From: scerir (scerir@libero.it)
Date: Tue Sep 17 2002 - 07:44:45 MDT
[Lee]
But to return to MWI again, probabilities there
retain their classical simplicity.
[me]
In MWI? Difficult to normalize probabilities
in there.
[Lee]
"In there"? The probabilities that I'm talking about are
those that simply arise in the laboratory when performing
any 1920's type calculation. If the new state can be any
one of a, b, c, then the classical probabilities are found
to be a^2, b^2, c^2 respectively, where these add up to 1.
Of course, for all I know, these could turn out to be
Bayesian probabilities ;-)
Hmm. Much worse than Bayesian I suppose ;-)
When a quantum system is in a superposition
of states (eigenstates) with different values (eigenvalues)
of some observable quantity X, the orthodox interpretation
associates probabilities to the various outcomes, via the
Born's rule (*).
Now the MWI is a deterministic theory (based on the deterministic
Schroedinger equation alone) which does 'not' have that randomness
coming from the 'collapse' of the probability packet (or reduction
of wave-packet, or projection postulate).
It makes very little sense to ask what is the probability that,
given a quantum system in a superposition of states, we obtain a
particular outcome. Because we get 'all' the possible outcomes
(for which there are non zero probabilities, according to
the orthodox interpretation).
It makes no sense to ask what is the probability of an
observer, in a particular branch (of the multiverse) corresponding
to a particular outcome, to obtain a particular outcome.
Because this probability is always = 1. The particular
observer can 'read' just one outcome. Another particular
observer (in another particular branch) can 'read' just
another outcome.
I leave aside the problem if all these observers keep
the same memories about events which happened before
the measurement which splitted the world in many branches.
For this see John Bell, "Quantum Mechanics for Cosmologists",
in his "Speakable and Unspeakable in Quantum Mechanics".
Anyway this is an interesting issue. Because if these observers
keep the 'same' memories about events which took place
before (and during?) the measurement performed by the original
observer it is possible (according to Lev Vaidman) to build a
consistent frame of reference for a MWI probability concept.
(A 'universal' wavefunction for each branch also seems to be
necessary, and this leads to a problem discussed by David
Bohm, about 'fluctuations', and to another one discussed by
Bell, about the time-asymmetrical nature of MWI, because
of 'branching', which has nothing to do with the time-symmetrical
Schroedinger equation, which is the heart of the MWI theory).
Different attemps were made by Everett, Graham, Lockwood,
Saunders, Deutsch, et al.
"On Schizophrenic Experiences of the Neutron or
Why We Should Believe in the Many-Worlds Interpretation
of Quantum Theory" by Lev Vaidman
http://arxiv.org/abs/quant-ph/9609006
This is a philosophical paper in favor of the Many-Worlds
Interpretation (MWI) of quantum theory. The concept of the `
`measure of existence of a world'' is introduced and some
difficulties with the issue of probability in the framework
of the MWI are resolved. Brief comparative analyses of t
he Bohm theory and the Many-Minds Interpretation are given.
(*) Born's 1926 paper about his rule was completely
un-noticed. And in that paper Born made two errors,
saying that the wavefunction gave the probability, then,
in a footnote, saying that the squared wavefunction gave
the probability. Independently Heisenberg wrote the same
rule, just a bit after. Was Pauli who gave the general, consistent,
best version of Born's rule, that one with the modulus.
The probabilistic interpretation states - as we all know - that
the wavefunction determines only the probability that a
'particle' (which brings energy and momentum, no energy
and momentum pertains to the 'wave') takes a path, or
can be found in some place. But amplitudes, not probabilities,
are fundamental. Amplitudes usually satisfy linear differential
equations and combine linearly, when different paths are given,
for the propagation. Intensities of wave-like phenomena are
given by energy fluxes, which are quadratic in amplitudes,
and are not additive. In QM the energy flux is related to the
'number' of particles per unit area, hence also to the 'probability
of arrival' of these particles. Poicarè already (Théorie Mathématique
de la Lumière, 1882) showed how to describe polarization states
by complex numbers. So it was easy to postulate the existence
of those complex amplitudes. And, perhaps, it was not so
difficult to link complex amplitudes, energy fluxes, number of
particles per unit area, 'probability' for particles to be found in
some place. See also M. Beller, Born's probabilistic interpretation:
a case study of concepts in flux, Studies in History
and Philosophy of Science, 21, p. 563-588, 1990).
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