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From: Christopher Maloney <dude.domain.name.hidden>

Date: Thu, 10 Jun 1999 06:28:05 -0400

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:

*>
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*> >
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*> > In Tegmark's paper,
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*> > in section 2G, he makes a crucial point that the fewer axioms
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*> > you use to define your mathematical structure, the larger is
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*> > the ensemble. This provides a concrete justification for the
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*> > principle of Occam's Razor. Similarly to the argument given
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*> > above, we would expect to find ourselves in worlds with fairly
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*> > few laws of physics, since those admit the most SAS's. You
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*> > can always add any bizarre behavior to the structure by adding
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*> > ad hoc axioms, but worlds in which that is the case
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*> > have a smaller measure than those that do not.
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*> >
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*> > This line of reasoning also explains why, in a general sense,
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*> > we find that our universe behaves sensibly from moment to moment.
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*> > Many philosophers have pondered the question of why everything
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*> > doesn't disintegrate into chaos in the next instant. What holds
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*> > the world together such that things persist and our memories
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*> > match our external reality? The answer is that the structure(s)
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*> > we are in obey physical laws, not because they were cast by
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*> > fiat from some omnipotent being, but simply because the structures
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*> > that do obey physical laws are more numerous than those that do
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*> > not, and hence we are likely to find ourselves in those.
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*> >
*

*>
*

*> I would take issue with this last statement. It seems that the above
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*> argument would imply that chaotic, unlawful universes should be more
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*> numerous than those obeying laws. The usual justification given for us
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*> finding ourselves in a universe with physical laws, is that such a
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*> universe is a minimum requirement for concious beings (or SASes, to
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*> use Tegmark's terminology) to exist. Unfortunately, because we don't
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*> understand the nature of conciousness enough, we cannot predict what
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*> the most general mathematical structure is to contain SASes.
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*>
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*> One approach I suggested earlier is to suggest that perhaps the
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*> structure underlying QM (ie a Hilbert space over the complex numbers)
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*> is the most general such structure, and work backwards, trying to find
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*> the reasons behind the properties.
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*>
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*> A Hilbert space is a Vector Space (ie it elements have a linearity
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*> property with respect to some field), and it has the additional
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*> property of having an inner product <x,y>.
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*>
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*> QM has an additional property of the universe being indexed by time
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*> "t". Schroedingers equation can be written as e^{iHt}, where H is a
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*> Hermitian operator. This basically follows from some reasonable
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*> assumption about probability (i.e. the probabilities of all
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*> possibilities must remain equal to 1 through time). I think it
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*> reasonable to assume that conciousness requires a time variable.
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*>
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*> The inner product is required to form "projections", which are the
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*> basis of observations: P(x is observed when A is measured)=<x,Ay>
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*> where y is the state of the universe. Something of this nature is
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*> required for conciousness.
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*>
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*> If one has a vector space, then the most general field possible is the
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*> set of complex numbers, so that explains why C is used.
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*>
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*> The problem is - why linearity? Is it related to Bayesian laws of
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*> probability - ie that probabilities of independent events are additive
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*> ( P(AuB) = P(a)+P(B) & P(AnB) = P(A)P(B) )? Is it necessary to have
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*> independent events for conciousness? Is it possible to get Bayesian
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*> laws with some generalisation of linearity?
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*>
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*> NB. QM is in fact inconsistent with our other big 20C theory of
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*> Physics, ie Relativity. One of the biggest problems is that in
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*> Relativity, there is no well defined concept of "now" - the locus of
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*> contemporary events depends on one's frame of reference. One of the
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*> most interesting attempts to reconcile these theories replaces them
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*> with motion on a fractal manifold, and time has a 2D character. It is
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*> only in the macroscopic limit that time has a one-dimensional
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*> nature. Perhaps the Anthropic Principle requires the universe only to
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*> be approximately described by QM to some level of accuracy. (We
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*> needn't worry about solutions that blow up over periods much longer
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*> than the current age of the universe).
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*>
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*> Enough ranting for now.
*

*>
*

*> >
*

Date: Thu, 10 Jun 1999 06:28:05 -0400

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:

-- Chris Maloney http://www.chrismaloney.com "Knowledge is good" -- Emil FaberReceived on Thu Jun 10 1999 - 15:23:01 PDT

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