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 writeIt 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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