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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