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.
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.
Bruce
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