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S11e. The preferred basis problem
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Born's rule, stated in the form that ||^2 is the probability
that a system prepared in state psi is, upon measurement, found in state
phi, is valid only if a complete set of commuting observables is
measured and phi belongs to the preferred basis determined by the
experimental setting (i.e., the family of projectors).
Given the present state of the universe (which fixes the experimental
setting), there is no choice in the preferred basis. Thus, in a
mathematical model of quantum mechanics in the large, it has to
be deduced from the assumptions about the initial state and the
dynamics.
The preferred basis is fully determined by Nature, and that's why we can
find it out. Given an unknown instrument, one finds out by
experimenting with the new piece, letting it interact with systems
of known properties, and matching the collected data to trial models
until one fits. This is how things are indeed done in practice.
The process is called model calibration (or parameter estimation if
the model is fixed up to adjustable parameters).
At first, one never knows a new instrument precisely, and has to check
out its properties. After sufficient experience with enough instruments,
one knows reasonably well what to expect of the next, similar one.
Then only fine-tuning is needed, which saves time. And this knowledge
can be used to create new instruments which are likely to behave a
certain way; but one still has to check to which extent they actually
do, since no theoretical design is realized exactly in practice.
Not even in the classical, macroscopic domain!
Nature's choice is systematic, hence after having
seen that a number of screens have a preferred position basis,
we conclude that this is the case generally. As for a spectrometer,
if it is built with a prism to analyze light, it is reduced by theory
to the observation of light or current at certain positions of the
screen, which is done in the preferred position basis. Something
similar can be said about the Stern-Gerlach experiment.
So once one knows _some_ of Nature's preferences and the general laws,
one can deduce other preferences.
The challenge posed in the measurement problem is to deduce
from first principles that a screen made of quantum matter,
with two slits in it, actually has a preferred position basis and
projects the incoming system to the part determined by the slits.