On 4/13/2015 4:35 PM, Bruce Kellett wrote:
LizR wrote:
On 14 April 2015 at 00:42, Bruce Kellett <bhkell...@optusnet.com.au
<mailto:bhkell...@optusnet.com.au>> wrote:
The expansion of the wave function in the einselected basis of the
measurement operator has certain coefficients. The probabilities are
the absolute magnitudes of these squared. That is the Born Rule. MWI
advocates try hard to derive the Born Rule from MWI, but they have
failed to date. I think they always will fail because, as has been
pointed out, the separate worlds of the MWI that are required before
you can derive a probability measure already assume the Born Rule.
The argument is at best circular, and probably even incoherent.
In an article published in the 60s (I think) Larry Niven pointed out that the MWI lead
to the following situation - if you throw a dice you have 6 outcomes, i.e. 6 branches.
But a loaded dice should favour (say) the branch where it lands on 6. Hence the MWI
doesn't work.
My reaction to this (when I first read it, probably several decades ago now) was that
you only have 6 MACROSCOPIC outcomes - like derivations of the second law of
thermodynamics, Niven's description of the system relies on microstates being
indistinguishable /to us/. But once you take this into account there are more
microstates ending with a 6 uppermost - and hence a lot more than 6 branches - the MWI
again makes sense using branch counting, at least for non-quantum dice (I may not have
known terms like microstates at the time, nor was it called the MWI, but that was
basically what I thought).
I do not think that classical analogies can ever get to the heart of quantum
probabilities.
Can't the same be true of any quantum event? The essential requirement is that any
quantum event leads to results which can be assigned a rational number, rather than an
irrational one. This gives us a finite number of branches, and counting to get the
probability. Or do quantum events lead to results with irrational numbered probabilities?
Quantum probabilities are not required to be rational: any real value between 0 and 1 is
possible. For example, if you prepare a Silver atom in a spin up state then pass it
through another S-G magnet oriented at an angle alpha to the original, the probability
that the atom will pass the second magnet in the up channel is cos^2(alpha/2). This can
take on any real value in the range.
One argument against branch counting is that if you have two equally likely outcomes
(which can be judged by symmetry) there are two branches; but if a small perturbation is
added then there must be many branches to achieve probabilities (0.5-epsilon) and
(0.5+epsilon) and the smaller the perturbation the larger the number required. Of course
the number required is bounded by our ability to resolve small differences in probability,
but in principle it goes as 1/epsilon.
I think Bruno's answer to this is that for every such experiment there are arbitrarily
many threads of the UD going throught at experiment and this provides the order 1/epsilon
ensemble. But this somewhat begs the question of why we should consider the probabilities
of all those threads to be equal since we have lost the justification of symmetry. I
think this is "the measure problem".
Brent
I know that a branch counting approach to quantum probabilities is disfavoured, though I
can't at the moment recall the standard argument. Clearly, the existence of real-valued
probabilities, not restricted to rational values, is a strong factor. But there is also
the fact that in the two-dimensional Hilbert space associated with spin 1/2 projections,
one only ever has two possible outcomes for an experiment -- why should the number of
branches one must consider depend on the angle of the magnet?
Bruce
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