As an exercise I've been trying to pinpoint exactly what is wrong with
Plaga's paper. For anyone who doubts that it *is* wrong, note that it
proposed 10 years ago an experiment which he said was feasible with what
was then state-of-the-art equipment. This technology has now massively
advanced. The experiment would guarantee a Nobel prize for anyone who
performed it successfully. In that time the paper has been cited in the
published literature only 3 times, and never by an experimental physicist.
And this is not because the paper was unnoticed by the community at the
time, e.g. it was publicised by John Baez, whose writings are widely read.
On careful reading, the paper is just littered with confusions and errors.
I guess this explains why no-one bothered to publish a rebuttal; this
falls in the class of "not even wrong". Probably the root problem is a
confusion about the true nature of decoherence. Decoherence is often
presented using the maths of density matrices, so I better explain
this briefly:
Density matrices allow you to handle the case when you don't know the
exact quantum state. The procedure is to divide your description into a
measurable "system" and a complex, not-measurable-in-detail "environment".
One can then define the density matrix of the combined system, and "trace
out" the uncertain state of the environment, giving a density matrix for
the system alone in the absence of information about the environment. A
test to see if the system has been decohered by its interaction with the
environment is that the off-diagonal terms in the system-only density
matrix go to zero. Plaga clearly accepts the usual position that
irreversible branching in MWI occurs when decoherence is (FAPP) total.
If you follow this through in Plaga's example, you do indeed find that the
density matrix for the states of his trapped ion, |A1> and |A2>, is
diagonal, confirming the obvious that once a macroscopic measurement has
taken place, we have total decoherence. But what Plaga does in his Eq 8
is to reverse the roles of system and environment (he actually does the
algebra wrong but the numerical answer is unaffected). Because at this
stage the ion knows nothing about the rest of the lab, he gets a density
matrix *for the lab* with large off-diagonal terms, corresponding to a
"pure" state:
(|W1> + |W2>) / sqrt(2).
So far, so correct (after all, in MWI the state is *always* pure).
But he now concludes that decoherence has not yet occurred. *WRONG*. The
condition "off-diagonal terms go to zero" is just a sufficient condition
for decoherence. It is only necessary if the "system" itself is so simple
that it could not decohere without the help of the environment. But Plaga
is treating the complex, macroscopic lab as the "system" and that
certainly can decohere without the help one more ion. The more basic
definition is that decoherence has occured once the states are permanently
orthogonal, so you cannot demonstrate quantum interference. Plaga
correctly states that |W1> and |W2> *are* permanently orthogonal, but does
not realise that this means that decoherence *is* complete, contrary to
what he says. Another way to put this is that the observer "Silvia"
doesn't need the density matrix in Eq. (8) because she knows for sure
already whether she detected the original photon or not, hence whether she
is in branch |W1> or |W2>.
Given this, the rest of Plaga's argument is just irrelevant. But he should
have noticed that his process blatantly violates the linearity of time
evolution, which is one of the fundamental assumptions of MWI QM. This is
manifest in his Eq. 6 which associates an excited ion with the |P2> term
in which no excitation took place (if you start with a photon in state
|P2>, when the photon is guaranteed not to be detected, the ion is never
excited). Hence Eq 6 is not a linear superposition of the two possible
histories. Hence, if we saw what he predicted, we would actually
*disprove* MWI QM, not confirm it as he thinks.
Paddy Leahy
======================================================
Dr J. P. Leahy, University of Manchester,
Jodrell Bank Observatory, School of Physics & Astronomy,
Macclesfield, Cheshire SK11 9DL, UK
Tel - +44 1477 572636, Fax - +44 1477 571618
Received on Fri May 27 2005 - 18:38:00 PDT
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