AI 中文总结
研究冯·诺依曼的希尔伯特空间无限张量积理论，在\(C_0\)序列等价类集合\(\Gamma\)上引入伪超度量\(d\)，证明其性质并应用于乘积酉算子。还发展规范不变变体\(\tilde d\)，将其解释为退相干指数，刻画态矢量分支的变化。
AI 中文摘要
我们重新审视冯·诺依曼的希尔伯特空间无限张量积理论。在标记完备张量积内不完全张量积的\(C_0\)序列等价类集合\(\Gamma\)上，引入自然伪超度量\(d\)：两类之间的距离是由任意一对代表元构成的级数\(\sum_j|\langle\varphi_j,\psi_j\rangle - 1|\)的收敛指数。证明\(d\)在等价类上定义良好，满足强三角不等式且完备。不同类可能距离为零，通过\(d = 0\)关系取商\(\widetilde\Gamma\)后，\((\widetilde\Gamma,d)\)是完备超度量空间。作为应用，表明因子\(U\)满足\(\inf_{\|x\| = 1}|\langle x,Ux\rangle - 1|>0\)的乘积酉算子\(\bigotimes_j U\)将每个类移到最大距离\(1\)。基于对埃弗雷特分支的刻画（其中无限张量积的扇区扮演世界的角色），还基于冯·诺依曼弱等价和拟局部代数上乘积态的拟等价发展了度量的规范不变变体\(\tilde d\)。由\(\tilde d\)测量的乘积酉算子下类的位移依赖于类并实现\([0,1]\)中的每个值。将\(\tilde d\)解释为退相干指数：它测量通用态矢量的两个分支随着监测环境的部分越来越大而在操作上变得不同的多项式速率。
英文摘要
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 $\sum_j|\langleφ_j,ψ_j\rangle-1|$ 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.
Comments45 pages
