On 7/16/2012 1:56 PM, meekerdb wrote:
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.
Hi Brent,
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.
Hi Brent,
What I am interested in, among other things, is to drill down into
this "environment" notion. So far it seems to be a repackaged version of
the "heat reservoir <http://en.wikipedia.org/wiki/Heat_reservoir>" of
old Clausius
<http://en.wikipedia.org/wiki/Second_law_of_thermodynamics#Clausius_statement>
and Boltzmann days. It is its infinity that troubles me; "It is an
effectively infinite pool of thermal energy
<http://en.wikipedia.org/wiki/Thermal_energy>at a given,
constanttemperature <http://en.wikipedia.org/wiki/Temperature>. The
temperature of the reservoir does not change irrespective of whetherheat
<http://en.wikipedia.org/wiki/Heat>is added or extracted.". Is there a
finite version that we could consider that if taken to an infinite limit
will give us the same nice ideal concept but that in a large but finite
case does not allow us to get away with "tracing out" and other cheats
that we do with StatMec
<http://en.wikipedia.org/wiki/Statistical_thermodynamics>. I think that
if we stop thinking of the environment as a ideal entity and instead
model it as a large but finite collection of systems - like oscillators
- that do indeed absorb the phase relations such that we end up with
systems with "mixed states".
The point is that we never should start of thinking of pure systems
that are completely cut off from each other and thus have no possibility
of representing interacting systems (there do not exist interaction
Hamiltonians for such!), we start off our models assuming that all the
systems are in mixed states and identify them in terms of things like
centers of mass, etc. The only real "pure state" system is the
observable universe itself (if it is actually closed!). I am trying to
link this back to the math, but I need to lay out some of my thinking in
informal terms...
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.
OK, gotcha. It is the basis that is made up of strictly orthogonal
vectors. One problem that I have noticed is that Hilbert spaces are too
simple for the kinds of questions that I am asking. They only allow a
very limited kind of functions and assume ZF type set theory. I cannot
see how to fit Streams
<http://plato.stanford.edu/entries/nonwellfounded-set-theory/#1.1.1>
into them. :_(
Schlosshauer explains this better and at greater length than I can.
I have printed that paper out and am carrying it around re-reading
it. It is quite good. I agree.
Brent
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?
--
Onward!
Stephen
"Nature, to be commanded, must be obeyed."
~ Francis Bacon
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