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REVIEW 3 major objections 9 minor 19 references

Finite-Precision Algebraic Quantum Field Theory

T0 review · 3 major / 9 minor · reviewed 2026-07-31 · grok-4.5

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Pith's one-line read Finite experimental data in relativistic quantum field theory is represented by open convex regions of state space, and the major structures of algebraic QFT still hold for those regions.

desk verdict Solid AQFT-to-parcels translation with real lattice and factor-type content, but Spectral Regularity is empty on type III local algebras, so the general modular/KMS half does not apply where the paper claims it does. read the letter →

arxiv 2607.24445 v1 pith:4LBAT75O submitted 2026-07-27 quant-ph math-phmath.MP

Abbas Edalat This is my paper · ORCID
classification quant-phmath-phmath.MP MSC 81T0546L6046L30 PACS 03.70.+k11.10.Cd03.65.Fd
keywords algebraicquantumfieldtheoryfiniteprecisionparcelsmodularKMSconditionUnruheffectReeh–SchliederMurray–vonNeumannclassification
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

Standard algebraic quantum field theory treats states as exact mathematical points, even though every real experiment only ever pins down finitely many expectation values to finite accuracy. This paper replaces those points, as the primary operational objects, by quantum parcels: weak-star open convex sets of states compatible with that finite data. It then shows that the central structures of the exact theory survive the change. Spacelike vacuum correlations and strict Bell violations persist throughout small enough parcels; the Reeh–Schlieder theorem becomes local reachability of parcels from the vacuum; Haag’s theorem becomes an obstruction to parcel equivalence; modular dynamics and the KMS condition lift to parcels (under a spectral-regularity hypothesis); the Unruh effect is recovered by refining Rindler parcels; lattice measurements induce continuum parcels consistently; and the Murray–von Neumann factor classification, together with the Type II₁ trace, can be read off the geometry and conjugation dynamics of limiting projection faces. The operator-algebraic backbone is left untouched: exact states remain as ideal limits of successive parcel refinement.

What carries the argument

The quantum parcel: a non-empty weak-star open convex subset of the (normal) state space, determined by finitely many finite-precision expectation constraints. Parcel reduction, measurement update, compatible parcel nets, modular action on parcels, KMS-defect, and limiting exposed faces of projection parcels carry the structural recasting.

What would settle it

Exhibit an operationally realistic parcel of normal states on a local algebra for which relative modular operators have spectrum accumulating at zero, and check whether the claimed modular continuity and parcel KMS reduction still hold; or construct vacuum-centred parcels on which a known CHSH violation fails to persist.

Watch

Extended reading notes

Core claim

Interval Algebraic Quantum Field Theory (IAQFT) gives a single finite-precision language in which the operational, modular and operator-algebraic structures of relativistic quantum theory—measurement update, locality, spacelike correlations and Bell violation, Reeh–Schlieder, Haag obstruction, modular/KMS/Unruh dynamics, lattice continuum limits, and Murray–von Neumann factor type plus II₁ trace—are all expressed in terms of the geometry and dynamics of quantum parcels rather than exact states.

Load-bearing premise

For the modular and KMS results, every operationally usable parcel of states is assumed to keep relative modular operators bounded away from zero spectrum—a regularity condition the paper adopts rather than derives.

Editorial extensions

If this is right

  • Finite laboratory data in relativistic QFT can be treated as parcels without losing spacelike correlations or strict Bell nonclassicality.
  • Reeh–Schlieder becomes an operational statement: local operations on the vacuum can reach any finite-precision neighbourhood of a pure normal state.
  • Haag-type no-go results already block parcel equivalence, so finite precision does not evade the free-versus-interacting obstruction.
  • Under parcel refinement, the Unruh temperature is recovered from boost KMS-defect collapse without requiring every intermediate state to be thermal.
  • Murray–von Neumann type and the II₁ trace can be read from parcel geometry and homogeneous face counting rather than from an a-priori dimension function.

Reading between the lines

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

  • If spectral regularity fails for physically natural parcels, the modular half of IAQFT would need a weaker continuity substitute or would be limited to the Bisognano–Wichmann/Unruh setting where dynamics is state-independent.
  • Parcel counting for the II₁ trace suggests a finite-precision route to continuous geometry that could be tested numerically inside hyperfinite approximations.
  • Extending parcel nets beyond the vacuum sector (thermal or charged sectors) would give a finite-precision language for superselection that the paper flags but leaves open.
  • The separation of lattice spacing from parcel width clarifies that continuum-limit numerics and observational uncertainty are independent refinement axes, which could guide how lattice QFT reports error bars.
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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 / 9 minor

Summary. The paper introduces Interval Algebraic Quantum Field Theory (IAQFT): finite-precision information is represented by "quantum parcels" — non-empty weak* open convex subsets of the state space of the quasi-local algebra (or of the normal state space of a von Neumann algebra) — rather than by exact states. Building on the author's earlier Interval Quantum Mechanics, it develops parcel measurement update and reduction theorems (Thms 4.1–4.2, 6.4), finite-dimensional volume contraction under Lüders updates (Prop 5.12, Thms 5.13–5.14), no-signalling for nonselective updates, open-ness of vacuum correlations and CHSH violation (Thms 7.2–7.3), Reeh–Schlieder as local parcel reachability (Thm 8.3), a parcel form of the Haag obstruction (Thm 9.4), lifts of modular dynamics and the J-conjugation to parcels (§10), a parcel KMS theory under a "Spectral Regularity Axiom" (§11), a parcel Unruh effect via Bisognano–Wichmann (§12), lattice compatibility (§13), and a parcel-geometric reconstruction of the Murray–von Neumann factor classification and the II_1 trace (§14, with Appendix C proofs). Most of the framework sections are technically sound, if often straightforward; the finite-dimensional Jacobian/volume results and the §14 factor-type criteria are the most substantial new mathematics. The load-bearing problem is the Spectral Regularity Axiom 11.1, which is unsatisfiable on type III factors — the AQFT case — for a classical reason (Connes' S-invariant). This vacates part of §11 and the

Significance. If the repairs are made, the paper delivers a coherent and mostly parameter-free finite-precision overlay of AQFT: reduction and local-reduction theorems from oscillation shrinkage alone, an exact Jacobian formula and general-n volume contraction for Lüders updates (new, quantitative), Reeh–Schlieder recast as a genuine equivalence with local parcel reachability, a parcel form of the Haag obstruction that is strictly stronger than vacuum equivalence, lattice/continuum compatibility, and a parcel-geometric reconstruction of the Murray–von Neumann classification with complete proofs. The Unruh section, once the empty hypothesis is removed, gives an honest finite-precision statement: thermality emerges from refinement plus Bisognano–Wichmann without assuming it of any approximating state. The work is a reinterpretation framework rather than new physics, but it is careful, and several constructions (KMS-defect, parcel nets, projection-face criteria) may prove useful independently.

major comments (3)
  1. [§11.1, Axiom 11.1 (Spectral Regularity)] Axiom 11.1 requires ε>0 with E_{Δ_{ψ,ω}}([0,ε))=0 for all ψ,ω∈O_f. Taking ψ=ω gives Δ_{ω,ω}=Δ_ω, so the axiom demands a spectral gap at 0 for every faithful state in the parcel. By Connes' S-invariant, M is type III iff 0∈S(M)=∩_φ spec Δ_φ; hence on any type III factor 0∈spec Δ_ω for EVERY faithful normal ω. Local and wedge algebras in AQFT are type III (III_1 in standard models; Buchholz–D'Antoni–Fredenhagen), and §11.1 itself proves every parcel meets S_f(M). So NO parcel on a local AQFT algebra satisfies the axiom: its scope is empty exactly in the relativistic setting the paper targets (it is satisfiable essentially only in finite-dimensional or suitably bounded semifinite contexts). This voids Prop 11.8, Lemma 11.3(iv), the modular-continuity route to Prop 11.5, and the 'operational regularity' framing of §15.6 as results about AQFT. Concrete test the authors cannot pass: exhibit on
  2. [§12.1, Definition of Rindler parcel] A Rindler parcel is defined as a parcel O⊆S_n(M(W_R)) 'satisfying the Spectral Regularity Axiom 11.1'. Since M(W_R) is type III, the preceding comment shows this class is empty, so Theorems 12.4–12.5 and Corollary 12.6 are formally statements about the empty class. Inspection of the proofs shows the axiom is never used: Thm 12.4(iii) says so explicitly, and Thm 12.5 uses only prescribed-state reduction (Thm 4.1), convexity, weak* continuity of FIXED boost-transformed observables, and Prop 12.2. The section is therefore repairable within the manuscript's scope: delete Axiom 11.1 from the definition of Rindler parcel and state plainly that the Unruh analysis rests on Bisognano–Wichmann plus fixed dynamics, not on spectral regularity (§15.6 already gestures at this distinction but §12.1 contradicts it).
  3. [§11.2, Corollary 11.16] Route (ii) ('down then right') claims the solid intervals I_{O_α}, K_{O_α} from Def 11.4 shrink to the two sides of the KMS identity for the limit state ω*. But Def 11.4 uses the STATE-DEPENDENT dynamics σ^ψ, while the KMS-defect (Def 11.10) and Thm 11.12 use the FIXED dynamics σ^{ω_0}. The claimed identification of the two limit points therefore requires an argument connecting σ^ψ→σ^{ω*} with σ^{ω_0} (e.g., uniqueness of normal KMS states on factors at fixed β), which is neither stated nor proved; it also inherits modular continuity, i.e. the vacuous Axiom 11.1 in the type III case. As written the commutativity is asserted, not established. Given that Thm 11.12 itself needs no regularity, the cleanest fix is to re-prove route (ii) entirely within the fixed-dynamics formalism or delete the corollary.
minor comments (9)
  1. [§11.2, Remark 11.13] Thm 11.12 as proved uses only interval shrinkage, vanishing defect, and non-empty intersection — the Spectral Regularity Axiom plays no role (the defect is defined via the fixed dynamics σ^{ω_0}). The sentence 'Under the Spectral Regularity Axiom, interval shrinkage, vanishing KMS-defect, and non-empty intersection together recover a unique normal KMS state' is a misattribution; correcting it actually strengthens the paper, since Thm 11.12 survives the vacuity problem of Axiom 11.1.
  2. [§5.2, proof of Theorem 5.13] Duplicated/garbled sentence: 'Let r = rank Π, 1≤r<n, Let r = rank(Π), the rank of the projection Π, that is, the dimension of its range Ran(Π).' Merge into one definition of r.
  3. [§9, proof of Theorem 9.4] The sentence 'By parcel equivalence, Φ*(V1) and V2 are cofinal under reverse inclusion' appears twice consecutively.
  4. [§11.1, Definition 11.7 vs Proposition 11.5] Prop 11.5 already uses the term 'modularly continuous parcel', but Def 11.7 giving its meaning appears afterwards; reorder so the definition precedes its first use.
  5. [§12.1.3, proof of Theorem 12.4] The part labels do not match the statement: the proof's 'Part (ii)' cites ω_0|M(W_R)∈O_f (should be ∈O, membership), and 'Part (iii) is the Bisognano–Wichmann KMS identity' actually proves statement (ii) (common vacuum boundary value). Also C^a_O uses α_{aτ+2πi}(B): it would help to note explicitly that B∈A_0 analytic makes this a fixed bounded element of M(W_R) for every ρ.
  6. [References, item [11] (Reed–Simon)] The descriptive note 'Sections VI.5–VI.6 treat compact and trace-class operators' does not match the actual citation (Thm. VIII.25 on core convergence/strong resolvent convergence); correct the note.
  7. [§8, Proposition 8.1 and Theorem 8.3] Prop 8.1(2) says 'weakly dense in the pure normal state space' but the proof works with pointwise convergence on M_0 (the σ(M*,M)-topology on states); since norm-density in M_* is a different (false-in-general) claim, state the topology precisely.
  8. [§12.1.3, Corollary 12.7] The symbols I_{O_α}, K_{O_α} are reused from Def 11.4 (state-dependent modular dynamics) although §12 uses the fixed boost dynamics; introduce distinct notation for the fixed-dynamics thermal intervals to avoid conflating the two settings (cf. the issue in Cor 11.16).
  9. [§16, Table 1] Rows 'KMS condition → Parcel KMS theorem' and 'Unruh effect → Parcel Unruh theorem' should be revisited after the §11–12 repairs so the table does not implicitly credit Spectral Regularity with results that hold without it.

Circularity Check

2 steps flagged · score 2.0 of 10

Mostly honest reformulation; factor-type/trace 'recovery' restates classical MV theory in face language, and IQM is prior self-work used as lemmas—not load-bearing circular prediction.

  1. renaming known result [§14.3 Thm 14.14; §14.4 Thm 14.19–Cor 14.21; Abstract final claim]
    "Finally, we show that the geometry and canonical conjugation dynamics of limiting projection faces recover minimality, finiteness, and the Murray–von Neumann type classification of factors, while homogeneous parcel counting in Type II₁ factors reconstructs the trace and Murray–von Neumann equivalence of projections."

    Atoms/Whole-finite/Some-finite are the classical type criteria (minimal projection; 1 finite; some nonzero finite projection) rewritten via exposed faces F_p and conjugation-fixed points, which the proofs identify with pure states and unique normal traces (Facts C.4, C.8). k_n(p) is defined using classical ∼, and the proof that k_n(p)/n→τ(p) uses τ and r≾p⇔τ(r)≤τ(p); Cor 14.21 then recovers p∼q from equal limits via the classical τ-criterion. This is equivalent rephrasing of MV theory, not an independent geometric derivation of type or trace.

  2. self citation load bearing [§5.1–5.2; Prop 5.4, Thm 5.6, Thm 5.13; Ref [6]]
    "The finite-dimensional case provides the link between IAQFT and the Interval Quantum Mechanics (IQM) developed in [6]. ... the operational content of finite-dimensional IAQFT and IQM is identical. ... The full details and proof are in [6]."

    Double-parcel updates, uniform positivity/separation, and volume-contraction theorems are imported from the author's IQM paper and transferred by Θ. This is load-bearing for the finite-dimensional information-gain section only; it does not underwrite the AQFT parcel nets, modular/KMS, Unruh, or factor-type claims. Ordinary prior-work citation, scored lightly.

full rationale

IAQFT is explicitly a recasting framework (Abstract: features of AQFT are 'recast in parcel-theoretic terms'). Reduction, parcel nets, Reeh–Schlieder-as-reachability, and Haag-as-obstruction-to-parcel-equivalence are equivalent reformulations or one-way implications from classical AQFT, not predictions forced by fitting or by defining the target into the input. Spectral Regularity is adopted as an axiom, not derived from the KMS conclusion. The finite-dimensional volume/double-parcel material cites the author's IQM paper [6] and transfers via the canonical homeomorphism Θ; that is ordinary self-citation of prior lemmas, and those lemmas are not the paper's central AQFT claims. The only mild circularity-adjacent pattern is §14's language of 'recovering' Murray–von Neumann type and the II₁ trace from parcel geometry: the criteria (Atoms / Whole-finite / Some-finite) and the counting function k_n(p) are translations of the classical definitions (minimal/finite projections; MV equivalence and the unique trace), and the proofs invoke standard facts (unique normal trace on finite factors, τ(p)=τ(q)⇔p∼q) rather than deriving type or τ from parcel data alone. That is renaming/equivalent characterization presented as geometric recovery, not a fitted or self-definitional prediction loop. No fitted-input-as-prediction pattern appears. Score 2 reflects minor renaming plus non-load-bearing self-citation; central operational development is self-contained.

Assumptions & free parameters 0 free parameters · 6 assumptions · 3 invented entities

The paper keeps standard Haag–Kastler/AQFT structure and adds parcels as the operational state objects. Load-bearing extras are Spectral Regularity for modular/KMS continuity, purity/irreducibility of the vacuum for the Reeh–Schlieder parcel form, and classical factor-theory facts used to read type off faces. No numerical free parameters are fitted. Parcels are invented mathematical entities with operational motivation, not new physical fields.

assumptions (6)
  • domain assumption Haag–Kastler net axioms (isotony, locality/Einstein causality, covariance) and quasi-local inductive limit for the continuum theory.
    Background AQFT structure assumed throughout §§2,6–9; not re-proved.
  • ad hoc to paper Spectral Regularity Axiom 11.1: uniform lower spectral gap ε>0 for relative modular operators on each operational parcel.
    Explicitly adopted, not derived; used to get modular continuity and parcel KMS reduction (Prop. 11.8, Thm. 11.12).
  • domain assumption Vacuum state ω₀ pure / GNS representation irreducible (M₀=B(H₀)) in the Reeh–Schlieder section.
    Stated in §8 as the single pure vacuum sector setting for Prop. 8.1 and Thm. 8.3.
  • domain assumption Bisognano–Wichmann identification of vacuum modular flow with Lorentz boosts on the Rindler wedge.
    Used as input for the parcel Unruh theorems (§12), not re-derived.
  • standard math Classical Murray–von Neumann factor theory (existence/uniqueness of traces on finite factors, type criteria, II₁ divisibility).
    Invoked via Facts C.4, C.8, C.9 and Thms 14.8–14.19 to match parcel criteria to type.
  • standard math Banach–Alaoglu weak* compactness of state space and standard GNS/Tomita–Takesaki apparatus.
    Used for reduction theorems, modular lifts, and KMS analytic cores.
invented entities (3)
  • Quantum parcel (weak* open convex subset of state space / normal state space)
    purpose: Primary operational object encoding finite-precision information from finitely many expectation constraints; carrier of measurement update and refinement.
    Defined in Def. 3.1; extends IQM parcels to AQFT. Mathematical construction with operational reading, not a new dynamical field.
  • Compatible parcel net
    purpose: Finite-precision local data assignment consistent under isotony/restriction maps.
    Def. 6.1; encodes locality of finite experiments on the Haag–Kastler net.
  • KMS-defect / boost KMS-defect of a parcel
    purpose: Quantitative failure of the KMS boundary identity across a parcel; drives parcel KMS and Unruh reduction.
    Defs. 11.10 and 12.1; operational surrogate for exact Tomita boundary relations.

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Cite this review

Pith. "Pith review of Finite-Precision Algebraic Quantum Field Theory." pith.science (2026). https://pith.science/paper/4LBAT75O

@misc{pith2026260724445,
  author       = {Pith},
  title        = {Pith review of: Finite-Precision Algebraic Quantum Field Theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4LBAT75O}},
  note         = {Machine review of arXiv:2607.24445}
}
abstract

Recent work introduced Interval Quantum Mechanics (IQM), a finite-precision framework in which physical information is represented by sets of possible states rather than exact states. We extend this approach to algebraic quantum field theory (AQFT) by introducing Interval Algebraic Quantum Field Theory (IAQFT), whose basic objects are quantum parcels: weak* open convex regions of state space encoding finite-precision information obtained from finitely many local observations. We develop parcel reduction and measurement update, establish finite-dimensional information-contraction results, and formulate locality through compatible parcel nets. Major structural features of AQFT are recast in parcel-theoretic terms, including the Reeh--Schlieder property, Haag's theorem, modular theory, the KMS condition, and the Unruh effect. We show that spacelike vacuum correlations and strict Bell violations persist under finite precision, and that lattice approximations are compatible with the parcel framework. Finally, we recover the Murray--von Neumann classification of factors together with the trace and projection equivalence in Type~$\mathrm{II}_1$ factors from the geometry of limiting parcels. IAQFT thus provides a unified finite-precision formulation of the operational, modular, and operator-algebraic structures of relativistic quantum theory.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

19 extracted references · 1 linked inside Pith

  1. [1]

    Oxford University Press, Oxford, 1999

    Huzihiro Araki.Mathematical Theory of Quantum Fields. Oxford University Press, Oxford, 1999

    1999

  2. [2]

    Bisognano and Eyvind H

    Joseph J. Bisognano and Eyvind H. Wichmann. On the duality condition for quantum fields.Journal of Mathematical Physics, 17(3):303–321, 1976

    1976

  3. [3]

    Robinson.Operator Algebras and Quantum Statistical Me- chanics 1: C*- and W*-Algebras

    Ola Bratteli and Derek W. Robinson.Operator Algebras and Quantum Statistical Me- chanics 1: C*- and W*-Algebras. Symmetry Groups. Decomposition of States. Texts and Monographs in Physics. Springer-Verlag, Berlin, second edition, 1987

    1987

  4. [4]

    Robinson.Operator Algebras and Quantum Statistical Mechan- ics 2: Equilibrium States

    Ola Bratteli and Derek W. Robinson.Operator Algebras and Quantum Statistical Mechan- ics 2: Equilibrium States. Models in Quantum Statistical Mechanics. Texts and Monographs in Physics. Springer-Verlag, Berlin, second edition, 1997

    1997

  5. [5]

    North-Holland Publishing Co., Amsterdam, 1981

    Jacques Dixmier.von Neumann Algebras, volume 27 ofNorth-Holland Mathematical Li- brary. North-Holland Publishing Co., Amsterdam, 1981. Translated from the French by F. Jellett

    1981

  6. [6]

    Finite-precision quantum mechanics, 2026

    Abbas Edalat. Finite-precision quantum mechanics, 2026. arXiv:2605.19706 [quant-ph], https://arxiv.org/abs/2605.19706

    arXiv 2026

  7. [7]

    Springer Science & Busi- ness Media, 2012

    Rudolf Haag.Local quantum physics: Fields, particles, algebras. Springer Science & Busi- ness Media, 2012

    2012

  8. [8]

    An algebraic approach to quantum field theory.Journal of Mathematical Physics, 5:848–861, 1964

    Rudolf Haag and Daniel Kastler. An algebraic approach to quantum field theory.Journal of Mathematical Physics, 5:848–861, 1964. 82

    1964

Show all 19 references
  1. [9]

    Kadison and John R

    Richard V. Kadison and John R. Ringrose.Fundamentals of the Theory of Operator Al- gebras, Volume I: Elementary Theory, volume 15 ofGraduate Studies in Mathematics. American Mathematical Society, Providence, RI, 1997. Reprint of the 1983 original

    1997

  2. [10]

    Kadison and John R

    Richard V. Kadison and John R. Ringrose.Fundamentals of the Theory of Operator Alge- bras, Volume II: Advanced Theory, volume 16 ofGraduate Studies in Mathematics. Amer- ican Mathematical Society, Providence, RI, 1997. Reprint of the 1986 original

    1997

  3. [11]

    I: Func- tional Analysis

    Michael Reed and Barry Simon.Methods of Modern Mathematical Physics, Vol. I: Func- tional Analysis. Academic Press, 2nd edition, 1980. Sections VI.5–VI.6 treat compact and trace-class operators

    1980

  4. [12]

    Springer-Verlag, Berlin, 1971

    Shˆ oichirˆ o Sakai.C∗-Algebras andW ∗-Algebras, volume 60 ofErgebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag, Berlin, 1971

    1971

  5. [13]

    Springer Science & Business Media, 2013

    Eduardo D Sontag.Mathematical control theory: deterministic finite dimensional systems, volume 6. Springer Science & Business Media, 2013

    2013

  6. [14]

    Streater and Arthur S

    Raymond F. Streater and Arthur S. Wightman.PCT, Spin and Statistics, and All That. W. A. Benjamin, New York, 1964

    1964

  7. [15]

    Summers and Reinhard F

    Stephen J. Summers and Reinhard F. Werner. The vacuum violates Bell’s inequalities. Physics Letters A, 110:257–259, 1985

    1985

  8. [16]

    Springer-Verlag, Berlin, 2002

    Masamichi Takesaki.Theory of Operator Algebras I, volume 124 ofEncyclopaedia of Math- ematical Sciences. Springer-Verlag, Berlin, 2002. Reprint of the 1979 original

    2002

  9. [17]

    William G. Unruh. Notes on black-hole evaporation.Physical Review D, 14(4):870–892, 1976

    1976

  10. [18]

    Princeton University Press, Princeton, NJ, 1960

    John von Neumann.Continuous Geometry, volume 25 ofPrinceton Mathematical Series. Princeton University Press, Princeton, NJ, 1960. Edited by Israel Halperin

    1960

  11. [19]

    Wightman

    Arthur S. Wightman. Quantum field theory in terms of vacuum expectation values.Physical Review, 101(2):860–866, 1956. 83

    1956

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