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 statevector is not continuous over its spectrum or what ever it iscalled ... the cover or span of the basis?It's continuous, but decoherence picks out different subspaces whichare almost perfectly orthogonal and correspond to differentclassical events. There's a different "you" in each of thesesubspaces 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 amvery interested in this process that we are calling decoherence.Where does it get this ability to "pick out different subspaces whichare almost perfectly orthogonal"?It's a consequence of the interaction Hamiltonian between a measuringdevice and the environment (which may just be part of the measuringdevice). Observable states that commute with this Hamiltonian will bestable and constitute a "record". So the stable subspaces are theeigenspaces of this interaction Hamiltonian. But you're right that isonly a partial solution to the measurement problem. That's becausethe division into systems - thing measured, measurement instrument (orobserver), and environment - is a choice we make in our description.And the operation of tracing (averaging) over the unknown environmentmodes is a mathematical operation we perform in our description - nota physical process. The cross terms in the reduced density matrixaverage to zero so then the matrix just has normal probabilitymeasures 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 theHilbert space involved) are strictly orthogonal and are so perpetually.No. They are no more necessarily orthogonal than vectors you mightdraw on a plane. Presumably the state of the universe/multiverse as awhole is a single ray in the Hilbert space of theuniverse/multiverse. But the application of the theory is always tosubspaces describing different things as the-thing-measured, themeasuring-apparatus, etc. and Hilbert spaces constructed as tensorproducts 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.`

BrentWhere do these "subspaces" come from? Are they defined by subsets ofthe state vectors (or eigenvectors)? How is the diffeomorphisminvariance (that the unitary evolution is equivalent to!) getpreserved in this process? How does "tracing out" eliminate things?

-- Onward! Stephen "Nature, to be commanded, must be obeyed." ~ Francis Bacon -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.