A complete ultrametric on von Neumann's incomplete tensor products

Technical Summary: A Complete Ultrametric on von Neumann's Incomplete Tensor Products

Problem Statement
The paper addresses the structural properties of the set $\Gamma$ of equivalence classes of $C_0$-sequences within von Neumann's theory of infinite tensor products of Hilbert spaces. In von Neumann's 1939 framework, the complete infinite tensor product $\bigotimes_{\iota \in I} \mathcal{H}_\iota$ decomposes into an orthogonal direct sum of "incomplete" tensor products, labeled by equivalence classes of $C_0$-sequences. Two sequences are equivalent if the series $\sum_\iota |\langle \phi_\iota, \psi_\iota \rangle - 1|$ converges. While this set $\Gamma$ is typically treated as an unstructured index set of large cardinality, the paper identifies a lack of quantitative structure to measure the "degree of inequivalence" between distinct classes. The central problem is to define a metric on $\Gamma$ that quantifies this separation and to analyze its topological and dynamical properties, particularly in the context of product unitary evolutions.

Methodology
The author assumes the index set $I$ is the natural numbers $\mathbb{N}$ with a standard enumeration. The methodology proceeds in three main stages:

1. Definition of the Pseudo-Ultrametric $d$:
   For two $C_0$-sequences $\phi$ and $\psi$, the distance $d(\phi, \psi)$ is defined as the convergence exponent of the series $\sum_j |\langle \phi_j, \psi_j \rangle - 1|$. Specifically:
   $$d(\phi, \psi) = \inf \left\{ p \ge 0 : \sum_{j=1}^\infty \frac{1}{j^p} |\langle \phi_j, \psi_j \rangle - 1| < \infty \right\}.$$
   This definition relies on the weighted transfer lemma (Lemma 2.5) relating the convergence of the inner product series to the convergence of the squared norm differences and imaginary parts.

2. Gauge-Invariant Variant $\tilde{d}$:
   Recognizing that physical states are often insensitive to component-wise phase changes, the author introduces a weak equivalence relation $\sim_w$ based on the convergence of $\sum_j ||\langle \phi_j, \psi_j \rangle| - 1|$. A corresponding metric $\tilde{d}$ is defined analogously to $d$ but using the absolute values of the inner products. This variant is invariant under "pure gauge" product unitaries.

3. Topological and Dynamical Analysis:
   The paper establishes the completeness of these spaces and analyzes the action of product unitaries $U = \bigotimes U_j$ on the classes. The analysis involves characterizing the permutations of the index set that preserve the metric and applying the results to a toy model of Everettian branching.

Key Contributions and Results

• Metric Structure on $\Gamma$:
  The function $d$ is proven to be a complete pseudo-ultrametric on $\Gamma$. It satisfies the strong triangle inequality $d(c, e) \le \max(d(c, d), d(d, e))$. However, $d$ is only a pseudo-metric because distinct classes can lie at distance zero (e.g., if the series diverges slower than any $j^{-p}$ for $p>0$). By quotienting $\Gamma$ by the relation $d=0$, the author constructs a complete ultrametric space $(\bar{\Gamma}, d)$.

• Gauge Invariance:
  The variant $\tilde{d}$ is shown to be a complete pseudo-ultrametric on the set of weak equivalence classes $\Gamma_w$. It descends to a genuine ultrametric on the quotient $\bar{\Gamma}_w$. The proofs for $\tilde{d}$ are noted to be simpler as they do not require controlling the imaginary parts of the inner products.

• Invariance under Reindexing:
  The paper characterizes the group of permutations $\pi$ of $\mathbb{N}$ that preserve the metric. A permutation preserves $d$ (and $\tilde{d}$) if and only if it satisfies "Condition (T)": only a subpolynomial number of indices can be displaced by more than a polynomial amount. This identifies a large gauge group of "polynomially tame" relabelings under which the metric structure is invariant.

• Dynamics of Product Unitaries:

  • Maximal Displacement: If a product unitary consists of a single unitary $U$ repeated on each factor, and $U$ satisfies $\inf_{\|x\|=1} |\langle x, Ux \rangle - 1| > 0$ (equivalent to $1 \notin \sigma(U)$ in finite dimensions), then $d(c, Uc) = 1$ for every class $c$. Such a unitary displaces every sector to the maximal possible distance.
  • Orbit Separation: If no eigenvalue of $U$ is a root of unity, the orbit $\{U^k c\}$ forms a 1-separated set in $(\Gamma, d)$.
  • Class-Dependent Displacement: In contrast, the gauge-invariant displacement $\tilde{d}(c, Uc)$ depends on the specific class. In a qubit setting, $\tilde{d}$ can realize any value in $[0, 1]$. Classes built from eigenvectors ("pointer states") have $\tilde{d}=0$, while those built from equal superpositions can have $\tilde{d}=1$.

• Physical Interpretation (Everettian Branching):
  The paper applies these results to a toy model of the many-worlds interpretation. The "worlds" are identified with the sectors (incomplete tensor products).

  • Superselection: The sector decomposition acts as a superselection structure relative to quasi-local observables; distinct sectors are orthogonal and indistinguishable by local measurements.
  • Decoherence Exponent: The metric $\tilde{d}$ is interpreted as a decoherence exponent. It measures the polynomial rate at which the overlap of two branches decays as an observer monitors an increasing number of environmental degrees of freedom. Specifically, the overlap decays as $\exp(-N^{\tilde{d}})$.
  • Branching Mechanism: Conditioned product unitaries split the universal state into branches. The value of $\tilde{d}$ determines whether these branches are distinct worlds ($\tilde{d} > 0$) or merely gauge-equivalent descriptions of the same world ($\tilde{d} = 0$). The ultrametric structure naturally encodes the hierarchical, tree-like structure of branching events.

Significance and Claims
The paper claims to provide a quantitative geometric framework for von Neumann's infinite tensor products, transforming the set of sectors from an unstructured index set into a complete ultrametric space. Its primary significance lies in:

1. Mathematical Rigor: It establishes that the space of inequivalent sectors possesses a natural, complete ultrametric topology, a property not previously highlighted in the literature.
2. Physical Interpretation: It offers a precise mathematical realization of "branching" in the Everett interpretation. The metric $\tilde{d}$ serves as a rigorous measure of decoherence, quantifying how quickly distinct branches become operationally disjoint for local observers.
3. Resolution of Phase Sensitivity: By distinguishing between the phase-sensitive metric $d$ and the gauge-invariant metric $\tilde{d}$, the paper clarifies the distinction between global orthogonality (visible to unrestricted observables) and local disjointness (visible to quasi-local observables), addressing critiques similar to Bell's regarding the Hepp model.
4. Discrete Time Evolution: The analysis supports the view that time evolution on the full infinite tensor product is intrinsically a jump process between sectors, necessitating discrete time steps unless the state components are eigenvectors of the Hamiltonian.

The author explicitly states that the model is a "mathematical caricature" intended to provide a controlled setting where the statement "the universe branches into inequivalent worlds" is a theorem rather than a metaphor, without claiming to resolve broader conceptual issues of the Everett interpretation such as the origin of probabilities.