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arxiv: 2604.26271 · v1 · submitted 2026-04-29 · 🪐 quant-ph · gr-qc· math-ph· math.MP

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Galilean Reeh--Schlieder Obstruction

Leonardo A. Pachon
Authors on Pith no claims yet

Pith reviewed 2026-05-07 13:47 UTC · model grok-4.3

classification 🪐 quant-ph gr-qcmath-phmath.MP
keywords Galilean quantum field theoryReeh-Schlieder propertyHaag-Kastler axiomsBargmann mass superselectionalgebraic QFTnon-relativistic QFTmodular theory
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The pith

Galilean quantum field theory axioms are incompatible with the Reeh-Schlieder property.

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

The paper establishes that no net of local field algebras satisfying the standard Galilean Haag-Kastler axioms together with Bargmann mass superselection can possess a vacuum vector that is cyclic and separating for every local algebra. This contrasts with relativistic quantum field theory, where the Reeh-Schlieder property holds as a theorem. The obstruction arises because Galilean Schrödinger fields annihilate the Fock vacuum, and mass superselection rules out the Hermitian combinations that preserve consistency in the relativistic case. Consequently, the modular flow from Tomita-Takesaki theory cannot be defined on these algebras.

Core claim

The standard Galilean Haag-Kastler axioms, augmented by Bargmann mass superselection, are inconsistent with the Reeh-Schlieder property: no such net admits a vacuum that is cyclic and separating for every local field algebra.

What carries the argument

Galilean Schrödinger fields that annihilate the Fock vacuum, together with Bargmann mass superselection forbidding Hermitian evasion.

Load-bearing premise

Canonical fields carry definite Bargmann mass charges and admit time-zero restrictions on a field-algebra-stable common dense domain.

What would settle it

Construction of one Galilean Haag-Kastler net obeying Bargmann superselection that admits a cyclic and separating vacuum for a local algebra would refute the claimed inconsistency.

Figures

Figures reproduced from arXiv: 2604.26271 by Leonardo A. Pachon.

Figure 1
Figure 1. Figure 1: FIG. 1. The structural difference between the relativistic and Galilean local field algebras. Relativistically, view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Mass spectrum after Step A of Proposition view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Logical structure of the strengthened obstruction. Grey nodes are axiomatic hypotheses; green nodes are derived intermediate results; view at source ↗
read the original abstract

We prove that the standard Galilean Haag--Kastler axioms, augmented by Bargmann mass superselection, are inconsistent with the Reeh--Schlieder property: no such net admits a vacuum that is cyclic and separating for every local field algebra. Two ingredients combine: Galilean Schr\"odinger fields annihilate the Fock vacuum, and Bargmann mass superselection forbids the Hermitian-combination evasion that keeps relativistic axioms consistent. The result extends beyond the Fock-representation hypothesis: any Galilean Haag--Kastler net whose canonical fields carry definite Bargmann mass charges and admit time-zero restrictions on a field-algebra-stable common dense domain is incompatible with Reeh--Schlieder. The bounded-below mass spectrum and the vacuum-at-spectral-minimum, usually imposed as separate axioms, are derived consequences -- of positive-energy boost positivity and a Bose-CCR algebraic descent, respectively. The Tomita--Takesaki modular flow is consequently unavailable on Galilean local field algebras. We identify the Reeh--Schlieder property as the precise structural ingredient distinguishing relativistic from Galilean algebraic quantum field theory: relativistic AQFT has it as a theorem, Galilean AQFT cannot.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper proves that the standard Galilean Haag-Kastler axioms, augmented by Bargmann mass superselection, are inconsistent with the Reeh-Schlieder property: no such net admits a vacuum that is cyclic and separating for every local field algebra. The argument combines the fact that Galilean Schrödinger fields annihilate the Fock vacuum with the prohibition on Hermitian combinations of creation and annihilation operators due to mass superselection. The result extends beyond Fock representations to any Galilean Haag-Kastler net whose canonical fields carry definite Bargmann mass charges and admit time-zero restrictions on a field-algebra-stable common dense domain. As consequences, the bounded-below mass spectrum follows from positive-energy boost positivity and the vacuum-at-spectral-minimum follows from a Bose-CCR algebraic descent. The Tomita-Takesaki modular flow is consequently unavailable on Galilean local field algebras.

Significance. If the central claim holds, the result identifies the Reeh-Schlieder property as the precise structural feature distinguishing relativistic from Galilean AQFT, explaining the unavailability of modular theory in the latter. Strengths include the derivation of spectrum conditions (usually imposed as separate axioms) as consequences of boost positivity and algebraic descent, the reliance on standard Galilean CCR and Bargmann superselection without free parameters or ad-hoc entities, and the explicit extension beyond the Fock case under a stated domain condition.

major comments (1)
  1. [general extension beyond Fock] The general extension beyond Fock (the statement that any Galilean Haag-Kastler net with canonical fields carrying definite Bargmann mass charges and admitting time-zero restrictions on a field-algebra-stable common dense domain is incompatible with Reeh-Schlieder): this domain-stability condition is load-bearing for the claim outside the Fock representation. A short argument showing it follows from the Haag-Kastler axioms (rather than being an extra restriction) would strengthen the result, particularly regarding invariance of the dense domain under the local algebras.
minor comments (2)
  1. [Abstract] The abstract is concise but could cross-reference the theorem number or section containing the main general statement.
  2. [Derivations of spectrum conditions] Notation for the net, fields, and domain should be checked for consistency across the derivations of the spectrum conditions.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading, positive evaluation, and constructive suggestion regarding the general extension. We address the major comment below.

read point-by-point responses

  1. Referee: The general extension beyond Fock (the statement that any Galilean Haag-Kastler net with canonical fields carrying definite Bargmann mass charges and admitting time-zero restrictions on a field-algebra-stable common dense domain is incompatible with Reeh-Schlieder): this domain-stability condition is load-bearing for the claim outside the Fock representation. A short argument showing it follows from the Haag-Kastler axioms (rather than being an extra restriction) would strengthen the result, particularly regarding invariance of the dense domain under the local algebras.

    Authors: We agree that the domain-stability condition merits explicit justification to clarify its status in the general case. In the Galilean Haag-Kastler framework, the canonical fields are densely defined operators on a common domain D that is required to be invariant under the local algebras by the very definition of the net (the fields generate the algebras, and the time-zero restrictions are part of the algebraic structure). Galilean covariance further preserves this domain. We will insert a short clarifying paragraph in Section 3 (or the relevant extension paragraph) deriving the stability from these standard axioms rather than treating it as an independent hypothesis. This strengthens the result without altering the core argument. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper establishes an inconsistency between Galilean Haag-Kastler nets (with Bargmann superselection) and the Reeh-Schlieder property by invoking standard external facts: Schrödinger fields annihilate the Fock vacuum and mass superselection precludes Hermitian combinations. It then derives the bounded-below mass spectrum and vacuum-at-minimum as consequences of boost positivity and Bose-CCR descent, without any reduction of the target claim to a fitted parameter, self-definition, or load-bearing self-citation. The domain condition is presented as a natural strengthening rather than an ad-hoc restriction that encodes the result. No ansatz is smuggled via citation and no known result is merely renamed. The argument therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claim rests on the standard Galilean Haag-Kastler axioms plus the additional Bargmann mass superselection rule; the paper treats positive-energy boost positivity and Bose-CCR algebraic descent as inputs that yield the bounded spectrum and vacuum-at-minimum as outputs.

axioms (3)
  • domain assumption Galilean Haag-Kastler axioms hold for the net of local algebras
    Invoked as the starting point for the inconsistency proof in the abstract.
  • domain assumption Bargmann mass superselection rule applies to the fields
    Explicitly augmented to the axioms and used to block Hermitian-combination evasion.
  • domain assumption Canonical fields carry definite Bargmann mass charges
    Required for the extension beyond Fock space in the abstract.

pith-pipeline@v0.9.0 · 5509 in / 1475 out tokens · 54496 ms · 2026-05-07T13:47:03.100609+00:00 · methodology

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Reference graph

Works this paper leans on

41 extracted references · 26 canonical work pages

  1. [1]

    author author J.-M. \ L\'evy-Leblond ,\ title title Galilean quantum field theories and a ghostless L ee model , \ 10.1007/BF01645427 journal journal Communications in Mathematical Physics \ volume 4 ,\ pages 157--176 ( year 1967 ) NoStop

    work page doi:10.1007/bf01645427 1967

  2. [2]

    author author L. A. \ Pach\' o n ,\ @noop title Algebraic quantum kinematics and SR -selection , \ ( year 2026 a ),\ note arXiv preprint; first of a five-paper series. Stop

    2026

  3. [3]

    author author L. A. \ Pach\' o n ,\ @noop title N ewton-- C artan limit of K lein-- G ordon AQFT and the collapse of G alilean modular structure , \ ( year 2026 b ),\ note arXiv preprint; third of a five-paper series. Stop

    2026

  4. [4]

    author author L. A. \ Pach\' o n ,\ @noop title Algebraic forcing of the semiclassical E instein equations , \ ( year 2026 c ),\ note arXiv preprint; third of a five-paper series. Stop

    2026

  5. [5]

    author author L. A. \ Pach\' o n ,\ @noop title An equivalence theorem for the algebraic forcing of the semiclassical E instein equations , \ ( year 2026 d ),\ note arXiv preprint; third of a five-paper series. Stop

    2026

  6. [6]

    author author L. A. \ Pach\' o n ,\ @noop title Gravity-dressed crossed products and the G alilean obstruction , \ ( year 2026 e ),\ note arXiv preprint; third of a five-paper series. Stop

    2026

  7. [7]

    author author J. Bain ,\ title title Quantum field theories in classical spacetimes and particles , \ 10.1016/j.shpsb.2011.01.001 journal journal Studies in History and Philosophy of Modern Physics \ volume 42 ,\ pages 98--106 ( year 2011 ) NoStop

    work page doi:10.1016/j.shpsb.2011.01.001 2011

  8. [8]

    author author M. Requardt ,\ title title Spectrum condition, analyticity, Reeh--Schlieder and cluster properties in non-relativistic G alilei-invariant quantum theory , \ 10.1088/0305-4470/15/12/021 journal journal Journal of Physics A: Mathematical and General \ volume 15 ,\ pages 3715--3723 ( year 1982 ) NoStop

    work page doi:10.1088/0305-4470/15/12/021 1982

  9. [9]

    Klaczynski ,\ title Haag's Theorem in Renormalised Quantum Field Theories ,\ 10.18452/17448 Ph.D

    author author L. Klaczynski ,\ title Haag's Theorem in Renormalised Quantum Field Theories ,\ 10.18452/17448 Ph.D. thesis ,\ school Humboldt-Universit\"at zu Berlin ( year 2016 ),\ note thesis DOI https://doi.org/10.18452/17448; preprint version arXiv:1602.00662 [hep-th] ,\ http://arxiv.org/abs/1602.00662 arXiv:1602.00662 NoStop

    work page doi:10.18452/17448 2016

  10. [10]

    author author G. D. \ Puccini \ and\ author H. Vucetich ,\ title title Axiomatic foundations of G alilean quantum field theories , \ 10.1023/B:FOOP.0000019584.39535.28 journal journal Foundations of Physics \ volume 34 ,\ pages 263--295 ( year 2004 ) ,\ note arXiv:quant-ph/0403181 ,\ http://arxiv.org/abs/quant-ph/0403181 arXiv:quant-ph/0403181 NoStop

    work page doi:10.1023/b:foop.0000019584.39535.28 2004

  11. [11]

    Falcone \ and\ author C

    author author R. Falcone \ and\ author C. Conti ,\ title title Localization in quantum field theory , \ 10.1016/j.revip.2024.100095 journal journal Reviews in Physics \ volume 12 ,\ pages 100095 ( year 2024 ) ,\ note arXiv:2312.15348 [hep-th] ,\ http://arxiv.org/abs/2312.15348 arXiv:2312.15348 [hep-th] NoStop

    work page doi:10.1016/j.revip.2024.100095 2024

  12. [12]

    author author R. Schrader ,\ title title On the existence of a local H amiltonian in the G alilean invariant L ee model , \ 10.1007/BF01654238 journal journal Communications in Mathematical Physics \ volume 10 ,\ pages 155--178 ( year 1968 ) NoStop

    work page doi:10.1007/bf01654238 1968

  13. [13]

    Théorie de la renormalisation

    author author K. Hepp ,\ 10.1007/BFb0108958 title Th\'eorie de la renormalisation ,\ series Lecture Notes in Physics , Vol. volume 2 \ ( publisher Springer ,\ year 1969 ) NoStop

    work page doi:10.1007/bfb0108958 1969

  14. [14]

    author author J.-P. \ Eckmann ,\ title title A model with persistent vacuum , \ 10.1007/BF01649436 journal journal Communications in Mathematical Physics \ volume 18 ,\ pages 247--264 ( year 1970 ) NoStop

    work page doi:10.1007/bf01649436 1970

  15. [15]

    Orbital Stability of Solitary Waves for Generalized Derivative Nonlinear Schrödinger Equations in the Endpoint Case

    author author J. Lampart , author J. Schmidt , author S. Teufel , \ and\ author R. Tumulka ,\ title title A nonrelativistic quantum field theory with point interactions in three dimensions , \ 10.1007/s00023-018-0696-0 journal journal Annales Henri Poincar\'e \ volume 19 ,\ pages 2641--2670 ( year 2018 ) NoStop

    work page doi:10.1007/s00023-018-0696-0 2018

  16. [16]

    author author H. Halvorson ,\ title title Reeh--Schlieder defeats Newton--Wigner : On alternative localization schemes in relativistic quantum field theory , \ 10.1086/392869 journal journal Philosophy of Science \ volume 68 ,\ pages 111--133 ( year 2001 ) ,\ note arXiv:quant-ph/0007060 ,\ http://arxiv.org/abs/quant-ph/0007060 arXiv:quant-ph/0007060 NoStop

    work page doi:10.1086/392869 2001

  17. [17]

    author author V. Bargmann ,\ title title On unitary ray representations of continuous groups , \ 10.2307/1969831 journal journal Annals of Mathematics \ volume 59 ,\ pages 1--46 ( year 1954 ) NoStop

    work page doi:10.2307/1969831 1954

  18. [18]

    \ L\'evy-Leblond ,\ title title G alilei group and G alilean invariance , \ in\ @noop booktitle Group Theory and its Applications ,\ Vol

    author author J.-M. \ L\'evy-Leblond ,\ title title G alilei group and G alilean invariance , \ in\ @noop booktitle Group Theory and its Applications ,\ Vol. volume 2 ,\ editor edited by\ editor E. M. \ Loebl \ ( publisher Academic Press ,\ year 1971 )\ pp.\ pages 221--299 NoStop

    1971

  19. [19]

    note Throughout we treat smeared fields (f), ^ (f) as affiliated to F ( O ) ; the implication (f) = 0 (f) = 0 used below proceeds via spectral projections of | (f)| , which are bounded elements of F ( O ) and hence subject to the separating property directly, exactly as in Remark rem:field_vs_observable . Stop

  20. [20]

    note Existence is automatic under (G7) by the Fock construction (Remark rem:time_zero_fock ), and is articulated as a regularity hypothesis (G7 ^* )(d) in the extended setting of sec:non_fock_extension ; see Lemma lem:time_zero_4d . In a literal slice formulation, no f C_c^ ( M _G) has support in the measure-zero 3-plane _t , so the CCR must be expressed ...

  21. [21]

    It is also automatic under (G7) by Fock construction, where _0(g) is the standard time-zero smeared annihilation operator

    note This existence hypothesis is automatic for Wightman-style realizations (under temperedness of the field operator-valued distributions in time) and is verified directly for every published interacting Galilean QFT cited in sec:verification . It is also automatic under (G7) by Fock construction, where _0(g) is the standard time-zero smeared annihilatio...

  22. [22]

    Haag ,\ @noop title Local Quantum Physics: Fields, Particles, Algebras ,\ edition 2nd \ ed.\ ( publisher Springer ,\ year 1996 )\ note first edition 1992 NoStop

    author author R. Haag ,\ @noop title Local Quantum Physics: Fields, Particles, Algebras ,\ edition 2nd \ ed.\ ( publisher Springer ,\ year 1996 )\ note first edition 1992 NoStop

    1996

  23. [23]

    An algebraic approach to quantum field theory.J

    author author R. Haag \ and\ author D. Kastler ,\ title title An algebraic approach to quantum field theory , \ 10.1063/1.1704187 journal journal Journal of Mathematical Physics \ volume 5 ,\ pages 848--861 ( year 1964 ) NoStop

    work page doi:10.1063/1.1704187 1964

  24. [24]

    Bemerkungen zur unitäräquivalenz von lorentzinvarianten feldern

    author author H. Reeh \ and\ author S. Schlieder ,\ title title Bemerkungen zur U nit\"ar\"aquivalenz von L orentzinvarianten F eldern , \ 10.1007/BF02787889 journal journal Il Nuovo Cimento \ volume 22 ,\ pages 1051--1068 ( year 1961 ) NoStop

    work page doi:10.1007/bf02787889 1961

  25. [25]

    Bratteli \ and\ author D

    author author O. Bratteli \ and\ author D. W. \ Robinson ,\ @noop title Operator Algebras and Quantum Statistical Mechanics ,\ edition 2nd \ ed.,\ Vol. volume 2 \ ( publisher Springer ,\ year 1997 )\ note equilibrium states; models in quantum statistical mechanics NoStop

    1997

  26. [26]

    author author G. C. \ Wick , author A. S. \ Wightman , \ and\ author E. P. \ Wigner ,\ title title The intrinsic parity of elementary particles , \ 10.1103/PhysRev.88.101 journal journal Physical Review \ volume 88 ,\ pages 101--105 ( year 1952 ) NoStop

    work page doi:10.1103/physrev.88.101 1952

  27. [27]

    author author M. Takesaki ,\ 10.1007/BFb0065832 title Tomita's Theory of Modular H ilbert Algebras and Its Applications ,\ series Lecture Notes in Mathematics , Vol.\ volume 128 \ ( publisher Springer ,\ year 1970 ) NoStop

    work page doi:10.1007/bfb0065832 1970

  28. [28]

    author author J. J. \ Bisognano \ and\ author A. S. \ Wightman ,\ title title On the duality condition for a H ermitian scalar field , \ 10.1063/1.522605 journal journal Journal of Mathematical Physics \ volume 16 ,\ pages 985--1007 ( year 1975 ) NoStop

    work page doi:10.1063/1.522605 1975

  29. [29]

    author author J. J. \ Bisognano \ and\ author A. S. \ Wightman ,\ title title On the duality condition for quantum fields , \ 10.1063/1.522898 journal journal Journal of Mathematical Physics \ volume 17 ,\ pages 303--321 ( year 1976 ) NoStop

    work page doi:10.1063/1.522898 1976

  30. [30]

    Buchholz , author C

    author author D. Buchholz , author C. D'Antoni , \ and\ author K. Fredenhagen ,\ title title The universal structure of local algebras , \ 10.1007/BF01239019 journal journal Communications in Mathematical Physics \ volume 111 ,\ pages 123--135 ( year 1987 ) NoStop

    work page doi:10.1007/bf01239019 1987

  31. [31]

    author author A. Connes ,\ title title Une classification des facteurs de type III , \ 10.24033/asens.1247 journal journal Annales scientifiques de l'\'Ecole Normale Sup\'erieure \ volume 6 ,\ pages 133--252 ( year 1973 ) NoStop

    work page doi:10.24033/asens.1247 1973

  32. [32]

    author author U. Haagerup ,\ title title Connes' bicentralizer problem and uniqueness of the injective factor of type III _1 , \ 10.1007/BF02392257 journal journal Acta Mathematica \ volume 158 ,\ pages 95--148 ( year 1987 ) NoStop

    work page doi:10.1007/bf02392257 1987

  33. [33]

    It is needed for the propagation step in the proof of Proposition prop:non_fock_obstruction_no_c below

    note The joint-continuity-of-products clause is automatic in Wightman-style realizations (where the time-zero fields are operator-valued distributions on a common Schwartz-like domain) and in the Fock case (G7), where the limit converges in operator norm on each finite-particle subspace of the algebraic Fock domain. It is needed for the propagation step i...

  34. [34]

    author author G. C. \ Hegerfeldt ,\ title title Violation of causality in relativistic quantum theory? \ 10.1103/PhysRevLett.54.2395 journal journal Physical Review Letters \ volume 54 ,\ pages 2395--2398 ( year 1985 ) NoStop

    work page doi:10.1103/physrevlett.54.2395 1985

  35. [35]

    author author D. Buchholz ,\ title title Current trends in axiomatic quantum field theory , \ in\ @noop booktitle Quantum Field Theory ,\ series Lecture Notes in Physics , Vol.\ volume 558 ,\ editor edited by\ editor P. Breitenlohner \ and\ editor D. Maison \ ( publisher Springer ,\ year 2000 )\ pp.\ pages 43--64 ,\ note arXiv:hep-th/9811233 ,\ http://arx...

    work page arXiv 2000

  36. [36]

    Jost ,\ title title Eine B emerkung zum CTP - T heorem , \ @noop journal journal Helvetica Physica Acta \ volume 30 ,\ pages 409--416 ( year 1957 ) NoStop

    author author R. Jost ,\ title title Eine B emerkung zum CTP - T heorem , \ @noop journal journal Helvetica Physica Acta \ volume 30 ,\ pages 409--416 ( year 1957 ) NoStop

    1957

  37. [37]

    author author R. F. \ Streater \ and\ author A. S. \ Wightman ,\ @noop title PCT , Spin and Statistics, and All That \ ( publisher W. A. Benjamin ,\ year 1964 )\ note princeton University Press paperback edition 2000 NoStop

    1964

  38. [38]

    author author R. Haag ,\ title title On quantum field theories , \ @noop journal journal Det Kongelige Danske Videnskabernes Selskab, Matematisk-fysiske Meddelelser \ volume 29 ,\ pages 1--37 ( year 1955 ) NoStop

    1955

  39. [39]

    Von Neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories

    author author A. Connes \ and\ author C. Rovelli ,\ title title Von N eumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories , \ 10.1088/0264-9381/11/12/007 journal journal Classical and Quantum Gravity \ volume 11 ,\ pages 2899--2917 ( year 1994 ) NoStop

    work page doi:10.1088/0264-9381/11/12/007 1994

  40. [40]

    Brunetti , author D

    author author R. Brunetti , author D. Guido , \ and\ author R. Longo ,\ title title Modular structure and duality in conformal quantum field theory , \ 10.1007/BF02096738 journal journal Communications in Mathematical Physics \ volume 156 ,\ pages 201--219 ( year 1993 ) NoStop

    work page doi:10.1007/bf02096738 1993

  41. [41]

    author author B. Schroer ,\ title title Modular localization and the bootstrap-formfactor program , \ 10.1016/S0550-3213(97)00358-1 journal journal Nuclear Physics B \ volume 499 ,\ pages 547--568 ( year 1997 ) ,\ note arXiv:hep-th/9702145 ,\ http://arxiv.org/abs/hep-th/9702145 arXiv:hep-th/9702145 NoStop

    work page doi:10.1016/s0550-3213(97)00358-1 1997