Probability in Many Worlds

In an Everettian account, unitary evolution retains the components associated with different measurement records. Probability cannot mean a fundamental lottery that leaves only one component. The challenge is to explain what an observer should expect and why squared amplitudes, rather than a count of branches, guide those expectations.

Required background. Born rule: Translate a specified state and measurement into outcome probabilities.

A measurement with several records

An idealized interaction yields

$$(c_0\lvert0\rangle+c_1\lvert1\rangle)\lvert R\rangle \longrightarrow c_0\lvert0\rangle\lvert R_0\rangle+ c_1\lvert1\rangle\lvert R_1\rangle.$$

Environmental correlations make interference between the record components inaccessible in ordinary observations. This provides the setting for Everettian relative states. It does not delete either nonzero component.

One proposal understands probability as self-locating uncertainty about which future record will be experienced. Another emphasizes how a present agent should care about future continuations. These accounts use related mathematics but should not be treated as identical philosophical claims.

Why branch counting is unstable

Suppose one macroscopic result becomes correlated with 100 unobserved environment alternatives while another becomes correlated with 1,000. Counting those arbitrary fine-grained components would make probabilities depend on the chosen environmental resolution. A further unitary on inaccessible degrees of freedom could change the count without changing the macroscopic experiment.

Squared norm is stable under such refinement: if a branch of amplitude $c$ is split into orthogonal pieces $c_j$, unitarity preserves $\sum_j|c_j|^2=|c|^2$. Approximate decoherence also lacks a unique exact world decomposition. These facts undercut naive counting; they do not alone establish the normative interpretation of norm weight.

Repeated measurements

For $N$ independent copies with single-copy weight $p$ for outcome 1, the total squared norm of records with $k$ successes is

$$W_N(k)=\binom Nk p^k(1-p)^{N-k}.$$

Orthogonality of distinct records makes their weights add. The resulting measure concentrates near $k/N=p$. For $p=0.9$, equal-counting of all binary strings instead concentrates near frequency $1/2$, showing that the choice of measure has explanatory work to do.

Low-weight atypical records still occur as nonzero components in this idealized state. A successful argument must explain why their existence does not make ordinary probability reasoning meaningless.

Proposed bridges

Deutsch–Wallace decision theory seeks Born-weighted preferences from rationality assumptions. Envariance uses symmetries of entanglement. Typicality and self-location approaches offer other bridges. Their premises, rather than the rhetoric of “deriving probability from nothing,” determine what each establishes.

The central open interpretive issue is the connection between mathematical weight, personal expectation, and empirical confirmation. The unitary dynamics and binomial norm calculation can be accepted even by a critic of a particular bridge principle.

Exercises

A branch of squared norm $0.3$ is split into five equal-amplitude orthogonal subbranches. Find each subbranch weight and explain why the parent event retains its weight.

Solution

Each subbranch has squared norm $0.3/5=0.06$. Their orthogonality gives total weight $5(0.06)=0.3$. Counting components would change the apparent relative share, but norm additivity preserves the coarse-grained event under the unitary refinement.

References

• H. Everett III, “‘Relative state’ formulation of quantum mechanics,” Reviews of Modern Physics 29, 454–462 (1957), doi:10.1103/RevModPhys.29.454.
• D. Wallace, “A formal proof of the Born rule from decision-theoretic assumptions” (2009), arXiv:0906.2718.