A complete ultrametric on von Neumann's incomplete tensor products
- Published
- Source
- arXiv
- Paper number
- 597
- Field
- Research
- arXiv ID
- 2607.09627
Key points
- We define a convergence-index-based pseudo-ultrametric d on the equivalence classes Gamma of infinite tensor products, and prove the strong triangle inequality and completeness.
- We define a gauge-invariant variant d~, which is insensitive to topological changes and consistent with von Neumann weak equivalence.
- We prove that the product of any unitary U with 1 not in sigma(U) moves every equivalence class to the maximum distance 1.
- We interpret d~ as a decoherence exponent: when monitoring N environmental degrees of freedom, the branching rate decays as exp(-N d~).
- In the Everett many-worlds model, the universe is an infinite qubit tensor product, worlds are sectors, and time is a discrete unitary step.
- The hierarchical structure of the ultrametric space corresponds exactly to the leaf space of the branching tree.
Paper links
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