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