On 7/3/2021 4:59 AM, Tomas Pales wrote:
On Saturday, July 3, 2021 at 1:10:43 AM UTC+2 Brent wrote:
On 7/2/2021 2:43 PM, Tomas Pales wrote:
On Friday, July 2, 2021 at 2:54:19 PM UTC+2 Lawrence Crowell wrote:
The GRW interpretation states there is with any quantum wave
a fundamental phenomenon of collapse. The collapse occurs
fundamentally by a stochastic rule.
Fundamental, irreducible probability seems like an incompletely
baked concept. Mathematically/structurally, probability can be
defined in terms of pure sets, like any other
mathematical/structural concept. Pure sets (combinations of
combinations of combinations etc. founded on the empty
combination) are the fundamental concept from which it is
possible, in principle, to build up any structure. MWI attempts
to define the quantum probability in terms of sets, whose most
straightforward interpretation seems to be worlds. But the
problem is that there seem to be infinitely many worlds in MWI
and a system of infinitely many objects may have different
probability measures that give different results, so it seems
that the Schrodinger equation is not sufficient to calculate
probabilities even in MWI and MWI also needs a probability
measure as an additional property of the quantum multiverse,
namely such that it results in the Born rule. There have been
some claims that such a measure is the only possible one
There's only one consistent measure on a Hilbert space and that's
the Born rule, as proven by Gleason's theorem.
If it is obvious that the Born rule is the only consistent probability
measure in QM, why is there not a generally accepted proof of it? What
is so controversial about proving that 1 = 1?
Wikipedia mentions several attempts to derive the Born rule from more
basic principles:
https://en.wikipedia.org/wiki/Born_rule#Derivation_from_more_basic_principles
and one of the referenced articles concludes at the end: "The
conclusion seems to be that no generally accepted derivation of the
Born rule has been given to date, but this does not implythat such a
derivation is impossible in principle."
https://www.math.ru.nl/~landsman/Born.pdf
Everett noted that every observer gets entangled with the result
and then exists in a superposition of different observed values.
He claimed this meant that any observer would observe the Born
rule probablity. But this depended on considering the observer in
one special basis of the Hilbert space (the pointer states) and
then zeroing out cross terms in the density matrix. By what
mechanism the observer or instrument gets into this state is unclear.
Isn't it the mechanism of decoherence that is contained in Schrodinger
equation?
No, because decoherence depends on there being a special basis, referred
to as "the pointer basis", in which decoherence operates to
approximately diagonalize the density matrix. CI just assumes this is a
projection onto classical variable values. Advocates of MWI want to
claim there are no projections (they aren't unitary) that instead the
the world "splits" and each approximately diagonal value is realized in
a subspace. But then one needs to explain what about those subspaces
corresponds to the probabilities, or in other words what does
"probability" mean when they all exist? It's not that what you suggest
is inconsistent; it's pretty much what Roland Omnes' says: QM is
probabilistic theory so it predicts some things happen and some don't.
But then it's just like the Copenhagen interpretation plus decoherence
to answer the problem of the Heisenberg cut.
Brent
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