On 17-06-2018 22:42, Jason Resch wrote:
Hi Lawrence,
Is the evolution of states of the wave function computable? If so then
the result of MRDP implies it is Diophantine.
Jason
Or you could try to see if QM could be a meta-theory that arises when
you try to give a statistical description of the set of all these
Diophantine sets. I tried to do something similar with the set of
algorithms a few years ago, getting a half-baked result, some hints at
how quantum field theory could arise from this.
You want to compute the probability that an observer that's encoded by
some mathematical structure has some given information content. So, if
you observe the outcome of an experiment, that's information in your
brain. But your brain is supposed to be some mathematical structure and
that then contains also that specific information about the outcome of
the experiment. Probabilities should presumably be obtained by counting
the number of states compatible with some observation, but we must then
impose the restriction that we're only going to count states that
correspond to some given observer making that observation. If observers
are specified algorithms that are specified by a set of input and
corresponding output states, then we must sum over all input and output
states, that fit each other. This is mathematically inconvenient, one
can replace such a summation by an unrestricted summation by including
Kronecker delta factors:
delta_{r,s} = 0 if r is not equal to s, otherwise it is 1.
One can then write:
delta_{r,s} = Integral from 0 to 1 of Exp[2 pi i (r-s) theta] dtheta
One can then sum over the variables freely, but one is then left with
integrations over many different theta variables. The idea is then that
in the limit of a large number of variables you can work with coarse
grained averages over the theta variables, you end up with something
similar to the path integral formulation of QFT.
Saibal
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