Can the Many Worlds interpretation state the Born Rule as a postulate?

  • #1
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Main Question or Discussion Point

Every derivation from the MWI of the born rule is circular. http://fmoldove.blogspot.com/search?q=MWI

So my question is, can the MWI state the born rule as a postulate (without deriving) and still be a coherent interpretation of probability?

The most famous argument against this notion is by Meir Hemmo.http://users.ox.ac.uk/~everett/docs/Hemmo Pitowsky Quantum probability.pdf

"We have made a full circle. In the many worlds theory the Born probability rule cannot be derived from the non-probabilistic part of quantum mechanics, with or without (the non probabilistic part of) rational decision theory.17 And the reason for this, as we just saw, boils down to the familiar problem we began with, namely that the branching is unrelated to the quantum mechanical probability. But, the problem goes even deeper. Not only is it the case that the Born probability cannot be derived in the many worlds theory, but also it is unintelligible to postulate as an additional empirical hypothesis any probability rule of the form: the probability of an observer to find herself on a post measurement branch is equal to the square of the amplitude of that branch in the universal quantum state.18 To see why, let us backtrack a bit and consider what it would mean to identify the quantum measure with probability in the many worlds theory as above. No matter how we understand probability (i. e. as measuring degrees of belief, limiting relative frequency, chance), if probability is supposed to do its job, it must be related at least a-posteriori to the statistical pattern in which events occur in our world in such a way that the relative frequencies that actually occur in our world turn out to be typical. We take this as a necessary condition on whatever it is that plays the role of probability in our physical theory. Now, the quantum probability rule cannot satisfy this condition in the many worlds theory (nor can any other non-trivial probability rule), since in this theory the dynamics logically entails that any combinatorially possible sequence of outcomes occurs with complete certainty, regardless of its quantum probability. As we argued in the previous sections, nothing in the unitary dynamics of the quantum state picks out the Born probability as related in such a way to the frequencies that occur in the worlds, and there are no other features of the many worlds theory which might pick out the Born probability in this sense as opposed to any other ‘probability’ rules. Even for agents like us, who observed up to now finite sequences which a-posteriori seem to conform to the quantum probability, adopting the quantum probability as our subjective probability for future action is completely arbitrary, since there are future copies of us who are bound to observe frequencies that don’t match the quantum probabilities. We know now and with certainty that for some of our future copies the quantum probability rule will turn out to be false. So, if we believe that the many worlds theory is true, it will be utterly irrational for us to adopt the quantum probability rule as our subjective probability for future action (nor any other non-trivial probability rule). Viewed in this way, it is not even clear that rational decision theory is applicable in the many worlds theory.

For these reasons we believe that in quantum mechanics the only way in which the subjective probabilities of observers could be guided by the quantum mechanical probabilities is by adding to the dynamics in the theory a genuine stochastic process. This can be done either by injecting chances into the dynamics (as in collapse theories) or by adding to the theory some stochastic (chance) process over and above the unitary dynamics of the state. However, as to the latter possibility, we have argued in our (2003) that it seems inevitable that adding a stochastic dynamics for the worlds or for the minds over and above the dynamics of the quantum state (e. g. of the kind proposed by Albert and Loewer (1988) in their many minds theory) would turn the many worlds theory into a hidden variables theory of a sort, in which subsets of branches (or minds) have quantitative properties that transcend, and cannot be inferred from quantum theory itself."

So taking account the argument above, can the MWI state the born rule as a postulate (without deriving) and still be a coherent interpretation of probability?
 

Answers and Replies

  • #2
Demystifier
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So taking account the argument above, can the MWI state the born rule as a postulate (without deriving) and still be a coherent interpretation of probability?
There is another problem with MWI, the preferred basis problem.
 
  • #4
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Anyone have an answer?
 
  • #5
DarMM
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Anyone have an answer?
Well more so it has to give a meaning connecting the branch coefficients to the "chance" to find yourself in a world.

So in the following state, with ##A_{\uparrow}## and ##A_{\downarrow}## being states of the measuring device:
$$|\psi\rangle = a|\uparrow,A_{\uparrow}\rangle + b|\downarrow,A_{\downarrow}\rangle$$

You have to give some reason why ##|a|^2## being larger means you are more likely to be in that world.

If you just have the Born Rule as an assumption in Many-Worlds the quantum state has a very weird meaning as its physical content would be simply the set of worlds:
$$\{|\uparrow,A_{\uparrow}\rangle,|\downarrow,A_{\downarrow}\rangle\}$$
with ##a## and ##b## just being coefficients indicating how likely you are to find yourself in a world, meaning the Hilbert Space formalism would be a strange mix of epistemic and ontological content. All basis kets in a basis with decoherence would be "worlds", but their coefficients in that basis are future predictions of agents to find themselves in the worlds. And what of bases that don't have classical worlds? Are the coefficients just meaningless there?

Better to have the coefficients be meaningful physically in all bases (which fits the main aim of Many-Worlds, to offer an objective, observer-independent account) and that in the case of a decohered semi-classical basis one can show that an agent prior to the split should assign:
$$Pr(I\:will\:enter\:world\: |\uparrow,A_{\uparrow}\rangle) = |a|^2$$

To me it would be like saying in a classical electromagnetic wave like:
$$E = a\vec{E_{k}} + b\vec{E_{q}}$$
where ##\vec{E_{k}}## and ##\vec{E_{q}}## are fields corresponding to light rays with momenta ##k,q##, that ##a## and ##b## are how much the light rays cause our devices to react, rather than saying they measure the beam's energy with this then explaining why our devices react to large amplitude ones more.
 
  • #7
DarMM
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The Quantum Darwinism based derivation, which Zurek was ultimately driven to after circularities were found in all his other derivations, is also circular since it requires a decompostion of the universal wavefunction into a preferred environment-pointer-system split and then assumes the phases in the environment are randomised.
 
  • #8
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The Quantum Darwinism based derivation, which Zurek was ultimately driven to after circularities were found in all his other derivations, is also circular since it requires a decomposition of the universal wavefunction into a preferred environment-pointer-system split and then assumes the phases in the environment are randomised.
There is no such assumption of random environmental phases, which are not needed in the derivation (and which can be removed by a unitary transformation, BTW). As for the decomposition business, if the universe is not complex enough for subsystems, that is hardly a problem for our universe.
 
  • #9
DarMM
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There is no such assumption of random environmental phases, which are not needed in the derivation (and which can be removed by a unitary transformation, BTW). As for the decomposition business, if the universe is not complex enough for subsystems, that is hardly a problem for our universe.
I'm speaking of the point Zurek was ultimately driven to in the most developed form of his proofs, that 2007 article has different issues.
 
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