# Centering the Born Rule

## Abstract

**:**

## 1. Introduction

## 2. The Centered Everett Interpretation

## 3. Utterances of Indexicals

- On Susie
_{1}’s branch and Susie_{2}’s branch, the electron is found to have x-spin up. - On Susie
_{3}’s branch and Susie_{4}’s branch, the electron is found to have x-spin down.

_{1}–Susie

_{4}all utter the sentence “I will see the electron in the x-spin up state”: so Susie

_{1}’s utterance is true, Susie

_{2}’s utterance is true, Susie

_{3}’s utterance is false, and Susie

_{4}’s utterance is false. In addition, suppose that the centered Born rule holds on all four branches in the figure; and suppose, moreover, that each of Susie

_{1}–Susie

_{4}has conducted thousands of experiments on their branch which confirm exactly that. So in the times before $t=1$, Susie

_{1}–Susie

_{4}have found that centered propositions expressed by sentences of the form “I am in one of the branches where systems with thus-and-so quantum state are found, after measurement, to have such-and-such quantum state” obtain with the frequencies that the centered Born rule predicts. Therefore, each of Susie

_{1}–Susie

_{4}knows that the centered Born rule holds on their branch.

_{1}–Susie

_{4}make; so long, that is, as Susie

_{1}–Susie

_{4}each have a distinct temporal part at the time of utterance. Susie

_{1}–Susie

_{4}are physical duplicates of each other, of course. But their temporal parts, at the time when they each utter the sentence “I will see the electron in the x-spin up state”, are distinct. That is why the problem discussed by Tappenden does not arise [16] (p. 311): Susie

_{1}–Susie

_{4}successfully refer to themselves, when the utterance of that sentence occurs, because they do not literally share temporal parts at that time. And all this vindicates the view, discussed in detail by Saunders and Wallace, that agents in the Everettian universe can experience de se uncertainty before measurement [17] (p. 301).

_{1}and Susie

_{2}find the electron to have x-spin up while Susie

_{3}and Susie

_{4}find the electron to have x-spin down, and (ii) the fact that the centered Born rule holds on the branches containing Susie

_{1}–Susie

_{4}. For the chance assigned to the proposition expressed by the sentence “I will see the electron in the x-spin up state”, one might claim, should be 0 or 1, depending on the agent Susie

_{1}–Susie

_{4}at issue. After all, on the branches containing Susie

_{1}and Susie

_{2}, that proposition obtains: therefore, one might claim that relative to Susie

_{1}and relative to Susie

_{2}—and relative to the wavefunction prior to measurement—that proposition’s chance is 1. Additionally, on the branches containing Susie

_{3}and Susie

_{4}, that proposition does not obtain: therefore, one might claim that its chance—relative to those agents, and to the wavefunction prior to measurement—is 0. So the proposition expressed by the sentence “I will see the electron in the x-spin up state” cannot be assigned chance $\frac{1}{2}$. But that is the chance that the centered Born rule, if true on the branches of Susie

_{1}–Susie

_{4}, would assign to that proposition. So given (i), it follows that (ii) is false: given what Susie

_{1}–Susie

_{4}actually observe, the centered Born rule does not hold on their branches.

_{1}and Susie

_{2}. But it does not follow that the chance of this proposition, relative to Susie

_{1}and relative to Susie

_{2}—and relative to the wavefunction prior to measurement—must be 1. Similarly, it is true that this proposition does not obtain on the branches containing Susie

_{3}and Susie

_{4}. But it does not follow that the chance of this proposition, relative to Susie

_{3}and relative to Susie

_{4}—and relative to the wavefunction prior to measurement—must be 0. Each of Susie

_{1}–Susie

_{4}utter the sentence “I will see the electron in the x-spin up state”. Susie

_{1}and Susie

_{2}speak truly: relative to them, the proposition which that sentence expresses is true. Susie

_{3}and Susie

_{4}speak falsely: relative to them, the proposition which that sentence expresses is false. But it simply does not follow, from any of this, that the chance of this proposition must be 1–relative to Susie

_{1}and Susie

_{2}(and to the wavefunction prior to measurement)–or 0–relative to Susie

_{3}and Susie

_{4}(and to the wavefunction prior to measurement).

_{1}. The objector might think that Susie

_{1}’s utterance of the sentence “I will see the electron in the x-spin up state” expresses the proposition that Susie

_{1}will see the electron in the x-spin up state; for after all, Susie

_{1}’s utterance of ‘I’ refers to Susie

_{1}. That proposition obtains deterministically: given the initial wavefunction, that proposition must hold. So the objector concludes that the proposition expressed by Susie

_{1}’s utterance of the sentence “I will see the electron in the x-spin up state” must have chance 1.

_{1}’s utterance of the sentence “I will see the electron in the x-spin up state” does not express the proposition that Susie

_{1}will see the electron in the x-spin up state. That proposition is uncentered: it is expressed without using indexicals. But the sentence “I will see the electron in the x-spin up state” contains an indexical and because of that—and for many other reasons too—expresses a centered proposition [18,19]. So the objector is wrong to claim that the proposition expressed by Susie

_{1}’s utterance of the sentence “I will see the electron in the x-spin up state” must have chance 1. The proposition that Susie

_{1}will see the electron in the x-spin up state does, of course, have chance 1 of obtaining. But the proposition expressed by Susie

_{1}’s utterance of the sentence “I will see the electron in the x-spin up state” is definitely not that proposition; so the proposition expressed by that utterance need not have chance 1, and so that proposition can have the chance assigned by the centered Born rule.

_{1}utters two sentences:

- (1)
- “I will see the electron in the x-spin up state”

- (2)
- “Susie
_{1}will see the electron in the x-spin up state.”

_{1}, the propositions expressed by those sentences are true. Nevertheless, the propositions expressed by those sentences are, despite the fact that Susie

_{1}is the speaker—and so Susie

_{1}’s utterance of ‘I’ refers to herself—different: (1) expresses a centered proposition, and (2) expresses an uncentered proposition. So while the truth value of the proposition expressed by (2) is completely fixed by the deterministic evolution of the wavefunction, the truth value of the proposition expressed by (1) is not.

_{1}–Susie

_{4}utter sentence (1), they each express the very same proposition, despite the fact that their respective uses of the indexical ‘I’ refer to different agents. This is simply how the standard semantics for centered propositions works [4]. If you and I both utter the sentence “I like apple pie”, for instance, then we both express the same proposition, despite the fact that our respective utterances of ‘I’ have different referents. This is related, of course, to the fact that truth values are only ever assigned—to centered propositions—relative to centers.

_{1}and Susie

_{2}find the electron to have x-spin up while Susie

_{3}and Susie

_{4}find the electron to have x-spin down, and (ii) the fact that the centered Born rule holds on the branches containing Susie

_{1}–Susie

_{4}. Relative to Susie

_{1}and Susie

_{2}—and relative to the wavefunction prior to measurement—the proposition expressed by the sentence “I will see the electron in the x-spin up state” is indeed true, and relative to Susie

_{3}and Susie

_{4}—and relative to the wavefunction prior to measurement—the proposition expressed by this sentence is indeed false. But it is not the case that this sentence, when uttered by Susie

_{1}say, expresses the proposition that Susie

_{1}will see the electron in the x-spin up state. So the proposition expressed by this sentence need not be assigned chance 1. It can be assigned the chance given by the centered Born rule.

## 4. Counting Branches

_{1}–Susie

_{4}are parts. There are continuum-many branches in the Everettian universe [21] (p. 20).

## 5. Metaphysics of Agents

_{1}–Susie

_{4}. Only Susie exists at that time. And so the metaphysics of agents, which I used to formulate the centered Everett interpretation, is false.

- (3)
- Susie will be such that she finds herself on branch $|{a}_{1}$⟩, and Susie will be such that she finds herself on branch $|{a}_{2}$⟩.

- (3.1)
- $F\left({A}_{1}s\right)\wedge F\left({A}_{2}s\right)$
- (3.2)
- $F({A}_{1}s\wedge {A}_{2}s)$

## 6. Branch Relativity

_{Earth}, the one-place property of being a law

_{our branch}, and so on.

## 7. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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Wilhelm, I. Centering the Born Rule. *Quantum Rep.* **2023**, *5*, 311-324.
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Wilhelm I. Centering the Born Rule. *Quantum Reports*. 2023; 5(1):311-324.
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Wilhelm, Isaac. 2023. "Centering the Born Rule" *Quantum Reports* 5, no. 1: 311-324.
https://doi.org/10.3390/quantum5010021