Multiverse with convergences

Interactive simulation of a multiverse in which the branches produced by quantum measurements can reconverge instead of only forming a tree. It has no dependencies and works offline.

Open it by double-clicking simulations/quantum-mechanics/many-worlds/index.html (or from any static server); nothing to install. Theory and references: many-worlds-convergence-theory.en.md; Italian version: many-worlds-simulation-guide.it.md.

Idea

In the many-worlds interpretation every measurement with N outcomes splits every world into N branches, and after t measurements the tree has N^t worlds. Here a world is described only by its record k, the sum of the outcomes (each outcome is worth 0..N−1). Two branches that reach the same k at the same step are the same world and converge, so the structure becomes a lattice with (N−1)·t+1 nodes at step t.

The record is drawn as x = 2k − (N−1)·t, centred on zero (for N=2 it changes by ±1 at every measurement, as in a random walk).

The three regimes

The lattice is computed once for all regimes; the regime only decides what to draw.

Model

The quantum regime is a coined quantum walk on the record k, exactly unitary.

• every node carries N amplitude components ψ(k, c), one per outcome;
• at every step the DFT coin U[c][d] = exp(2πi·c·d/N)/√N (Hadamard for N=2) mixes the components of each node, then component c moves to node k+c: ψ_{t+1}(k+c, c) = Σ_d U[c][d]·ψ_t(k, d);
• the system starts in state |0⟩ of node k=0;

• node probability p(k) = Σ_c |ψ(k, c)|²; flow along branch c of node k:

  |(Uψ)_c|², computed after the coin and before the shift.

Component c of node k+c comes only from node k. The amplitudes of converging branches therefore sit in different components and interfere at the coin of the next step. The sum of the probabilities stays 1 at every step.

The phase disorder D ∈ [0, 1] rotates every node by a random phase

|φ| ≤ π·D before the coin (seeded generator, reproducible). The evolution stays unitary. With large D the interference becomes incoherent and the spreading turns diffusive again (σ ∝ √t), as in the classical regime.

In the classical case the flow along each branch is p(k)/N and the probabilities add.

Parameters and query string

Example: index.html?n=2&T=30&reg=quantum&step=30.

Language and theme (System, Light, Dark) are set with the gear button at the top right. They are saved in the browser and apply to the other simulations too.

Readings

They all refer to the displayed step t.

• Tree worlds N^t and distinct worlds (N−1)·t+1.
• Effective worlds 1/Σp²: how many worlds really "weigh" (N^t if all the worlds of the tree had the same weight, much less in the lattice).
• Spread σ: standard deviation of the record x.
• Branches cancelled by interference: branches leaving step t−1 with quantum flow below 10⁻⁹.
• Worlds with zero probability: nodes of step t with quantum probability below 10⁻⁹.

Classical and quantum readings appear in every regime, so they can be compared.

Graph legend

• Time horizontally, record x vertically (tree view: fan of worlds).
• Node size and colour grow with probability, relative to the maximum of the step. Branch opacity is proportional to the flow.
• Dashes and empty circles: branches and worlds cancelled by interference.
• Pointer on a node: the node, the branches converging into it and the nodes they come from turn orange. The tooltip shows the record, outcome sequences and the two probabilities.
• Histogram at the bottom: distribution at step t. In the quantum regime the orange outline is the classical distribution.
• Table view: the same numbers for step t, with the ratio quantum p / classical p.

Structure

simulations/quantum-mechanics/many-worlds/
├── index.html      page and control panel
├── css/style.css   theme tokens (light, dark, system) and responsive layout
├── js/i18n.js      it/en dictionaries and translation functions
├── js/model.js     lattice, quantum walk, statistics (also usable from Node)
├── js/graph.js     Canvas renderer, tooltip, histogram
└── js/app.js       controls, query string, animation

The settings button (language and theme) is shared by all simulations and lives in simulations/shared/ (js/settings.js, css/settings.css).

Simplifications

• Convergence is imposed by construction: two histories are "the same world" if they have the same record. The criterion is deliberately coarse and allows an exact, computable lattice.
• If the amplitudes of converging branches were added directly, without a coin, the dynamics would not be unitary. The coined quantum walk is the standard way to make propagation on a lattice unitary.
• The initial state |0⟩ with the Hadamard coin gives an asymmetric distribution, a known property of the Hadamard walk.
• The tree view is only counting and simulates nothing beyond the lattice.

Verification

cd simulations/quantum-mechanics/many-worlds && node -e 'const M=require("./js/model.js"); const I=require("./js/i18n.js").I18n; const L=M.buildLattice({outcomes:2,steps:3}); console.log(L[3].map(n=>n.pQuantum.toFixed(3)).join(" "), I.check().length)'

Expected: 0.125 0.625 0.125 0.125 0 (known distribution of the Hadamard walk at t=3 and no missing keys in the dictionaries).