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I am still not covinced that the Born rule is sufficient. It misses what I called "bottom-up" perspective.
The Born rule says that
- results of a measurement of an observable A will always be one of its eigenvalues a
- the probability for the measurement of a in an arbitrary state psi is given by a projection to the eigenstate
##p(a) = \langle\psi|P_a|\psi\rangle##
This is a probability formulated on the full Hilbert space.
I still do not see how this is answers the question:
What is the probability p(Ba) that I find me as an observer in a certain branch Ba where a is realized as measurement result?
One could reformulate the problem as follows: The Born rule says that the probability to find a is p(a). What I am asking for is the probability to find a, provided that I am in a certain branch Ba where a is realized (the expectation is 100%, so we need some kind of Bayesian argument to extract the probability p(Ba) the branch)
I would like to see a mathematical expression based on the MWI assumptions which answers this question.
The Born rule as stated above is formulated on the full Hilbert space and therefore provides a top-down perspective, but I as an observer within one branch do have a bottom-up perspective. I still don't see why these two probabilities are identical and how this can be proven to be a result of the formalism. There is some additional (hidden) assumption.
The Born rule says that
- results of a measurement of an observable A will always be one of its eigenvalues a
- the probability for the measurement of a in an arbitrary state psi is given by a projection to the eigenstate
##p(a) = \langle\psi|P_a|\psi\rangle##
This is a probability formulated on the full Hilbert space.
I still do not see how this is answers the question:
What is the probability p(Ba) that I find me as an observer in a certain branch Ba where a is realized as measurement result?
One could reformulate the problem as follows: The Born rule says that the probability to find a is p(a). What I am asking for is the probability to find a, provided that I am in a certain branch Ba where a is realized (the expectation is 100%, so we need some kind of Bayesian argument to extract the probability p(Ba) the branch)
I would like to see a mathematical expression based on the MWI assumptions which answers this question.
The Born rule as stated above is formulated on the full Hilbert space and therefore provides a top-down perspective, but I as an observer within one branch do have a bottom-up perspective. I still don't see why these two probabilities are identical and how this can be proven to be a result of the formalism. There is some additional (hidden) assumption.