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S11a. A concise formulation of the measurement problem of QM
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Quantum mechanics asserts in the Born rule (also called Lueder's rule)
that when a particle prepared in a pure state passes an ideal
measuring instrument characterized by a finite family of mutually
orthogonal projectors P_k (with P_k = P_k^*, P_k P_l = delta_kl P_l
and sum_k P_k = 1), it transforms the pure state psi into the pure
state psi_k = P_k psi/p_k with probability p_k= psi^* P_k psi.
This is a consistent rule in a purely statistical interpretation
in which psi is an objective property of a source (describing the
statistical behavior of an ideal - stationary and pure - source of
particles) rather than an objective property of each individual
particle.
The measurement problem arises when (as is commonly informally assumed)
the wave function is regarded as an objective property of a particle.
Then the stochastic transformation demanded by the Born rule, called
the collapse of the wave function, conflicts with the deterministic,
unitary dynamics of the wave function demanded by quantum mechanics
of the joint system consisting of particle+instrument+environment.
The unitary dynamics predicts that the joint system is in a macroscopic
superposition, which is not observed.
Note that a measurement does not need a conscious observer.
A measurement is any permanent record of an event, whether or not
anyone has seen it. Thus the terabytes of collision data collected
by CERN are measurements, although most of them have never been
looked at by anybody. We human beings only look at crude summaries
of such high tech data, but the collapse (which gives rise to
individual particle tracks) is clearly independent of whether or when
we look at them.