On 7/15/2012 9:21 PM, Stephen P. King wrote:
Interesting. So the unitary evolution of the SWF or state vector is not continuous
over its spectrum or what ever it is called ... the cover or span of the basis?
It's continuous, but decoherence picks out different subspaces which are almost
perfectly orthogonal and correspond to different classical events. There's a different
"you" in each of these subspaces corresponding to seeing Schrodinger's cat alive or dead.
Does not seem as if decoherence is a bit too clever by half? I am very interested in
this process that we are calling decoherence. Where does it get this ability to "pick
out different subspaces which are almost perfectly orthogonal"?
It's a consequence of the interaction Hamiltonian between a measuring device and the
environment (which may just be part of the measuring device). Observable states that
commute with this Hamiltonian will be stable and constitute a "record". So the stable
subspaces are the eigenspaces of this interaction Hamiltonian. But you're right that is
only a partial solution to the measurement problem. That's because the division into
systems - thing measured, measurement instrument (or observer), and environment - is a
choice we make in our description. And the operation of tracing (averaging) over the
unknown environment modes is a mathematical operation we perform in our description - not
a physical process. The cross terms in the reduced density matrix average to zero so then
the matrix just has normal probability measures along its diagonal.
I was operating under the belief that all of the vectors (in the Hilbert space involved)
are strictly orthogonal and are so perpetually.
No. They are no more necessarily orthogonal than vectors you might draw on a plane.
Presumably the state of the universe/multiverse as a whole is a single ray in the Hilbert
space of the universe/multiverse. But the application of the theory is always to
subspaces describing different things as the-thing-measured, the measuring-apparatus, etc.
and Hilbert spaces constructed as tensor products of these.
Schlosshauer explains this better and at greater length than I can.
Where do these "subspaces" come from? Are they defined by subsets of the state vectors
(or eigenvectors)? How is the diffeomorphism invariance (that the unitary evolution is
equivalent to!) get preserved in this process? How does "tracing out" eliminate things?
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