Everett Many Worlds

Everettian quantum mechanics applies unitary dynamics to systems, apparatus, and observers without a fundamental collapse. A measurement produces correlated relative records. Modern versions use decoherence to explain why those records support approximately autonomous classical descriptions, while differing about the precise ontology of worlds and probability.

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

Relative states

Write a correlated state as

$$\lvert\Psi\rangle =\alpha\lvert0\rangle\lvert R_0\rangle+ \beta\lvert1\rangle\lvert R_1\rangle.$$

Relative to record $R_0$, the system state is $\lvert0\rangle$; relative to $R_1$, it is $\lvert1\rangle$. This is obtained by projecting the joint vector on a record and normalizing the remaining vector. It is a relation within the global state, not a physical removal of the other term.

Everett’s 1957 relative-state formulation introduced this strategy. Later “many worlds” accounts often describe worlds as emergent patterns or branches, not as an additional exact list inserted into the Schrödinger equation.

Decoherence and approximate branches

An environment evolves into distinguishable records,

$$\lvert\Psi\rangle =\alpha\lvert0,R_0,E_0\rangle+ \beta\lvert1,R_1,E_1\rangle.$$

Small $\langle E_1|E_0\rangle$ suppresses locally observable interference. Repeated amplification and environmental monitoring can stabilize record structures. The classicality chapter examines when this is sufficient for a classical description.

Branch structure depends on coarse-graining and approximate dynamical independence. It is usually not a unique exact decomposition into an integer number of worlds. Arbitrary changes of basis in Hilbert space do not by themselves generate equally stable macroscopic record structures.

Outcomes and probability

A branch contains an observer record correlated with a definite outcome relative to that branch. The global state retains alternatives. Thus Everett gives up a unique absolute outcome while aiming to preserve ordinary definite experience within each record structure.

Why should an observer use $|\alpha|^2$ rather than count alternatives? Probability in many worlds explains why branch counting is unstable and distinguishes norm concentration from its interpretation. Decision theory, symmetry, and self-location proposals address different aspects of the remaining problem.

A claim that “every imaginable event happens” is inaccurate. Dynamics, initial conditions, and zero amplitudes constrain which records occur.

Locality and empirical status

Unitary dynamics can be formulated using local interactions in relativistic theories, but that does not make the full account Bell-local in the hidden-variable factorization sense. Entangled states and Bell correlations persist. The meaning of locality must be specified before comparing Everett with other interpretations.

Within its usual domain and with Born probabilities, Everett reproduces standard quantum predictions. It is not established experimentally as the unique interpretation. Possible quantum-gravity extensions are additional theoretical work, not consequences of the name “many worlds.”

Strengths and costs

The proposal treats observers within the same dynamics as other systems and avoids a fundamental measurement boundary. Its explanatory costs include the emergence and ontology of branches, the probability bridge, and the relationship between global structure and experienced records. Decoherence substantially constrains these questions without settling every philosophical version.

Exercises

If $\langle E_1|E_0\rangle=0$, is the total state a classical mixture? What observable distinguishes the pure superposition from that mixture in principle?

Solution

The total state is still pure. Write $\lvert u\rangle=\lvert0,R_0,E_0\rangle$ and $\lvert v\rangle=\lvert1,R_1,E_1\rangle$. When both amplitudes are nonzero, let $\theta=\arg(\alpha^*\beta)$ and choose

$$O=e^{-i\theta}\lvert u\rangle\langle v\rvert +e^{i\theta}\lvert v\rangle\langle u\rvert.$$

Its expectation is $2|\alpha\beta|$ in the superposition and zero in the corresponding incoherent mixture. The phase choice matters: a fixed real off-diagonal observable can miss a relative phase of $\pi/2$. If either amplitude vanishes, the two states already coincide. Implementing that global measurement may be impractical, but local indistinguishability does not imply global equality.

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, The Emergent Multiverse: Quantum Theory according to the Everett Interpretation, Oxford University Press (2012).