Disjoint additivity and local quantum physics

TL;DR

Locality in quantum systems with higher-form symmetries cannot be captured by microcausality or standard additivity alone. The authors propose Haag duality together with a weaker, covariant disjoint additivity as robust locality criteria, and prove these hold for broad lattice constructions with gauge constraints, including lattice gauge theories and stabilizer codes. They also exhibit continuum and lattice examples where Haag duality and disjoint additivity are violated, illustrating nonlocal behavior in theories with nontrivial symmetry structures, and show how a higher-dimensional SymTFT framework can restore disjoint additivity by reinterpreting certain nonlocal boundary theories as boundary conditions of a bulk topological theory. This work sharpens the algebraic notion of locality and links symmetry constraints to higher-dimensional topological constructions, with implications for both condensed matter and quantum field theory.

Abstract

Quantum systems of physical interest are often local, but there are at least three competing perspectives on how "locality" should be formalized: an algebraic framework, a path-integral framework, and a lattice framework. One puzzle in this competition is that systems with higher-form symmetries, which are perfectly local from the path-integral and lattice perspectives, can violate the algebraic principle of "additivity". In this paper, we propose a resolution to this puzzle by introducing a weaker locality principle, "disjoint additivity", which together with Haag duality should always be satisfied in local quantum systems. As evidence, we give examples in which disjoint additivity is preserved when ordinary additivity is violated; we show that Haag duality and disjoint additivity are satisfied in rather general lattice systems with local symmetry constraints; we give examples of nonlocal theories in which either disjoint additivity or Haag duality is violated; and finally we give examples of systems with nonlocal symmetry constraints in which disjoint additivity is violated, but can be restored by passing to a local "SymTFT" system in one higher dimension.

Figures (21)

• Figure 1: Defining a region $R$ and its spatial complement $R'$ in non-relativistic and relativistic systems. On the left we have a Hamiltonian lattice at fixed time, while on the right we have a spacetime diagram where light moves on 45-degree lines. The dashed boundaries are not included in either $R$ or $R'$.
• Figure 2: A non-contractible region $R=R_1\cup R_2$ that encloses a line operator $W$ (green). Here $R_1$ (red) and $R_2$ (blue) are overlapping, contractible regions. If $W$ carries a nontrivial topological 1-form global symmetry charge, it cannot end on local operators. In this case, we have $W\in {\cal A}(R_1\cup R_2)$ but $W \notin {\cal A}(R_1) \vee {\cal A}(R_2)$. This violates additivity. Here we have shown spatial regions; to get relativistic regions, we can take the domain of dependence of each spatial region.
• Figure 3: A line operator carrying a topological one-form symmetry charge must be unbreakable. Here we demonstrate this in three spacetime dimensions for a $U(1)$ one-form symmetry generated by a topological line operator $\cal L$. Let $W$ be a line operator that carries charge $q$ under ${\cal L}$. This means that if we locally unlink $W$ and ${\cal L}$, this process gives a phase $e^{iq\theta}$. If $W$ can end on a pair of operators $\psi,\psi^\dagger$, then we reach a contradiction by deforming the topological line $\cal L$ in two different ways.
• Figure 4: In one spatial dimension, $\mathbb{Z}_2$ gauge theory contains an "$X$" operator that acts as the Pauli $X$ on local edges, and a "$Z$" operator that acts as a Pauli-$Z$ chain on all edges simultaneously.
• Figure 5: In the ground space of the toric code, there are two kinds of nontrivial operators, which are invariant under topological deformations. The $X$-type line operators must pass through a face of the lattice when moving from one segment of the path to the next segment. By contrast, the $Z$-type line operators must pass through a vertex.

Theorems & Definitions (12)

• Lemma 1
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• Lemma 2
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• Theorem 1
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• Theorem 2
• Lemma 3
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• proof