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.
Bruce
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