Dmitry67 said:
Again, it is possible (and very likely) that in act different observers do not agree on what you call an 'observables', and even on the number of the elementary particles, but they Do agree on the microscopic events. Unruh effect is a good example
In standard semiclassical gravity they can agree about the observables as well as in classical relativity, where everybody agrees about the showings of a traveling clock from A to B along a given trajectory.
One can associate some observables with some observers, but this association is quite arbitrary. If one considers a particle detector in some state of movement, say, along a given trajectory, the prediction about the resulting average particle numbers measured by this detector is unique, well-defined.
in the Universe without YOU there is no need to calculate a decoherence in some basis at all: you can be satisfied with a unitary evolution of the 'universe' wavefunction.
You have not got the point. The wave function is a function on the space of all possible universes. From this birds view there are only particular branches with or without me, the multiverse is not with or without me.
Now I have a wave function, and want to know what I can expect to observe. How does this work? You would like to look at branches which contain me. But the branches are simply not defined before decoherence has finished its job. You need decoherence to define the branches. To define decoherence, you need a decomposition into systems. This decomposition into systems has to be defined on the full Hilbert space, the one which contains all these superpositions of something close to our own branch (better, what becomes our own branch, everything else appropriately defined) to something where Earth does not exist.
The "decomposition into systems" means some H=H_{rest}\otimes H_{obs}. Then, every basic state (branch) is a product \psi=\psi_{rest}\psi_{obs}. What is \psi_{obs} supposed to describe if the Earth does not exist? H_{obs} is always the same, independent of the question if \psi_{rest} describes a state where the Earth exists or not.
Note the words I highlighted
So, you do not get the values of these 'observables' directly.
At first, you must decoherence your system (or a particle) with some macroscopic device, right?
If you want to talk about the standard applications of decoherence outside many worlds, no problem. You have some classical Copenhagen part of the world, and this part defines nicely a decomposition into various systems, in part classical, in part quantum. The observer is, in this case, classical, thus, fixed at every moment of time. States where the Earth does not exist are simply not part of such considerations.
MWI has the problem how to obtain all this background in a consistent way. It cannot start with me or some other systems here on Earth to define a decomposition into systems to start decoherence, because these systems are only defined in some small subset of the small part of the multiverse which contains our Earth.
Now about the usual way to apply decoherence: I measure what I like, by rotating variouos devices in various ways. If the measurement device is rotated in one way, the decoherence-preferred observable is, say, S
1, if it is rotated in another way it may be something different, S
2. Thus, for small quantum systems all quantum observables may appear observable, one simply has to use some appropriate environment, with appropriately rotated devices, where it appears decoherence-preferred.
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