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jbergman
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- TL;DR Summary
- The equal probability case of the Born rule is relatively easy to grok in the MWI discussion. I wanted to have a focused discussion on the case where the magnitude of the amplitudes are unequal and derivations from some of the literature in the field.
In Caroll and Seben's paper, Many Worlds, the Born Rule, and Self-Locating Uncertainty, they present a derivation of the Born rule.
For the equal probability case their derivation is based on the following principles.
$$| \Psi \rangle = | O \rangle (|\uparrow \rangle + \sqrt{2} | \downarrow \rangle)| E \rangle$$
According to Carroll, entanglement of the measurement happens before the observer registers so the state evolves to
$$| \Psi \rangle = | O \rangle (|\uparrow \rangle | E_{\uparrow} \rangle + \sqrt{2} | \downarrow \rangle | E_{\downarrow} \rangle)$$
Finally the observer sees the measurement ant the state of the observer changes on each branch.
$$| \Psi \rangle = | O_{\uparrow} \rangle |\uparrow \rangle | E_{\uparrow} \rangle + \sqrt{2} | O_{\downarrow} \rangle| \downarrow \rangle | E_{\downarrow} \rangle$$
Carroll himself points out that the principle of indifference might suggest that branch counting would give an equal probability for each outcome. However, Carroll adds another principle, "Epistemic Separability", which he argues that this isn't so.
Epistemic Separability is basically interpreted as saying that a unitary transformation that acts only on the environment doesn't affect the probabilities of the Observer/System subsystem. So in the unequal probabilities case they consider the state we described above which is after measurement but before the observer has registered the measurement (where we have relabeled ##E_{\downarrow}## as ##E_1##, etc., because we will need to consider more orthonormal states of the environment besides up and down.
$$| \Psi \rangle = | O \rangle (|\uparrow \rangle | E_1 \rangle + \sqrt{2} | \downarrow \rangle | E_2 \rangle)$$
They then consider a unitary transformation,
$$U = | \hat{E}_1 \rangle \langle E_1 | + \frac{(| \hat{E}_2 \rangle + | \hat{E}_3 \rangle )}{\sqrt{2}} \langle E_2 | + \frac{(| \hat{E}_2 \rangle - | \hat{E}_3 \rangle )}{\sqrt{2}} \langle E_3 | + \sum_{\mu > 3} | \hat{E}_{\mu} \rangle \langle E_{\mu} |$$
Then
$$ U | \Psi \rangle = | O \rangle (|\uparrow \rangle | \hat{E}_1 \rangle + | \downarrow \rangle | \hat{E}_2 \rangle + | \downarrow \rangle | \hat{E}_3 \rangle)$$
So, in essence they have applied a unitary transformation that only affects the environment that results in three states with equal amplitude and hence equal probabilities for each of the worlds. This leads to their conclusion that since ##\uparrow## occurs in one of the worlds and ##\downarrow## occurs in the other 2 that,
$$P(\uparrow | \Psi) = 1/2 P(\downarrow | \Psi) = 1/3$$
This argument is similar to Zurek's and essentially can be seen as saying that their is a freedom to introduce extra state from the environment such that we can arrange to have a set of orthonormal states/worlds such that they all have equal amplitude.
Overall, the argument seems logical, but a few things bother me. For one, it implies that branch counting is ok when we have equal probabilities but not when we have unequal probabilities. It also suggests that every state that has unequal probabilities is equivalent to a state with equal magnitude amplitudes when looking at a subsystem. Lastly, it doesn't seem to fit nicely with the description of worlds as decoherent orthonormal states. We have somehow had to introduce this additional unitary operator which we applied after to get more decoherence to achieve the appropriate branch counts.
I feel like the Principle of Indifference is not solid enough and instead we need a more physical explanation that allows to rely only on branch counts.
For the equal probability case their derivation is based on the following principles.
- Self Locating Uncertainty - "the condition of an observer who knows that the environment they experience occurs multiple times in the universe, but doesn’t know which example they are actually experiencing"
- The Principle of Indifference - "if all you know is that you are one of N occurrences of a particular set of observer data, you should assign equal credence, 1/N, to each possibility"
$$| \Psi \rangle = | O \rangle (|\uparrow \rangle + \sqrt{2} | \downarrow \rangle)| E \rangle$$
According to Carroll, entanglement of the measurement happens before the observer registers so the state evolves to
$$| \Psi \rangle = | O \rangle (|\uparrow \rangle | E_{\uparrow} \rangle + \sqrt{2} | \downarrow \rangle | E_{\downarrow} \rangle)$$
Finally the observer sees the measurement ant the state of the observer changes on each branch.
$$| \Psi \rangle = | O_{\uparrow} \rangle |\uparrow \rangle | E_{\uparrow} \rangle + \sqrt{2} | O_{\downarrow} \rangle| \downarrow \rangle | E_{\downarrow} \rangle$$
Carroll himself points out that the principle of indifference might suggest that branch counting would give an equal probability for each outcome. However, Carroll adds another principle, "Epistemic Separability", which he argues that this isn't so.
Epistemic Separability is basically interpreted as saying that a unitary transformation that acts only on the environment doesn't affect the probabilities of the Observer/System subsystem. So in the unequal probabilities case they consider the state we described above which is after measurement but before the observer has registered the measurement (where we have relabeled ##E_{\downarrow}## as ##E_1##, etc., because we will need to consider more orthonormal states of the environment besides up and down.
$$| \Psi \rangle = | O \rangle (|\uparrow \rangle | E_1 \rangle + \sqrt{2} | \downarrow \rangle | E_2 \rangle)$$
They then consider a unitary transformation,
$$U = | \hat{E}_1 \rangle \langle E_1 | + \frac{(| \hat{E}_2 \rangle + | \hat{E}_3 \rangle )}{\sqrt{2}} \langle E_2 | + \frac{(| \hat{E}_2 \rangle - | \hat{E}_3 \rangle )}{\sqrt{2}} \langle E_3 | + \sum_{\mu > 3} | \hat{E}_{\mu} \rangle \langle E_{\mu} |$$
Then
$$ U | \Psi \rangle = | O \rangle (|\uparrow \rangle | \hat{E}_1 \rangle + | \downarrow \rangle | \hat{E}_2 \rangle + | \downarrow \rangle | \hat{E}_3 \rangle)$$
So, in essence they have applied a unitary transformation that only affects the environment that results in three states with equal amplitude and hence equal probabilities for each of the worlds. This leads to their conclusion that since ##\uparrow## occurs in one of the worlds and ##\downarrow## occurs in the other 2 that,
$$P(\uparrow | \Psi) = 1/2 P(\downarrow | \Psi) = 1/3$$
This argument is similar to Zurek's and essentially can be seen as saying that their is a freedom to introduce extra state from the environment such that we can arrange to have a set of orthonormal states/worlds such that they all have equal amplitude.
Overall, the argument seems logical, but a few things bother me. For one, it implies that branch counting is ok when we have equal probabilities but not when we have unequal probabilities. It also suggests that every state that has unequal probabilities is equivalent to a state with equal magnitude amplitudes when looking at a subsystem. Lastly, it doesn't seem to fit nicely with the description of worlds as decoherent orthonormal states. We have somehow had to introduce this additional unitary operator which we applied after to get more decoherence to achieve the appropriate branch counts.
I feel like the Principle of Indifference is not solid enough and instead we need a more physical explanation that allows to rely only on branch counts.
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I believe there have been some previous PF threads on the topic, including some good references to the literature; it might be worth searching PF.