A complete ultrametric on von Neumann's incomplete tensor products

• URL: http://arxiv.org/abs/2607.09627v3
• Date: Fri, 17 Jul 2026 01:18:56 GMT
• Title: A complete ultrametric on von Neumann's incomplete tensor products
• Abstract summary: We revisit von Neumann's theory of infinite tensor products of Hilbert spaces.<n>The distance between two classes is the convergence exponent of the series $sum_j|langlevarphi_j,_jrangle-1|$ formed from any pair of representatives.<n>We show that $d$ is well defined on equivalence classes, satisfies the strong triangle inequality, and is complete.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We revisit von Neumann's theory of infinite tensor products of Hilbert spaces. On the set $Γ$ of equivalence classes of $C_0$-sequences, which labels the incomplete tensor products inside the complete tensor product, we introduce a natural pseudo-ultrametric $d$: the distance between two classes is the convergence exponent of the series $\sum_j|\langle\varphi_j,ψ_j\rangle-1|$ formed from any pair of representatives. We show that $d$ is well defined on equivalence classes, satisfies the strong triangle inequality, and is complete. Distinct classes may lie at distance zero, so $d$ separates points only after passing to the quotient $\widetildeΓ$ of $Γ$ by the relation $d=0$; the pair $(\widetildeΓ,d)$ is then a complete ultrametric space. As an application, we show that a product unitary $\bigotimes_j U$ whose factor $U$ satisfies $\inf_{\|x\|=1}|\langle x,Ux\rangle-1|>0$ (in particular, a unitary on a finite dimensional space with $1\notinσ(U)$) displaces every class to the maximal distance $1$. Guided by the intended application -- a caricature of Everettian branching, in which the sectors of the infinite tensor product play the role of worlds -- we also develop a gauge-invariant variant $\tilde d$ of the metric, based on von Neumann's weak equivalence and matched to the quasi-equivalence of product states on the quasi-local algebra. The displacement of a class under a product unitary, measured by $\tilde d$, is class dependent and realizes every value in $[0,1]$. We interpret $\tilde d$ as a decoherence exponent: it measures the polynomial rate at which two branches of the universal state vector become operationally distinct as ever larger portions of the environment are monitored.

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