From: *Lawrence Crowell* <[email protected]
<mailto:[email protected]>>
On Wednesday, June 13, 2018 at 11:18:10 PM UTC-5, Bruce wrote:
From: *Brent Meeker* <[email protected]>
No. There's a preferred basis in which this "world" and it's
spots on the screen, is spanned by basis vectors which are
orthogonal to the basis vectors of the "worlds" in which the
spots are in different places on the screen. But in each world
there are different (not necessarily position) bases, but they
describe the same physics.
I don't think that is correct. The preferred basis is selected as
the eigenvectors of the operator that commutes with the
interaction Hamiltonian. If you choose a different basis for the
Hilbert space, even by a simple rotation of your present basis,
you are going to get eigenvectors (and eigenvalues) of a different
operator. Since this operator must also be dominant in the
interaction Hamiltonian, the physics is necessarily going to be
different. A different position basis is going to result in more
than different places on the screen for the spots.
Bruce
I would agree, and that you are invoking the Hamiltonian segues into
what I wrote yesterday. I can set an apparatus to measure the spin of
an electron in any orientation.
That is true; but that is just making a choice about what to measure --
equivalent to the choice of whether to measure the position or momentum
of a free particle. These measurements are mutually exclusive, but they
do not set the measurement basis. When you use a S-G magnet to measure
the spin projection of a spin-half particle you chose an orientation,
but the actual measurement that gives you the required result is a
position measurement -- whether the particle emerges in the up or down
channel. That is why this was originally referred to as "space
quantization".
I can't make a measurement of energy that is something other than the
eigenstates or the diagonal form of the Hamiltonian. Energy is the
physical quantity which defines the einselected basis that is stable
in a classical-(like) outcome or for the emergence of classicality.
That is incorrect. If you are making a position measurement, energy does
not come into it. Certainly, for many physical system, such as atoms and
molecules, the energy eigenstates are what one measures. But one
measures these in the preferred energy basis, which is quite similar to
the preferred position basis. We are used to a position basis with
eigenstates as position delta functions along the real line. The
preferred energy basis is similar, energy delta functions along the real
line (remember that we can get any real value as the result of a generic
energy measurement. Energies are quantized only for specific physical
systems.)
I think people get trapped into thinking that our usual delta-function
basis for either position or energy is the only possible basis, because
that is the only basis in which we are able to measure anything. But
that itself is just a consequence of einselection to a preferred basis
-- attempting to measure in some other basis is not a position or energy
measurement as we know it, and the eigenfunctions of the alternative
operators decohere into our known basis extremely rapidly. But the fact
that the usual basis is ubiquitous, made so by decoherence, does not
explain why it is that basis, rather than some other basis, which is
stable against decoherence. We could easily choose another basis, and as
I pointed out to Brent, the split into separate branches on the MWI
would be very different in a different basis. Eigenfunctions and
probabilities would be different with a different basis, so the physics
would be different. The real question is "Why is physics the way it is?
It could easily have been different."
Bruce
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