Actually, unitary pseudo-collapse?
Von Neuman said quantum mechanics proceeds by two processes: unitary evolution and nonunitary reduction, also now called projection, collapse and splitting.
Collapse is non-unitary by definition, and the impossibility of unitary collapse is also a basic mathematical theorem. But we no longer believe, as Bohr did, in two Kingdoms, one classical and one quantum. Instead, we now talk about the emergence of the pseudo-classical. Similarly, I suggest, we can now talk about a pseudo-collapse, which is pseudo-unitary.
In a collapse two things happen: First, the matrix diagonalizes, and then, it becomes one dimensional, ie one element of the main diagonal becomes 1 and all the others become 0. This second part is the nonunitary projection part. (In a slight variation, we have two corresponding matrices, labelled system and apparatus, and the coresponding "pointer state" in the apparatus goes to 1, and all the other pointer states go to 0.) Now along comes decoherence, and we have three matrices, S,A, and E: System, Apparatus and Environment. Actually, these three matices are submatrices, ie diagonal blocks in one bi gger matrix. You can still diagonalize everything by unitary operations. But you still can't project it to a one dimensional definite outcome by unitary means. However, I think you can make the S block one dimensional by unitary operations if you push the off diagonal terms into the S-E quadrant. Is this correct?
This seems almost obvious to me, but I can't find confirmation. In fact, as I searched, I found lots of discussion on the diagonalization process, but almost nothing about the selection step, which is the real crux of the process, as far as `I am concerned. Why is this? Can someone point me to a good discussion of the selection process? And confirm that my pseudo-unitary pseudo-collapse is mathematically possible?