly to the argument given
> > above, we would expect to find ourselves in worlds with fairly
> > few laws of physics, since those admit the most SAS's.  You
> > can always add any bizarre behavior to the structure by adding
> > ad hoc axioms, but worlds in which that is the case
> > have a smaller measure than those that do not.
> >
> > This line of reasoning also explains why, in a general sense,
> > we find that our universe behaves sensibly from moment to moment.
> > Many philosophers have pondered the question of why everything
> > doesn't disintegrate into chaos in the next instant.  What holds
> > the world together such that things persist and our memories
> > match our external reality?  The answer is that the structure(s)
> > we are in obey physical laws, not because they were cast by
> > fiat from some omnipotent being, but simply because the structures
> > that do obey physical laws are more numerous than those that do
> > not, and hence we are likely to find ourselves in those.
> >
> 
> I would take issue with this last statement. It seems that the above
> argument would imply that chaotic, unlawful universes should be more
> numerous than those obeying laws. The usual justification given for us
> finding ourselves in a universe with physical laws, is that such a
> universe is a minimum requirement for concious beings (or SASes, to
> use Tegmark's terminology) to exist. Unfortunately, because we don't
> understand the nature of conciousness enough, we cannot predict what
> the most general mathematical structure is to contain SASes.
> 
> One approach I suggested earlier is to suggest that perhaps the
> structure underlying QM (ie a Hilbert space over the complex numbers)
> is the most general such structure, and work backwards, trying to find
> the reasons behind the properties.
> 
> A Hilbert space is a Vector Space (ie it elements have a linearity
> property with respect to some field), and it has the additional
> property of having an inner product <x,y>.
> 
> QM has an additional property of the universe being indexed by time
> "t". Schroedingers equation can be written as e^{iHt}, where H is a
> Hermitian operator. This basically follows from some reasonable
> assumption about probability (i.e. the probabilities of all
> possibilities must remain equal to 1 through time). I think it
> reasonable to assume that conciousness requires a time variable.
> 
> The inner product is required to form "projections", which are the
> basis of observations: P(x is observed when A is measured)=<x,Ay>
> where y is the state of the universe. Something of this nature is
> required for conciousness.
> 
> If one has a vector space, then the most general field possible is the
> set of complex numbers, so that explains why C is used.
> 
> The problem is - why linearity? Is it related to Bayesian laws of
> probability - ie that probabilities of independent events are additive
> ( P(AuB) = P(a)+P(B) & P(AnB) = P(A)P(B) )? Is it necessary to have
> independent events for conciousness? Is it possible to get Bayesian
> laws with some generalisation of linearity?
> 
> NB. QM is in fact inconsistent with our other big 20C theory of
> Physics, ie Relativity. One of the biggest problems is that in
> Relativity, there is no well defined concept of "now" - the locus of
> contemporary events depends on one's frame of reference. One of the
> most interesting attempts to reconcile these theories replaces them
> with motion on a fractal manifold, and time has a 2D character. It is
> only in the macroscopic limit that time has a one-dimensional
> nature. Perhaps the Anthropic Principle requires the universe only to
> be approximately described by QM to some level of accuracy. (We
> needn't worry about solutions that blow up over periods much longer
> than the current age of the universe).
> 
> Enough ranting for now.
> 
> >

-- 
Chris Maloney
http://www.chrismaloney.com

"Knowledge is good"
-- Emil Faber


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