From: *Brent Meeker* <meeke...@verizon.net <mailto:meeke...@verizon.net>>
On Friday, September 21, 2018 at 12:11:01 AM UTC-5, Bruce Kellett wrote:
Adrian Kent (arXiv:1408.1944) makes some interesting comments
about the recent argument by Sebens and Carroll (arXiv:1405.7577)
that probability in MWI can be understood in terms of
self-locating uncertainty -- when all outcomes of a measurement
are realized in unitary quantum mechanics, probabilities might
arise because one is does not know in which branch of the
universal wave function one is located. Kent points out that this
raises questions about how branches are formed in unitary quantum
mechanics.
The usual Everettian argument is that when one measures a state
with two possible outcomes, say a spin-1/2 particle, unitary
evolution takes the states representing the apparatus, observer,
and environment to a FAPP orthogonal set of states branched
according to each of the possible measurement results.
Schematically, one writes the interaction with
|psi> = (|+> + |->)/sqrt(2)
as |psi>|O>, where O is the "ready" state of the observer
(including apparatus and environment). Thus:
(|+> + |->)|O>
At this point there is just one observer who has not become
entangled with the apparatus or the rest of the environment. To
take this to the next stage, Kent points out that we use the
distribution law of algebra to eliminate the above brackets, and
write
It seems that you are treating this mathematical rewriting as a
physical process. Why insert it between
(|+> + |->)|O> and |+>|O+> + |->|O-> and create the appearance of a
problem?
There is a lacuna in the physical narrative at this point. Each
component of the superposition acts on the apparatus/observer in the
same 'ready' state in order to get |O+> as different from |O->. This
differentiation must take place before decoherence acts to diagonalize
the density matrix. Otherwise all terms in the density matrix would be
the same and there would be no distinction between outcomes. You can't
just paper over this explanatory gap by calling it a mathematical rewriting.
Bruce
Brent
|+>|O> + |->|O> (O is uncertain which result he will see)
which, by unitary evolution, becomes entangled with the rest of
the wave function:
|+>|O+> + |->|O-> ( O has a definite result>
representing observers who record '+' or '-' results,
respectively. Before the last step, the observer does not know
which branch he is on, hence the self-locating uncertainty that
is presumed to be the origin of quantum probabilities.
But Kent points out that there is a problem with this -- in the
line in which O is uncertain, the observer has already split:
there is a copy on each branch of the wave function, even though
the observer has not yet interacted with the apparatus or the
environment, so what caused the observer to split and appear on
both branches in this way? We have used the distribution law of
algebra to expand the brackets in such as way as to naively
indicate that such a split has taken place. But how does this
actually happen, physically? Above we are just talking about
equations -- these have to be related to the physics in some
unambiguous way.
Kent comments on the problem that this causes for the Sebens and
Carroll idea of probability as self-locating uncertainty. But it
would seem that the problem is deeper than this. We commonly
divide the Hilbert space into the tensor product of subspaces
representing the apparatus and the environment, as well as the
observer. Then unitary evolution is supposed to act on each
component of this product space so that, ultimately, decoherence
renders the branches FAPP orthogonal, and we can then talk of
separate "worlds". But there is no reason to suppose that this
division into convenient classical components corresponds to any
actual factorization of the quantum Hilbert space -- there is no
clear separation into apparatus-observer-environment, so it is
reasonable to call them all the one thing, as I have done above.
Kent comments on this situation as follows:
"...these are just statements about ink on paper. To translate
them into statements about one or more observers, who are
uncertain about some relevant fact about their location on
branches, requires some principled general account of how we
start from the universal wave function and derive an ontology
that includes (at least) observers and branches.....and observers
must be split into copies before they observe the relevant
event." Kent sees several problems with any such approach to
understanding the above, apparently simple, mathematical relations.
His conclusion is: "Fifty-seven years of sometimes careful work
on trying to make scientific sense of Everettian quantum theory
ought, surely, to have persuaded the theoretical physics
community that one cannot make useful progress this way. Whatever
one thinks of the scientific status of many worlds quantum
theory, one cannot reasonably, at this point, think it is so
obvious how to translate equations into statements about a
many-worlds reality that arguments and explanations are redundant."
And again: "Moreover, it is worth underlining again here that, if
we /were/ able to find reasonably natural postulates that
respected physical symmetries and defined an objective branching
structure for the universal wave function, it would be
superfluous to postulate many independent real worlds. It would
be simpler and more natural to postulate that precisely one of
the branches is randomly chosen (using the Born wright
distribution) and realized in nature."
That idea would certainly overcome the problem of the apparent
need for apparatus, observers, and the environment to split
/before/ there is any interaction -- one potential branch is
randomly chosen, and then that branch develops in the standard
way. In reality there would be no splitting -- just a stochastic
process.
Bruce
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