On 04 Nov 2015, at 00:55, Bruce Kellett wrote:
On 4/11/2015 4:49 am, Brent Meeker wrote:
On 11/3/2015 4:49 AM, 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.
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 think that would have been better expressed if you had noted that
the robust basis is not necessarily the position basis. As
Schlosshauer notes, for atomic size things it is often the energy
eigenvalues that are robust. But we didn't predict that; we have
discovered it empirically.
This is the mistake made by both Zurek and Schlosshauer: Position
and energy are variables, not bases, and for both you have the same
basis problem (both are operators in infinite dimensional Hilbert
spaces, but *different* spaces). The question as to whether a
measurement is primarily one of position or of energy is depends on
the system under study and the experimental set up. But the basis in
either position or energy space still has to be chosen. Einselection
then turns out to be rather trivial because all the measurements and
interaction Hamiltonians are always expressed in terms of the
classical counterparts of the relevant variables.
So would it be a complete solution of the measurement problem if we
could predict which basis choice would provide robust eigenvalues?
Would this prediction start from a very complex instrument/
environment interaction Hamiltonian? It seems that if it did we'd
be in the position of having to do stat mech on the interaction,
which would again involve assumptions about chaos and averaging? I
think we might run into Chris'es "cat in the tree" problem.
The argument that is made is that the 'classical' world emerges from
the quantum, in the sense that the quantum is more fundamental. But
when we look into it, we find that Bohr was quite perceptive with
his Correspondence Principle: since we are essentially 'classical'
beings, we have to start with classical concepts even to begin to
build a quantum theory. I doubt that it would be even possible to
construct a quantum theory ab initio, without reference to classical
ideas.
OK.
Now, with computationalism, we assume classical logic, but only on the
arithmetical propositions, and the quantum is retrieved from the logic
of "probability one" on the sigma_1 sentences (the pieces of
computation), or the logic of the yes/no experimental question in that
context. The quantum appears as describing our first person (plural)
view emerging from our distribution in the Universal Dovetailing.
The quantum is given by the digital seen from inside by digital finite
creature. We can get an intuition of this directly from the UD
Argument, and the math confirms that we get a quantum logic, a priori
different from the quantum logic coming from physics, but up to now it
fits well.
Bruno
Bruce
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