On 03 Nov 2015, at 13:49, Bruce Kellett wrote:
On 3/11/2015 8:50 pm, Bruno Marchal wrote:
On 02 Nov 2015, at 11:23, Bruce Kellett wrote:
On 2/11/2015 7:10 pm, Bruno Marchal wrote:
The choice of the bases is what Zurek have explained. That
extends Everett.
I think you should study Zurek at little more closely. He did not
actually explain the choice of basis. His result was that the
basis had to be robust against environmental decoherence -- which
is true, but does not give the actual basis. Position space is a
Hilbert space, not a basis for a Hilbert space.
?
Position provides a basis for the Hilbert space, which is
independent of the choice of that basis. we could use momentum
instead, and described the position by linear combination of
momentum, and Zurek wil still justify that the subject with a brain
will handle the position more easily than using the momentum.
I am not sure I can make sense of "Position space is a Hilbert
space". An Hilbert space is closed for linear combination (even
infinite) of any element belonging to any bases chosen in the
Hilbert space.
Basic quantum mechanics. Observables are represented by hermitian
operators, possible measurement results are the eigenvalues of such
operators. The operators act in the Hilbert space spanned by the
complete orthonormal set of eigenvectors. The problem is that the
form of the operator is not determined by the theory, and the
eigenvectors of each possible operator in the space provide a
different possible basis. By completeness, each possible basis can
be expressed as linear combinations of the vectors of any other
basis -- these are the (infamous) quantum superpositions.
OK.
The choice of basis (or equivalently, the choice for the form of the
operator) is the basis problem of quantum mechanics. Zurek's
einselection is an attempt to solve this problem by finding a basis
(and associated set of eigenvalues) that is robust against
environmental disturbance. It turns out, by Bohr's Correspondence
Principle, that the basis corresponding to the classical position
variable is robust in this sense, but that scarcely solves the
general basis problem because it is essentially a circular argument
-- stable classical values come out if we build in classical
variables.
I don't think this is circular if you accept classical
computationalism (even without its immaterialist consequence). To
develop cognitive ability, the machine needs to be able to make enough
clear distinction between its mental states, its memories, and that
with Everett+Zurek justifies the classical behavior of what we are
usually talking about, like when believing that it rains, or not.
The evolution of the brains in our branches has selected the base for
us, from our point of view. That whole process does not depend on the
choice of the basis. We can write the universal wave in any base, or
in any picture, but some base and picture will be preferred for
practical reason, as we do have a long history behind us. Then the
robustness of the position base against environmental disturbance is
explained by the necessity of having the classical means of the
computationalist constraint.
Classical logic (at least for arithmetic) is part of the
computationalist assumption, although that assumption can be weakened
a lot (but then the proofs get longer and more complex).
Bruno
Bruce
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