Galilean Reeh-Schlieder obstruction in the vacuum and the thermal Reeh-Schlieder property

arXiv:2604.26271 · quant-ph, gr-qc, math-ph, math.MP · Submitted 2026-04-29 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: Today's paper: "Galilean Reeh-Schlieder obstruction in the vacuum and the thermal Reeh-Schlieder property".

Mira: No Galilean Haag–Kastler net admits a vacuum that is cyclic and separating for every local field algebra.

Kai: First, who's behind it and why it matters.

Title and authors: Kai: So, we're diving into the paper "Galilean Reeh-Schlieder obstruction in the vacuum and the thermal Reeh–Schlieder property." It looks like this piece is really digging into why standard Galilean quantum field theory runs into trouble when you try to make it behave like relativistic quantum field theory.

Mira: Exactly. The title tells us right away that they are looking at the Reeh–Schlieder property in the context of Galilean theories, specifically focusing on how it relates to thermal properties. It’s suggesting a fundamental incompatibility between the two frameworks when you stick to certain basic rules.

Lev: From an error correction standpoint, this is interesting because if we can't even have a vacuum that is cyclic and separating for every local algebra, it makes building stable systems in these non-relativistic models much harder to justify.

Kai: Right, Mira? So what’s the actual core idea here? I want to make sure I get the simplest version of this complex argument down for the listeners.

Mira: The summary explains that the paper proves that standard Galilean Haag–Kastler axioms, when you add Bargmann mass superselection, are simply inconsistent with having a vacuum that is both cyclic and separating for every local field algebra. They show this happens because Galilean Schrödinger fields annihilate the Fock vacuum, which then creates a problem when you try to maintain locality.

Lev: That annihilation part sounds like it connects directly to how we define the state in practice, doesn't it? If the bare fields just wipe out the vacuum, that’s a big hurdle for any physical construction.

Kai: It is, Lev. And then they go further by showing that because of Bargmann mass superselection, there's no way to find a clever Hermitian combination of modes to sneak around this issue while keeping the relativistic axioms consistent.

Mira: That’s the structural ingredient they pinpoint: the difference in how Hermitian combinations work between relativistic and Galilean settings, which is why this paper says it’s where the Reeh–Schlieder obstruction gets triggered in Galilean AQFT but remains vacuous in relativistic AQFT.

Lev: It makes sense that if you can't find that evasion mechanism, the system just breaks down into a contradiction with the equal-time canonical commutation relations, which is what they point to as a violation of "

ψˆ0(g1),ψˆ†zero(g2): = Z R3 g1(x)g2(x)d3x ·⊮" in one specific context <ref:2604.26271#pg1>.

Kai: It’s like the mathematical machinery just falls apart because the fields vanish where they shouldn't, which is a very concrete failure mode for the theory. So, what are these suggested improvements they mention?

Title and authors: Mira: The paper discusses how they extended their results beyond just the Fock representation hypothesis and look at two specific ways to strengthen their argument. Proposition one uses a "natural spectral weakening" where they require canonical fields to have definite Bargmann mass charges, the mass spectrum needs to be bounded below, and the vacuum must sit at the spectral minimum <ref:2604.26271#pg1>.

Lev: Bounded below mass spectrum seems crucial for any physical model because you need a stable ground state energy level to start with; if you don't have that bound, you can just keep going down into negative infinity.

Kai: And Proposition two is more interesting because it removes that specific vacuum-at-spectral-minimum clause, replacing it with an algebraic induction argument involving the Bose–CCR algebraic descent to show that even a high-power vanishing of operators propagates all the way down to zero, contradicting the c-number CCR <ref:2604.26271#pg1>.

Mira: So these improvements demonstrate that the structural divider isn't just about whether you use a Fock space or not, but rather about how the Bargmann mass charge structure of those canonical fields interacts with their time-zero regularity.

Lev: That suggests that if we can control those mass charges and ensure the vacuum stays at the bottom of that spectrum, we might be able to keep things consistent in some non-relativistic settings.

Kai: It’s a lot to take in, but it really paints a clearer picture of what's required for consistency in these theories. So, where does this leave us regarding the broader implications?

Mira: The direct consequence they draw is that the Tomita–Takesaki modular flow on the Fock space relative to becomes undefined for any Galilean Haag–Kastler net that satisfies those standard axioms (G1) through (G7). This has big implications because approaches in quantum gravity that rely on modular structure, like the Connes–Rovelli thermal time hypothesis, are fundamentally relativistic.

Lev: That means if you're looking at non-relativistic gravity or thermodynamics, you can't use those tools unless the theory is actually relativistic from the start. That’s a strong constraint for any model we try to build.

Kai: It really tells us that the thermal physics we expect in these systems is inherently tied to spacetime structure, which reinforces why they look so different from standard relativity. So, what about verifying this against actual models?

Mira: They did check this by examining five published interacting Galilean models: Lévy-Leblond, Schrader, Hepp, Eckmann, and Lampart–Schmidt–Teufel–Tumulka. In every case where the construction followed the axioms strictly—specifically for the Lévy-Leblond, Schrader, and Hepp models—the failure of Reeh–Schlieder was confirmed because those bare fields just annihilate the Fock vacuum.

Lev: And even in cases that didn't follow strict Galilean covariance, like Eckmann’s construction, they still showed the same underlying mechanism is at play: the dynamical ground state ends up being exactly what you’d expect from a bare Fock vacuum and gets annihilated by those canonical fields.

Title and authors: Kai: So it confirms that the failure isn't just a fluke in one specific model; it seems to be baked into the structure when you combine Galilean kinematics with these basic algebraic rules. It’s quite sobering for experimentalists because it suggests that if we try to build something purely Galilean, we might run into this fundamental vacuum annihilation issue.

Mira: That's exactly what they are pointing toward: the structural divider is the precise combination of Bargmann mass-charge structure and time-zero regularity, not just the Fock representation hypothesis itself. It’s a very specific algebraic constraint that separates the two regimes.

Lev: For error correction, this means any non-relativistic error correction scheme needs to fundamentally account for how these mass charges are handled because they aren't just simple internal degrees of freedom; they are tied to the spacetime structure itself in this framework.

Kai: So, looking ahead, what does this mean for future research? Are there other directions they’re pointing toward with these improvements?

Mira: The paper is clearly pushing researchers to focus on those weakened sets of axioms mentioned in Proposition two and its related results <ref:2604.26271#pg1>. It suggests that if we can't satisfy the full set, we need to look at how these weaker conditions still restrict the vacuum properties enough to maintain some consistency.

Lev: From my side, I see this as a roadmap: first try to enforce those mass charge restrictions on any candidate model you design, and second, test it against the algebraic descent argument they used—that’s a rigorous way to check if your proposed system will eventually contradict the c-number CCR.

Kai: It sounds like we should be looking for models that inherently possess that spectral minimum property, because those are the things keeping things mathematically sound in this discussion about Galilean Reeh–Schlieder obstruction in the vacuum and the thermal Reeh–Schlieder property.

Mira: That’s right; it directs our focus toward models where these necessary structural conditions are met, which is a huge step forward in understanding the boundaries of non-relativistic AQFT.

Lev: So, to wrap up this discussion on "Galilean Reeh–Schlieder obstruction in the vacuum and the thermal Reeh–Schlieder property," we see that relativistic theories possess this property naturally due to microcausality and spectral conditions, while Galilean theories cannot have it under these specific constraints.

Kai: It’s a clear delineation of where the algebraic structure of quantum field theory fundamentally diverges based on its kinematic symmetry.

Mira: Indeed, and the unavailability of the Tomita–Takesaki modular flow is a very concrete consequence for any Galilean net satisfying these standard rules.

Lev: And for error correction, this paper shows us exactly what structural obstacles we face when trying to force non-relativistic systems into a framework that requires relativistic vacuum properties.

The paper's summary: Kai: So, we're looking at how this paper explains that standard Galilean theories just can't have that vacuum property we see in relativistic ones, which is really about that Reeh–Schlieder obstruction and its link to modular flow.

Mira: Exactly, Kai; the core finding is that for a Galilean Haag–Kastler net satisfying those basic axioms, you simply can't have a vacuum that's both cyclic and separating for every local algebra. They show this isn't just a technicality but stems from the way Hermitian combinations of fields are treated in non-relativistic settings compared to relativistic ones.

Lev: From an error correction standpoint, that structural barrier is significant because it means the underlying state space itself doesn't support the necessary mathematical machinery for certain types of QEC protocols if you stay strictly in the Galilean regime.

Kai: That makes sense, Lev; so when Mira talks about this "structural ingredient," she’s emphasizing that it’s not just about whether you use a Fock space or not, but how the mass charges interact with the time-zero regularity of those fields.

Mira: Precisely; they pinpoint that specific interaction between Bargmann mass superselection and time-zero regularity as the exact place where relativistic AQFT succeeds while Galilean AQFT fails this test. This failure directly leads to the consequence that you can't define the Tomita–Takesaki modular flow on the Fock space for these standard Galilean nets.

Lev: That unavailability of modular structure is a big deal because it means approaches to quantum gravity that rely on thermalization or holographic principles, like Connes–Rovelli’s ideas, are fundamentally relativistic in nature if you stick to those standard axioms.

Kai: It really hammers home the idea that observer-dependent thermodynamic phenomena just aren't going to be there in a purely Galilean vacuum setup under these conditions.

Mira: And the verification part is pretty telling; they checked five different models, and even when they used slightly weaker versions of the axioms, like Proposition two it still showed that this structural divider remains tied to those mass-charge constraints rather than just the initial Fock structure hypothesis.

Lev: I think for practical hardware realization, this means any attempt to build a truly Galilean system will have to be carefully engineered so that those mass charges are managed in a way that respects the spectral minimum condition they highlighted, otherwise you're headed toward a contradiction with the c-number CCR.

Kai: So, this paper isn't just an abstract math exercise; it’s setting hard physical boundaries for what we can expect from non-relativistic quantum field theory when we try to impose standard algebraic rules.

Mira: It is; the implication is that if you want a non-relativistic QFT that has deep connections to relativistic physics, you have to find a way to satisfy those structural requirements—like the spectral minimum—which are inherently more demanding than just satisfying microcausality alone.

Lev: That points us toward focusing on those specific mass-charge constraints when designing any future error correction codes or physical systems aiming for non-relativistic description.

Kai: It’s fascinating how this paper uses such a fundamental property as the Reeh–Schlieder condition to draw a line between two very different kinematic symmetries, and that's something we need to keep in mind as we look at experimental setups.

Mira: Absolutely; understanding where these theories diverge is crucial for deciding which physical systems are actually consistent with the algebraic framework they aim to describe.

The paper's improvements: Tom: So, we're looking at how the authors suggest ways to strengthen their argument about overcoming that vacuum obstruction in Galilean AQFT, which they present through two specific propositions and an algebraic induction proof.

Kai: That sounds like a roadmap for someone trying to actually build something non-relativistic; they’re essentially telling us exactly what spectral conditions we need to enforce if we want the theory to stay consistent.

Mira: Exactly, Kai; Proposition one involves requiring canonical fields to have definite Bargmann mass charges and ensuring the mass spectrum is bounded below while keeping the vacuum at that spectral minimum, which is a very concrete way to add physical constraints.

Lev: Bounded below mass spectrum makes sense because if you don't have that lower bound, you’re dealing with states that can just run off to negative energy indefinitely, which would be physically unstable for any hardware implementation.

Kai: So it’s about making sure the underlying physics has a stable ground state level before we even worry about the dynamics of excitations.

Mira: And Proposition two is interesting because it removes that strict vacuum-at-minimum clause and replaces the spectral argument with a Bose–CCR algebraic induction, showing that operator vanishing propagates down to zero, which contradicts the c-number canonical commutation relations if you don't have those mass charge restrictions.

Lev: That algebraic descent argument sounds like a very rigorous consistency check; it’s hard to argue against when you show that high-power vanishing of operators actually forces a contradiction with the basic commutation rules.

Kai: It means the structural divider isn't just about Fock structure anymore; it’s really about how those mass charges interact with the time-zero regularity, and these improvements are telling us exactly what algebraic constraints to look for in any candidate model.

Mira: The real implication is that instead of just saying Galilean theories fail, they are giving us a precise set of conditions—the conjunction of Bargmann mass-charge structure and time-zero regularity—that must be met for a non-relativistic theory to avoid this specific obstruction.

Lev: For quantum error correction, this means we should treat the mass charges not just as labels but as integral parts of the system's fundamental algebraic structure that needs careful management during encoding or decoding.

Kai: So, if we want to explore experimental realizations in this space, the focus shifts toward designing models that inherently possess those spectral minimum properties so they don't immediately run into this kind of contradiction.

Mira: Precisely; it directs future research away from just checking the standard G1 through G7 axioms and toward enforcing these more nuanced structural requirements to see if we can even have a consistent Galilean framework at all.

Conclusion: Kai: So, to wrap up, we've seen that this paper on "Galilean Reeh–Schlieder obstruction in the vacuum and the thermal Reeh–Schlieder property" shows that Galilean AQFT fundamentally cannot have a vacuum with those specific cyclic and separating properties under standard axioms.

Mira: That’s right; the main implication is that this result sets a hard line between relativistic theories, which naturally possess those vacuum properties due to microcausality, and Galilean theories, which don't.

Lev: It means for error correction research, we can stop assuming we can just use the same tools across both kinematic regimes; if you want to build something non-relativistic, you have to respect this algebraic distinction.

Kai: It really forces us to be much more specific about the underlying symmetry when designing any quantum system that aims for a Galilean description.

Mira: And they showed that even in their weakened versions, the structural divider remains tied to those mass charges and time-zero regularity, so we need to focus on enforcing those constraints if we want consistency.

Lev: That makes sense; it gives us a clear target for what needs to be controlled when designing any physical implementation, whether it's for quantum computation or other systems.

Kai: It’s fascinating how this paper uses such a fundamental vacuum property to delineate the boundaries between kinematic symmetries in quantum field theory.

Mira: Indeed, and the unavailability of the Tomita–Takesaki modular flow for standard Galilean nets is a very concrete consequence that has major implications for any QFT approach involving thermal physics.

Lev: For my work, it suggests that any non-relativistic attempt at thermalization won't be as straightforward as in relativistic models unless you manage these constraints perfectly.

Kai: We’re going to keep an eye on how this paper influences the next generation of experimental setups and see if we can find systems that naturally satisfy those necessary structural conditions.

quant-ph, gr-qc, math-ph, math.MP

Submitted: 2026-04-29

Updated: 2026-10-02

Comments: 17 pages, 2 figures, 1 table. v2: rewritten, new title. The locality axiom (G2) of v1 required commutation at unequal times, which even the free field violates; v2 uses nets generated by time-zero fields. The non-separating vacuum is kept. Withdrawn: the claims of v1 that Galilean theories cannot have the Reeh-Schlieder property or modular flow, and its Corollary 3 on five published models

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 79/100

The gist: No Galilean Haag–Kastler net admits a vacuum that is cyclic and separating for every local field algebra.

Key concepts

Reeh–Schlieder property: This property requires that the vacuum state can be used to create any local state in the theory. The paper shows this cannot hold simultaneously with cyclic conditions in Galilean theories, proving its absence there.
Galilean Haag–Kastler axioms: These are a set of rules (G1-G7) that define how local field algebras behave in Galilean quantum field theory. The authors demonstrate that these standard rules are mathematically inconsistent with the Reeh–Schlieder property.
Relativistic mechanism vs. Galilean failure: The difference lies in how Hermitian combinations of fields are treated. In relativity, certain combinations can live inside the local algebra, which allows the Reeh–Schlieder property to hold. In Galilean theories, structural constraints prevent this evasion mechanism.
Modular structure: This is a mathematical framework used to describe entanglement and thermal properties in quantum field theory. The obstruction proves that for Galilean theories satisfying the axioms, this modular flow on the Fock space is undefined.

Terminology

Axioms — 40× in this paper · explained in 2 episodes
Modular flow — 15× in this paper · explained in 2 episodes
Verification — 9× in this paper · explained in 2 episodes
Hilbert Space — 6× in this paper · explained in 2 episodes
Consistency — 5× in this paper · explained in 3 episodes
No-Go Theorem — 4× in this paper · explained in 2 episodes
Core — 3× in this paper · explained in 2 episodes
Additivity — 1× in this paper · explained in 1 episode
Generalization — 1× in this paper · explained in 13 episodes
Physical Constraints — 1× in this paper · explained in 7 episodes
Quantum Physics — 1× in this paper · explained in 10 episodes
Trace — 1× in this paper · explained in 2 episodes

Summary

No Galilean Haag–Kastler net admits a vacuum that is cyclic and separating for every local field algebra. This result establishes that the Reeh–Schlieder property is the structural ingredient distinguishing relativistic from Galilean algebraic quantum field theory, proving its unavailability in the latter.

The Core Obstruction

The paper proves that standard Galilean Haag–Kastler axioms (G1)–(G7) are inconsistent with the Reeh–Schlieder property (Definition 2). This inconsistency is established through a two-step process: first, showing that Galilean Schrödinger fields annihilate the Fock vacuum, which implies that ψˆ(f)omega = 0 for every f ∈ C∞c(MG) (Step 1); and second, showing that the locality of the canonical fields in (G4) forces a contradiction when combined with the separating property of the vacuum (Step 2). The final contradiction arises from violating the equal-time canonical commutation relations (G4), as [ψˆ0(g1),ψˆ†0(g2)] = Z R3 g1(x)g2(x)d3x ·⊮ is violated by the vanishing of the fields.

The Relativistic Mechanism vs. Galilean Failure

The structural difference between relativistic and Galilean local field algebras lies in how Hermitian combinations are treated. In relativistic AQFT, the evasion mechanism allows for a Hermitian combination of annihilation and creation modes, such as φˆ(f) = aˆ(f) + aˆ†(¯f), to lie within the local algebra R(G), while the individual modes aˆ(f) and aˆ†(f) are not. This is because the complex structure J on the one-particle space fails to preserve S(G), meaning that for Galilean fields, axiom (G4) places ψˆ(f) and ψˆ†(f) in F(O) as individual sector-graded elements, and Bargmann mass superselection forbids the Hermitian-sum substitute. This structural difference is the precise locus where the Reeh–Schlieder obstruction is rendered vacuous in the relativistic setting but triggered in Galilean AQFT.

The Role of Axiomatic Weakening

The obstruction theorem is robust and extends beyond the Fock representation hypothesis (G7). The authors present two extensions: Proposition 1, which uses a natural spectral weakening (G7∗) requiring that canonical fields carry definite Bargmann mass charges, the mass spectrum be bounded below, and the vacuum sit at the spectral minimum. Proposition 2 removes this vacuum-at-spectral-minimum clause (G7∗)(c) by replacing the spectral argument with a Bose–CCR algebraic induction, showing that a high-power vanishing of operators propagates down to zero, contradicting the c-number CCR. This demonstrates that the structural divider is not Fock structure, but rather the conjunction of Bargmann mass-charge structure of canonical fields and time-zero regularity.

Consequences for Modular Structure

A direct consequence of the obstruction is that the Tomita–Takesaki modular flow on F(O) relative to omega is undefined for any Galilean Haag–Kastler net satisfying (G1)–(G7). This implies that approaches to quantum gravity relying on modular structure, such as the Connes–Rovelli thermal time hypothesis, are intrinsically relativistic. The paper also establishes Corollary 2, stating that under a slightly weakened set of axioms ((G7∗)(a) and (G7∗)(d)), the vacuum omega is not separating for F(O), meaning the modular flow is undefined.

Verification against Published Models

The theorem is verified by examining five published interacting Galilean models (Lévy-Leblond, Schrader, Hepp, Eckmann, and Lampart–Schmidt–Teufel–Tumulka). In all cases where the construction satisfies (G1)–(G7) literally—specifically the Lévy-Leblond, Schrader, and Hepp models—the failure of Reeh–Schlieder is confirmed because bare fields annihilate the Fock vacuum, leading to a contradiction with the equal-time CCR. Even constructions that violate strict Galilean covariance (like Eckmann’s) illustrate the same underlying mechanism: the dynamical ground state coincides with the bare Fock vacuum and is annihilated by the construction’s canonical fields.

The Structural Conclusion

The paper concludes that "relativistic AQFT has it [Reeh–Schlieder] as a theorem, derived from microcausality and the spectrum condition; Galilean AQFT, in every setting in which the canonical fields carry definite Bargmann mass charges and the vacuum sits at the bottom of the Bargmann mass spectrum, cannot have it. The result identifies the Reeh–Schlieder property as the precise structural divider between relativistic and Galilean algebraic quantum field theory.

Improvements for AI systems

As a fastidious researcher, I have analyzed the provided paper, Galilean Reeh–Schlieder Obstruction, which establishes a fundamental structural distinction between relativistic and Galilean Algebraic Quantum Field Theory (AQFT).

The primary contribution of this work is the identification of the Reeh–Schlieder property as the precise structural ingredient distinguishing relativistic from Galilean algebraic quantum field theory.

Here are specific, actionable improvements for AI systems based on this scientific knowledge:

)

Improve AI systems by incorporating a formal framework that distinguishes between kinematic symmetries (Galilean vs. Poincaré/Lorentz) and their corresponding algebraic structures (Haag–Kastler nets).
Improve the ability of AI to evaluate theoretical models in non-relativistic regimes by explicitly testing for the presence or absence of the Reeh–Schlieder property, rather than just checking standard axioms like microcausality.
Improve AI systems to perform structural proofs or consistency checks on non-relativistic QFTs by systematically searching for Hermitian combination evasion (as described in Section III C) that might allow a net to evade the Reeh–Schlieder obstruction, thereby pinpointing exactly why Galilean theories fail where relativistic ones succeed.
Improve the ability of AI to understand and predict consequences in quantum gravity models by recognizing that modular flow—a key ingredient for thermalization and holographic principles—is fundamentally unavailable in any Galilean Haag–Kastler net satisfying the standard axioms (G1)–(G7). This allows for a more nuanced assessment of which QFT approaches are intrinsically relativistic.
Improve AI systems to handle complex mathematical structures involving Bargmann mass superselection by correctly identifying how this mechanism forces the necessity of separate, non-Hermitian canonical fields in Galilean settings, which is the structural locus where the Reeh–Schlieder obstruction triggers.

The improved AI system can:

Perform rigorous classification of quantum field theories based on their kinematic structure (Galilean vs. Relativistic) by checking for the presence of the Reeh–Schlieder property rather than just standard axioms.
Diagnose the specific algebraic mechanisms (like Bargmann mass superselection) that cause a system to evade or trigger fundamental structural obstructions in quantum field theory proofs.
Validate whether a proposed non-relativistic model is fundamentally incompatible with relativistic AQFT by testing if its local algebra structure aligns with the known requirements for modular flow and vacuum properties in Galilean settings.
Automatically predict the absence of observer-dependent thermodynamic phenomena (like KMS thermalization) in any system described by a Galilean Haag–Kastler net, as these require the unavailable Tomita–Takesaki modular flow.
Differentiate between various non-relativistic constructions (e.g., those satisfying the full axiom set G1–G7 versus those satisfying a weakened spectral hypothesis G7∗) to determine their actual physical relevance and structural consistency within the Galilean framework.

Abstract

We ask in which states, and for which local algebras, Galilean quantum fields have the Reeh-Schlieder property. The nets are generated by time-zero canonical fields on spatial regions and by the dynamics. The Bargmann mass over the charge of the field is a number operator for the time-zero fields, bounded from below by positivity of the energy and boost covariance, and Chaiken's theorem gives Galilean Fock rigidity: these fields form a direct sum of Fock representations, and the time-zero annihilation fields annihilate every translation-invariant vector. If the vacuum is the only vector invariant under spatial translations (not implied by the usual uniqueness axiom), the vacuum sector of the observable net is one-dimensional. No vector of finite mass is cyclic or separating for an algebra of a spatial region at fixed time. In the vacuum of Bose gases with bounded or Coulomb-type pair potentials (with Buchholz's renormalised dynamics if not H-stable), the fields of any region over any time interval generate all bounded operators. By contrast, a KMS state is cyclic and separating for the algebra of every region B times I with temporal extent whenever the spatial algebras at all times generate the global algebra, as for free gases. For the free Schrödinger field in the vacuum, the thermal states of Bose and Fermi gases and the Fermi sea, fixed-time field algebras are type I factors, for which thermal vectors and the Fermi sea are separating but not cyclic, and every field algebra with temporal extent is the global one: type I in pure states, type III 1 in thermal states, with the grand-canonical dynamics as modular group. Galilean vacua are thus never separating for local algebras. For the free field the type of the algebras of regions with temporal extent depends on the state, and what is specifically relativistic is the locality of type III 1.