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Disjoint additivity and local quantum physics

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Pith's one-line read Locality in quantum theory is Haag duality plus disjoint additivity.

desk verdict A serious, well-built case for replacing additivity with disjoint additivity as the algebraic mark of locality—the lattice theorems are real and the examples are telling, but the universal continuum claim is a conjecture resting on an explicitly unproven bridge from finite lattice systems. read the letter →

arxiv 2509.03589 v1 pith:RMXKBWYR submitted 2025-09-03 hep-th cond-mat.str-elmath.OAquant-ph

Daniel Harlow , Shu-Heng Shao , Jonathan Sorce , Manu Srivastava This is my paper
classification hep-thcond-mat.str-elmath.OAquant-ph MSC 81T0581T1346L10
keywords algebraicquantumfieldtheoryHaagdualitydisjointadditivityhigher-formsymmetrieslatticegaugeMajoranachainSymTFTlocality
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proposes a precise algebraic criterion for what counts as a local quantum theory: any local theory should satisfy Haag duality together with a weaker form of additivity called disjoint additivity. Ordinary additivity is too strong, because topological higher-form global symmetries, as in free Maxwell theory, force unbreakable line operators that cannot be generated by local operators in overlapping contractible regions. The paper argues that Haag duality plus disjoint additivity holds in broad classes of lattice systems with local symmetry constraints, including lattice gauge theories and stabilizer code ground spaces, and fails in the nonlocal constructions of hyperplane-restricted theories, generalized free fields, global-symmetry invariant sectors, Virasoro identity multiplets, and odd Majorana chains. If correct, this gives a single algebraic rule that admits higher-form-symmetric field theories as local while excluding the usual suspects.

What carries the argument

The machinery is the pair of algebraic conditions — Haag duality, A(R)′ = A(R′), and disjoint additivity, A(R1∪R2) = A(R1)∨A(R2) for spatially disjoint R1,R2 — applied to von Neumann algebras assigned to causally complete regions. In the lattice proof, the load-bearing identity is the factorization of the group-averaged operator: for a compact Lie group S that factorizes as S = S1 × S2 × N relative to two non-adjacent regions, (O1O2)_S = (O1)_S (O2)_S, where (O)_S = ∫ ds U(s) O U(s)†. This makes the constraint projection factorize and forces the algebra of the union to be generated by the two subregion algebras. The short-range adjacency rule decides which pairs of regions the axiom is impos

What would settle it

Compute the region algebras of free U(1) Maxwell theory on a spatial torus using covariant regions R = R′′ and check disjoint additivity for a pair of spatially disjoint regions whose union is a non-contractible annulus containing a Wilson loop. A single pair with A(R1∪R2) ≠ A(R1)∨A(R2) would refute the main claim; alternatively, find one ordinary local QFT with a short-range lattice definition whose continuum limit violates either axiom.

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Extended reading notes

Core claim

The central claim is that a local quantum system is one whose region algebras obey two axioms: Haag duality, A(R)′ = A(R′), and disjoint additivity, A(R1∪R2) = A(R1)∨A(R2) whenever R1 and R2 are spatially disjoint. Ordinary additivity is dropped because topological higher-form symmetries create unbreakable extended operators that lie in the algebra of a non-contractible region but are not generated by the algebras of overlapping contractible pieces. The paper proves on the lattice that projecting a tensor-product Hilbert space onto the invariant subspace of a compact Lie group constraint preserves Haag duality, and preserves disjoint additivity for regions that are non-adjacent under a short

Load-bearing premise

Every continuum local quantum field theory is assumed to arise as a continuum limit of a finite lattice system whose constraint group action factorizes between complementary regions and whose adjacency rule is short-range; the paper's theorems are proved exactly only for such lattice systems.

Editorial extensions

If this is right

  • Free Maxwell theory and other continuum QFTs with topological higher-form global symmetries count as local: they violate ordinary additivity but satisfy Haag duality and disjoint additivity.
  • The four classic microcausal but nonlocal constructions — hyperplane restrictions, generalized free fields, invariant sectors under a global symmetry, and Virasoro identity multiplets — are excluded because they violate at least one of the two axioms.
  • Odd-numbered Majorana chains fail Haag duality because they secretly forget a fermion, while even chains satisfy both axioms, giving a sharp algebraic diagnosis of odd-chain nonlocality.
  • Lattice gauge theories and stabilizer-code ground spaces with factorizing compact Lie constraints satisfy both axioms for non-adjacent regions, providing a large class of explicitly local models.
  • Invariant-sector nonlocality is sometimes curable by realizing the system as the boundary of a topological theory in one higher dimension, restoring disjoint additivity through bulk Wilson lines.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The proposal suggests locality should be viewed as a property of the algebra-to-region assignment rather than of the Hamiltonian or path integral alone: two theories with the same Hilbert space but different region algebras can differ in locality.
  • A testable extension would apply the two axioms to lattice systems with long-range or non-local adjacency rules; the paper's theorems predict disjoint additivity fails exactly when the constraint group is not locally generated.
  • Following the paper's use of modular invariance as a completeness criterion, one could scan 2D CFT data: any sector whose torus partition function is not modular invariant must violate at least one of the two axioms.
  • The SymTFT restoration implies that apparent nonlocalities can be dimensional illusions: a system that is nonlocal in d dimensions may be exactly local as a boundary theory in d+1, so spacetime dimension is part of the data of locality.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper proposes a universal locality criterion for quantum systems: any local quantum theory must satisfy Haag duality together with a weaker replacement for additivity, called disjoint additivity. The motivation is that higher-form global symmetries, e.g. in free Maxwell theory, violate ordinary additivity while still being local. The paper defines disjoint additivity for lattice, nonrelativistic continuum, and relativistic regions, proves Haag duality and disjoint additivity for finite lattice systems with a compact Lie group constraint whose action factorizes as in Eq. (3.1), gives examples of nonlocal theories that violate one or both axioms (hyperplane restrictions, generalized free fields, invariant sectors, Virasoro identity multiplets, odd Majorana chains), and shows how invariant sectors can sometimes be reinterpreted as local boundary theories of a SymTFT in one higher dimension. Appendix A gives a detailed covariant definition of regions in Lorentzian spacetimes.

Significance. If the central proposal is correct, the paper resolves a long-standing tension between algebraic locality axioms and the physical locality of theories with higher-form symmetries, and it sharpens the algebraic diagnosis of nonlocal 'sub-theories' such as invariant sectors and generalized free fields. The lattice proofs in Secs. 3.2-3.3 and App. B are genuine derivations, not merely heuristic; the Majorana-chain analysis in App. C is explicit and checkable; and the covariant region definitions in App. A are a useful contribution independent of the main proposal. The main weakness is that the paper's universal claim extends considerably beyond the proved lattice statements, and the flagship continuum example, free Maxwell theory, is asserted but not directly verified.

major comments (3)
  1. [Sec. 3.3, after Eq. (3.36)] The central proposal (Sec. 1) is universal, but the proof of disjoint additivity is proved only for finite lattice systems with a factorized compact Lie group action and regions declared non-adjacent through subgroups S1,S2,N satisfying (i)-(iii). The text explicitly states that the theorem does not promise that continuum spatially disjoint regions arise as continuum limits of non-adjacent lattice regions, and that a finite lattice has no unambiguous spatial dimension. The added 'short-range' condition is an assumption, not a theorem about arbitrary local continuum QFTs. Thus the claim that every local continuum QFT obeys the two axioms is not established. This is load-bearing for the paper's main claim. Please either prove a continuum statement, or explicitly frame the continuum part as a conjecture and state what evidence or falsification would count.
  2. [Sec. 2.3] Free Maxwell theory is advertised in the introduction and abstract as the main positive example of a theory that violates ordinary additivity but obeys disjoint additivity. However, Sec. 2.3 only gives explicit verifications for the 1+1D Z2 gauge chain and the 2+1D toric code ground space. No argument is supplied for free Maxwell theory with the continuum definitions (2.14)-(2.16), either on R^4 or on a compact spatial manifold. Given that the additivity violation is topological, a direct check that closures-disjoint unions cannot create the relevant non-contractible cycles is needed for the flagship example.
  3. [Sec. 4.4] The argument that the Virasoro identity multiplet violates at least one of the two axioms relies on the assertion that the modular-invariance theorem of [10] only requires disjoint additivity, stated as 'straightforward to see' without proof. This is a nontrivial modification of a published theorem and is load-bearing for one of the four main violation examples. Please provide the argument, or restrict the claim to the explicitly analyzed RCFT cases where the violation of disjoint additivity/Haag duality is shown directly.
minor comments (6)
  1. [Sec. 2.3] Typo: 'operaotrs' should be 'operators'. Also, notation is inconsistent: Eq. (2.11) uses A[R1 ∪ R2] while Eqs. (2.14) and elsewhere use A(R1 ∪ R2).
  2. [Sec. 3.3] Typo: 'non-relavistic' should be 'non-relativistic'.
  3. [Sec. 4.2] The sentence 'A(R1 ∪ R2) is generated by field operators in R1 and R2 and thus by elements of A(R2) and A(R2)' presumably should end with 'A(R1) and A(R2)'.
  4. [References] References [15] and [31] are the same paper (Shao, Sorce, Srivastava, arXiv:2503.20863) and should be merged or cross-referenced.
  5. [Fig. 6] The caption says 'The regions R1 and R3 are spatially disjoint,' but the figure labels are not fully legible; please clarify which shaded regions are R1, R2, R3 and their complements.
  6. [Sec. 6.2] The symbol e+1/2 is used in Eq. (6.16) before it is defined. Please define the 'vertical edge' notation explicitly.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the lattice theorems are genuine conditional derivations; the only self-citation is minor and non-load-bearing.

full rationale

The paper's central claim is explicitly a proposal, not a derivation from assumed conclusions. The main positive evidence is the lattice theorem in Secs. 3.2–3.3 and App. B, which derives Haag duality and disjoint additivity from explicit hypotheses: a compact Lie group S acting faithfully with factorization (3.1), and the non-adjacency conditions (i)–(iii). None of these hypotheses contains disjoint additivity; the proof uses Haar-measure manipulations and von Neumann algebra arguments to establish the factorization identity (3.36), which then yields disjoint additivity. The non-adjacency predicate is admittedly defined so that the proof works ('These are the most general conditions under which we are able to prove disjoint additivity'), but this is a normal conditional theorem, not a definitional circularity. The paper is also explicit that the continuum extrapolation is not proved: 'it does not promise that continuum regions which are spatially disjoint... arise as the continuum limits of non-adjacent regions on the lattice.' That is an acknowledged gap, a correctness risk, not a circular step. The only self-citation by the current authors is [15] in Sec. 4.4 for the Virasoro identity multiplet of rational CFTs; this supports one illustrative violation example, is corroborated for c>=1 by the external result [10], and is not the load-bearing core of the paper's central claim. No fitted data are renamed as predictions. Overall, the derivation chain is self-contained at the lattice level, and the universal continuum claim rests on an unproven bridge rather than on circular reasoning.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

No numerical fits or hand-tuned constants enter the arguments; the paper is purely analytic. The paper introduces a new axiom and definitions but no new physical objects (no new particles, forces, dimensions, or conserved quantities). The SymTFT construction uses existing topological field theory concepts. The adjacency relation is a definitional structure, not a fitted number.

assumptions (8)
  • standard math Local algebras are von Neumann algebras satisfying microcausality A(R') ⊆ A(R)' (eq. (2.4)).
    Standard algebraic QFT background used throughout; stated in §2.1.
  • domain assumption The lattice symmetry group S is a faithful, strongly continuous unitary representation of a compact Lie group whose action factorizes between complementary regions: U(s) = U_R(s) ⊗ U_R'(s) (eq. (3.1)).
    This on-site factorization is assumed in §3.1 for every lattice theorem; it rules out symmetry actions that are not locally generated, including global and subsystem symmetries.
  • domain assumption For non-adjacent regions R1, R2, S is generated by Lie subgroups S1, S2, N with U(S1) ⊆ A(R2)', U(S2) ⊆ A(R1)', S1,S2 commuting, and S1,S2,N generating S (conditions (i)-(iii) in §3.3).
    This is the operative definition of non-adjacency; disjoint additivity is proven only when this generation condition holds, with the special product case in the main text and the general case in Appendix B.
  • ad hoc to paper The lattice adjacency rule is short-range: regions whose minimal separation in lattice units goes to infinity with system size are non-adjacent for all but finitely many system sizes (§3.3).
    Needed to connect finite-lattice theorems to the continuum claim; the paper notes this fails for global symmetries, which indeed violate disjoint additivity.
  • domain assumption Global symmetries are splittable: for each region R there is a localized symmetry operator U(g,R) acting like the global symmetry on R and as the identity on R' (§4.3).
    Assumed to formalize the invariant-sector violation; the paper cites [66-68] for continuum QFT and notes splittability can fail on other manifolds.
  • ad hoc to paper Haag duality and disjoint additivity are the correct axioms of locality: every local quantum system obeys both (Sec 1 proposal).
    This is the paper's central hypothesis, supported by examples and theorems but not proven.
  • domain assumption The [10] result that additivity plus Haag duality implies modular invariance in 2D CFT extends to disjoint additivity (§4.4, footnote 21).
    External theorem plus a claimed straightforward extension, used to conclude the Virasoro identity multiplet violates one of the two axioms.
  • domain assumption The timelike tube theorem: the algebra of operators in a full hypersurface equals all of B(H) (§4.1).
    Cited to [50,51] and used to show the hyperplane-restricted theory violates Haag duality.

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Pith. "Pith review of Disjoint additivity and local quantum physics." pith.science (2026). https://pith.science/paper/RMXKBWYR

@misc{pith2026250903589,
  author       = {Pith},
  title        = {Pith review of: Disjoint additivity and local quantum physics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RMXKBWYR}},
  note         = {Machine review of arXiv:2509.03589}
}
read the original 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

Figures reproduced from arXiv: 2509.03589 by the authors.

Figure 1
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′ . In continuum relativistic quantum field theory, however, there is ambiguity in the literature for how regions should be de… view at source ↗
Figure 2
Figure 2. A non-contractible region R = R1 ∪ R2 that encloses a line operator W (green). Here R1 (red) and R2 (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 ∈ A(R1 ∪ R2) but W /∈ A(R1) ∨ A(R2). This violates additivity. Here we have shown spatial regions; to get relativistic regions, we can take the do… view at source ↗
Figure 3
Figure 3. A line operator carrying a topological one-form symmetry charge must be unbreak [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (18 more)
Figure 4
Figure 4. Figure 4: In one spatial dimension, Z2 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. and is thus unbreakable. Additivity is violated because the circle S 1 can be writte…
Figure 5
Figure 5. Figure 5: In the ground space of the toric code, there are two kinds of nontrivial operators, [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: Illustrating spatial disjointness. The regions [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: The black circles denote two non-adjacent regions on a lattice system acted on by a [PITH_FULL_IMAGE:figures/full_fig_p020_7.png]
Figure 8
Figure 8. Figure 8: Violating Haag duality in the hypersurface theory: if a local operator [PITH_FULL_IMAGE:figures/full_fig_p023_8.png]
Figure 9
Figure 9. Figure 9: Disjoint additivity is violated in the invariant sector under a compact global sym [PITH_FULL_IMAGE:figures/full_fig_p025_9.png]
Figure 10
Figure 10. Figure 10: A symmetry insertion under which the bilocal operator (4.5) transforms nontrivially. [PITH_FULL_IMAGE:figures/full_fig_p025_10.png]
Figure 11
Figure 11. Figure 11: Restoration of disjoint additivity by coupling to a bulk gauge field. Left: A pair of [PITH_FULL_IMAGE:figures/full_fig_p032_11.png]
Figure 12
Figure 12. Figure 12: Left and middle: Local constraints of a 2+1D topological [PITH_FULL_IMAGE:figures/full_fig_p034_12.png]
Figure 13
Figure 13. Figure 13: Restoration of disjoint additivity in toric code. Left: Disjoint additivity is violated [PITH_FULL_IMAGE:figures/full_fig_p035_13.png]
Figure 14
Figure 14. Figure 14: Left: An open set S, its causal complement S ′ , and two points p, q with the set J +(p) ∩ J −(q) shaded in. Right: A choice of point q ′ in the chronological future of q, and a point p ′ in the chronological past of p, such that p ′ and q ′ are still in S ′ . Every p…
Figure 15
Figure 15. Figure 15: Left: A point p that is assumed to lie on the boundary D(T), together with an open set that contains points outside of D(T). Each of these points must contain an inextendible causal curve that does not intersect T. Right: By taking smaller and smaller open sets around…
Figure 16
Figure 16. Figure 16: Illustration of the proof of lemma 3. The point [PITH_FULL_IMAGE:figures/full_fig_p041_16.png]
Figure 17
Figure 17. Figure 17: An example of a region R (shaded blue) in the Schwarzschild geometry that obeys R = R′′ but is not the domain of dependence of an open subset of a Cauchy surface. An achronal set T of the type promised by theorem (1) is shaded orange, but it is not a subset of any Cau…
Figure 18
Figure 18. Figure 18: An example of a globally hyperbolic spacetime with two spatially disjoint regions [PITH_FULL_IMAGE:figures/full_fig_p044_18.png]
Figure 19
Figure 19. Figure 19: Using the integral curves of a a timelike vector field to define a homeomorphism [PITH_FULL_IMAGE:figures/full_fig_p045_19.png]
Figure 20
Figure 20. Figure 20: Constructing an achronal Cauchy surface containing [PITH_FULL_IMAGE:figures/full_fig_p045_20.png]
Figure 21
Figure 21. Figure 21: (a) In a metric space, for every open set Ω containing a compact set [PITH_FULL_IMAGE:figures/full_fig_p051_21.png]

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Forward citations

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