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arxiv: 2606.27387 · v1 · pith:HPMMKPE2new · submitted 2026-06-15 · ⚛️ physics.gen-ph

On the Meaning of Localization in Non-Local Quantum Field Theory

E. J. Thompson This is my paper

Pith reviewed 2026-06-29 01:49 UTC · model grok-4.3

classification ⚛️ physics.gen-ph
keywords nonlocal quantum field theorylocalizationuncertainty principleminimal lengthdetector response kernelLorentz covariancemicrocausalityultraviolet structure
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The pith

Nonlocal quantum field theory implies a minimal localization length through a modified uncertainty relation.

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

The paper examines localization in an ultraviolet complete nonlocal quantum field theory. It assumes an induced equal time detector response kernel and proves that the observed localization width follows an exact variance addition law. When this law is combined with the standard Heisenberg inequality, a nonlocal uncertainty relation emerges. This relation matches the usual one at low energies but enforces a minimal localization length at high energies of order the nonlocality scale. The analysis concludes that spacetime stays a continuous Lorentz covariant manifold while pointlike localization becomes unobservable below that scale.

Core claim

Under the hypothesis of an induced equal time detector response kernel, the observed localization width obeys an exact variance addition law. Combined with the Heisenberg inequality this yields a nonlocal uncertainty relation whose UV bound implies a minimal localization length of order L_M, while spacetime remains a Lorentz covariant continuum but pointlike localization ceases below the nonlocality scale.

What carries the argument

The induced equal time detector response kernel, which is hypothesized to produce an exact variance addition law for localization widths that, when added to the Heisenberg relation, generates the nonlocal uncertainty principle.

If this is right

  • The standard Heisenberg uncertainty relation is recovered in the local limit as the nonlocality scale approaches zero.
  • Pointlike localization is not a physically realizable notion below the nonlocality scale.
  • The spacetime manifold remains Lorentz covariant and continuous at all scales.
  • The ultraviolet structure of the theory is modified while preserving microcausality in the appropriate sense.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The result distinguishes the mathematical manifold structure of spacetime from the physical observability of point localization.
  • This formulation may connect to other nonlocal approaches that retain continuum spacetime but alter short-distance observables.

Load-bearing premise

The existence of an induced equal time detector response kernel whose functional form produces an exact variance addition law for localization widths.

What would settle it

An explicit calculation of the detector response in a specific nonlocal model that violates the assumed variance addition law, or an observation of localization at scales smaller than the predicted minimal length.

Figures

Figures reproduced from arXiv: 2606.27387 by E. J. Thompson.

Figure 1
Figure 1. Figure 1: FIG. 1. The underlying local localization density [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Comparison between the local uncertainty [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. One-loop vacuum-polarization contributions. [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
read the original abstract

In this paper we explore and derive an uncertainty principle for an ultraviolet complete nonlocal quantum field theory where under our hypothesises of an induced equal time detector response kernel, we then prove that the observed localization width obeys an exact variance addition law. Then when we combine this with the ordinary Heisenberg inequality and we obtain a nonlocal uncertainty relation. The bound reduces to the usual local relation in the infrared or local limit when $E_M \to \infty$, while in the ultraviolet it implies a minimal localization length of order $L_M$. We go on to explain what this means for locality, microcausality, the interpretation of spacetime points, and the ultraviolet structure of quantum field theory. In this formulation we note and prove that spacetime will remain a Lorentz covariant continuum at the level of the manifold description but pointlike localization ceases to be a physically realizable observable notion below the nonlocality scale.

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.

Circularity Check

1 steps flagged

Variance addition law is enforced by the hypothesized detector kernel whose form is chosen to produce it

specific steps
  1. self definitional [Abstract, paragraph beginning 'under our hypothesises of an induced equal time detector response kernel']
    "under our hypothesises of an induced equal time detector response kernel, we then prove that the observed localization width obeys an exact variance addition law. Then when we combine this with the ordinary Heisenberg inequality and we obtain a nonlocal uncertainty relation."

    The kernel hypothesis is stated as the premise, and the variance addition law is presented as a derived result under that hypothesis. The law is not obtained from the nonlocal QFT dynamics independently; the hypothesis is introduced with a functional form selected to enforce the exact addition law, so the 'proof' and all downstream conclusions (nonlocal uncertainty relation, minimal length L_M) reduce to the input assumption by construction.

full rationale

The paper's central derivation begins by hypothesizing an induced equal time detector response kernel and then 'proves' that localization width obeys an exact variance addition law under that hypothesis. The subsequent nonlocal uncertainty relation, UV bound of order L_M, and claims about cessation of pointlike localization all follow directly from this step. No independent derivation or external benchmark for the kernel's specific functional form is provided; the hypothesis is introduced precisely to yield the variance addition law, making the result equivalent to the input assumption by construction. This matches the self-definitional pattern with load-bearing impact on the entire claim.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The derivation depends on one central hypothesis (the induced equal-time detector response kernel) whose functional form is not independently derived or constrained by external data; no free parameters are explicitly fitted in the abstract, but L_M and E_M function as characteristic scales introduced by the nonlocality.

free parameters (2)
  • L_M
    Characteristic minimal localization length introduced as the UV scale; its value is stated as 'of order L_M' without independent determination.
  • E_M
    Energy scale at which nonlocality becomes important; appears as the cutoff where the bound deviates from Heisenberg.
axioms (2)
  • ad hoc to paper Induced equal time detector response kernel exists and takes a form that produces exact variance addition for localization width.
    Invoked explicitly in the abstract as the starting hypothesis for the derivation.
  • domain assumption Standard Heisenberg uncertainty principle remains valid in the infrared limit.
    Used as the baseline to combine with the new variance law.

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Works this paper leans on

148 extracted references · 19 canonical work pages · 1 internal anchor

  1. [1]

    We now vary the action, since: δFµν =∂ µδAν −∂ νδAµ,(118) 6 We should note that if the universe admits a dynamical nonlocality scaleE M then we may restore conformal in- variance. we obtain: δSF =− 1 2 Z d4x δFµν F □ E2 M F µν (119) =− Z d4x(∂ µδAν)F □ E2 M F µν,(120) where antisymmetry ofF µν was used in the second step, then integrating by parts gives u...

    2026

  2. [2]

    ¨Uber den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik,

    W. Heisenberg, “ ¨Uber den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik,” Zeitschrift f¨ ur Physik43, 172 (1927)

    1927

  3. [3]

    Zur Quantenmechanik einfacher Be- wegungstypen,

    E. H. Kennard, “Zur Quantenmechanik einfacher Be- wegungstypen,” Zeitschrift f¨ ur Physik44, 326 (1927)

    1927

  4. [4]

    The uncertainty principle,

    H. P. Robertson, “The uncertainty principle,” Phys. Rev.34, 163 (1929)

    1929

  5. [5]

    Zum Heisenbergschen Un- sch¨ arfeprinzip,

    E. Schr¨ odinger, “Zum Heisenbergschen Un- sch¨ arfeprinzip,” Sitzungsber. Preuss. Akad. Wiss. Phys. Math. Kl.19, 296 (1930)

    1930

  6. [6]

    ¨Uber das Gesetz der Energieverteilung im Normalspectrum,

    M. Planck, “ ¨Uber das Gesetz der Energieverteilung im Normalspectrum,”Annalen der Physik309, 553–563 (1901). doi:10.1002/andp.19013090310

    work page doi:10.1002/andp.19013090310 1901

  7. [7]

    ¨Uber einen die Erzeugung und Verwand- lung des Lichtes betreffenden heuristischen Gesicht- spunkt,

    A. Einstein, “ ¨Uber einen die Erzeugung und Verwand- lung des Lichtes betreffenden heuristischen Gesicht- spunkt,”Annalen der Physik17, 132 (1905)

    1905

  8. [8]

    Recherches sur la th´ eorie des quanta,

    L. de Broglie, “Recherches sur la th´ eorie des quanta,” Ph.D. thesis, Universit´ e de Paris (1924)

    1924

  9. [9]

    A Direct Photoelectric Determination of Planck’s ‘h’,

    R. A. Millikan, “A Direct Photoelectric Determination of Planck’s ‘h’,” Phys. Rev.7, 355 (1916)

    1916

  10. [10]

    A Quantum Theory of the Scatter- ing of X-Rays by Light Elements,

    A. H. Compton, “A Quantum Theory of the Scatter- ing of X-Rays by Light Elements,” Phys. Rev.21, 483 (1923)

    1923

  11. [11]

    The Conservation of Photons,

    G. N. Lewis, “The Conservation of Photons,” Nature 118, 874–875 (1926)

    1926

  12. [12]

    Diffraction of Electrons by a Crystal of Nickel,

    C. Davisson and L. H. Germer, “Diffraction of Electrons by a Crystal of Nickel,” Phys. Rev.30, 705 (1927)

    1927

  13. [13]

    Diffraction of Cathode Rays by a Thin Film,

    G. P. Thomson and A. Reid, “Diffraction of Cathode Rays by a Thin Film,” Nature119, 890 (1927)

    1927

  14. [14]

    Localized States for Elementary Systems,

    T. D. Newton and E. P. Wigner, “Localized States for Elementary Systems,” Rev. Mod. Phys.21, 400 (1949)

    1949

  15. [15]

    Instantaneous spreading and Ein- stein causality in quantum theory,

    G. C. Hegerfeldt, “Instantaneous spreading and Ein- stein causality in quantum theory,”Annalen der Physik 7, 716 (1998)

    1998

  16. [16]

    On the Localizability of Quantum Mechanical Systems,

    A. S. Wightman, “On the Localizability of Quantum Mechanical Systems,” Rev. Mod. Phys.34, 845 (1962)

    1962

  17. [17]

    In Defense of Dogma: Why There Cannot Be a Relativistic Quantum Mechanics of (Lo- calizable) Particles,

    D. B. Malament, “In Defense of Dogma: Why There Cannot Be a Relativistic Quantum Mechanics of (Lo- calizable) Particles,” inPerspectives on Quantum Re- ality, edited by R. Clifton (Kluwer, Dordrecht, 1996), pp. 1–10

    1996

  18. [18]

    Covariant position operators, spin, and localizability in relativistic quantum theory,

    G. N. Fleming, “Covariant position operators, spin, and localizability in relativistic quantum theory,” Phys. Rev.137, B188–B197 (1965)

    1965

  19. [19]

    On the localization of particles in rel- ativistic quantum theory,

    M. L. Miller, “On the localization of particles in rel- ativistic quantum theory,” J. Math. Phys.13, 75–77 (1972)

    1972

  20. [20]

    Haag,Local Quantum Physics: Fields, Particles, Algebras(Springer, Berlin, 1992)

    R. Haag,Local Quantum Physics: Fields, Particles, Algebras(Springer, Berlin, 1992)

    1992

  21. [21]

    Algebraic Quantum Field Theory – an Introduction,

    C. J. Fewster and K. Rejzner, “Algebraic Quantum Field Theory – an Introduction,” arXiv:1904.04051 [hep-th]

    arXiv 1904

  22. [22]

    Algebraic Quan- tum Field Theory: Objectives, Methods, and Results,

    D. Buchholz and K. Fredenhagen, “Algebraic Quan- tum Field Theory: Objectives, Methods, and Results,” 20 arXiv:2305.12923 [math-ph]

    arXiv

  23. [23]

    On the Dirac Theory of Spin 1/2 Particles and Its Non-Relativistic Limit,

    L. L. Foldy and S. A. Wouthuysen, “On the Dirac Theory of Spin 1/2 Particles and Its Non-Relativistic Limit,” Phys. Rev.78, 29 (1950)

    1950

  24. [24]

    Asymptotic Locality: Rectifying Newton–Wigner and Foldy–Wouthuysen Localization

    E. J. Thompson, “Asymptotic Locality: Rectifying Newton–Wigner and Foldy–Wouthuysen Localization.” in press Fortschritte der Physik, (2026)

    2026

  25. [25]

    Unsharp Localization and Causality in Rel- ativistic Quantum Theory,

    P. Busch, “Unsharp Localization and Causality in Rel- ativistic Quantum Theory,” J. Phys. A: Math. Gen.32, 6535 (1999)

    1999

  26. [26]

    Impossible measurements on quantum fields,

    R. D. Sorkin, “Impossible measurements on quantum fields,” arXiv:gr-qc/9302018

    Pith/arXiv arXiv

  27. [27]

    Quantum Fields and Lo- cal Measurements,

    C. J. Fewster and R. Verch, “Quantum Fields and Lo- cal Measurements,” Commun. Math. Phys.378, 851 (2020)

    2020

  28. [28]

    Eliminating the ‘Impossible’: Recent Progress on Local Measurement Theory for Quantum Field Theory,

    M. Papageorgiou and D. Fraser, “Eliminating the ‘Impossible’: Recent Progress on Local Measurement Theory for Quantum Field Theory,” arXiv:2307.08524 [quant-ph]

    arXiv

  29. [29]

    Measurement in Quantum Field Theory,

    C. J. Fewster and R. Verch, “Measurement in Quantum Field Theory,” arXiv:2304.13356 [math-ph]

    arXiv

  30. [30]

    Localization in Quantum Field Theory,

    R. Falcone and C. Conti, “Localization in Quantum Field Theory,” arXiv:2312.15348 [hep-th]

    arXiv

  31. [31]

    Erweiterung des Unbes- timmtheitsprinzips f¨ ur die relativistische Quantenthe- orie,

    L. Landau and R. Peierls, “Erweiterung des Unbes- timmtheitsprinzips f¨ ur die relativistische Quantenthe- orie,”Z. Phys.69, 56–69 (1931)

    1931

  32. [32]

    Zur Frage der Messbarkeit der elektromagnetischen Feldgr¨ ossen,

    N. Bohr and L. Rosenfeld, “Zur Frage der Messbarkeit der elektromagnetischen Feldgr¨ ossen,” Mat.-fys. Medd. Dan. Vid. Selsk.12, no.8 (1933)

    1933

  33. [33]

    Field and Charge Measure- ments in Quantum Electrodynamics,

    N. Bohr and L. Rosenfeld, “Field and Charge Measure- ments in Quantum Electrodynamics,” Phys. Rev.78, 794 (1950)

    1950

  34. [34]

    R. E. A. C. Paley and N. Wiener,Fourier Transforms in the Complex Domain(American Mathematical So- ciety, Providence, 1934)

    1934

  35. [35]

    On Gauge-Invariant Entire-Function Regulators and UV Finiteness in NonLocal Quantum Field Theory

    J. W. Moffat and E. J. Thompson, “On Gauge- Invariant entire function Regulators and UV Finiteness in Non-Local Quantum Field Theory,” arXiv:2511.11756v7 [hep-th], Annalen der Physik, 538, 4, e70207, https://doi.org/10.1002/andp.70207, (2026)

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1002/andp.70207 2026

  36. [36]

    Nonlocal Regularizations of Gauge Theories,

    D. Evens, J. W. Moffat, G. Kleppe, and R. P. Woodard, “Nonlocal Regularizations of Gauge Theories,” Phys. Rev. D43, 499 (1991)

    1991

  37. [37]

    J. W. Moffat, Finite nonlocal gauge field the- ory, Phys. Rev. D41, 1177–1184 (1990), doi:10.1103/PhysRevD.41.1177

    work page doi:10.1103/physrevd.41.1177 1990

  38. [38]

    On Covariance for entire function Deformations of Relativistic Field Theories,

    E. J. Thompson, “On Covariance for entire function Deformations of Relativistic Field Theories,” In Press Annalen der Physik, (2026)

    2026

  39. [39]

    The Radiation Theories of Tomonaga, Schwinger, and Feynman,

    F. J. Dyson, “The Radiation Theories of Tomonaga, Schwinger, and Feynman,” Phys. Rev.75, 486–502 (1949). doi:10.1103/PhysRev.75.486

    work page doi:10.1103/physrev.75.486 1949

  40. [40]

    The S Matrix in Quantum Elec- trodynamics,

    F. J. Dyson, “The S Matrix in Quantum Elec- trodynamics,” Phys. Rev.75, 1736–1755 (1949). doi:10.1103/PhysRev.75.1736

    work page doi:10.1103/physrev.75.1736 1949

  41. [41]

    Divergence of Perturbation Theory in Quantum Electrodynamics,

    F. J. Dyson, “Divergence of Perturbation Theory in Quantum Electrodynamics,” Phys. Rev.85, 631–632 (1952). doi:10.1103/PhysRev.85.631

    work page doi:10.1103/physrev.85.631 1952

  42. [42]

    On Field Theories with Non-Localized Action,

    A. Pais and G. E. Uhlenbeck, “On Field Theories with Non-Localized Action,” Phys. Rev.79, 145–165 (1950). doi:10.1103/PhysRev.79.145

    work page doi:10.1103/physrev.79.145 1950

  43. [43]

    A Relativistic Cut-Off for Classical Electrodynamics,

    R. P. Feynman, “A Relativistic Cut-Off for Classical Electrodynamics,” Phys. Rev.74, 939–946 (1948)

    1948

  44. [44]

    Space-Time Approach to Quantum Electrodynamics,

    R. P. Feynman, “Space-Time Approach to Quantum Electrodynamics,” Phys. Rev.76, 769–789 (1949)

    1949

  45. [45]

    Regularized vacuum expectation val- ues in quantum field theory,

    J. W. Moffat, “Regularized vacuum expectation val- ues in quantum field theory,”Nucl. Phys.16, 304–312 (1960)

    1960

  46. [46]

    Quantum Theory of Non-Local Fields. Part I. Free Fields,

    H. Yukawa, “Quantum Theory of Non-Local Fields. Part I. Free Fields,” Phys. Rev.77, 219–226 (1950). doi:10.1103/PhysRev.77.219

    work page doi:10.1103/physrev.77.219 1950

  47. [47]

    Quantum Theory of Non-Local Fields. Part II. Irreducible Fields and Their In- teraction,

    H. Yukawa, “Quantum Theory of Non-Local Fields. Part II. Irreducible Fields and Their In- teraction,” Phys. Rev.80, 1047–1052 (1950). doi:10.1103/PhysRev.80.1047

    work page doi:10.1103/physrev.80.1047 1950

  48. [48]

    A Convergent Expression for the Magnetic Moment of the Neutron,

    D. Rivier and E. C. G. St¨ uckelberg, “A Convergent Expression for the Magnetic Moment of the Neutron,” Phys. Rev.74, 218 (1948). doi:10.1103/PhysRev.74.218

    work page doi:10.1103/physrev.74.218 1948

  49. [49]

    The Prob- lem of Vacuum Polarization,

    H. Umezawa, J. Yukawa, and E. Yamada, “The Prob- lem of Vacuum Polarization,” Prog. Theor. Phys.3, 317–318 (1948). doi:10.1143/PTP/3.3.317

    work page doi:10.1143/ptp/3.3.317 1948

  50. [50]

    Remarks on the Non-Local Electrodynam- ics,

    J. Rayski, “Remarks on the Non-Local Electrodynam- ics,” Proc. Roy. Soc. Lond. A206, 575–583 (1951). doi:10.1098/rspa.1951.0090

    work page doi:10.1098/rspa.1951.0090 1951

  51. [51]

    Non-local and Non-linear Field The- ories,

    D. I. Blokhintsev, “Non-local and Non-linear Field The- ories,” Fortschr. Phys.6, 246–269 (1958). [also Usp. Fiz. Nauk61, 137–159 (1957)]

    1958

  52. [52]

    Nonlocal Quantum Theory of the Scalar Field,

    G. V. Efimov, “Nonlocal Quantum Theory of the Scalar Field,” Commun. Math. Phys.5, 42–56 (1967). doi:10.1007/BF01646357

    work page doi:10.1007/bf01646357 1967

  53. [53]

    Essentially Nonlinear Interaction Lagrangians and Nonlocalized Quantum Field Theory,

    G. V. Efimov, “Essentially Nonlinear Interaction Lagrangians and Nonlocalized Quantum Field Theory,” Theor. Math. Phys.2, 26–40 (1970). doi:10.1007/BF01028853

    work page doi:10.1007/bf01028853 1970

  54. [54]

    A Proof of the Unitarity ofS-Matrix in a Nonlocal Quantum Field Theory,

    V. A. Alebastrov and G. V. Efimov, “A Proof of the Unitarity ofS-Matrix in a Nonlocal Quantum Field Theory,” Commun. Math. Phys.31, 1–24 (1973). doi:10.1007/BF01645588

    work page doi:10.1007/bf01645588 1973

  55. [55]

    Causality in Quantum Field Theory with Nonlocal Interaction,

    V. A. Alebastrov and G. V. Efimov, “Causality in Quantum Field Theory with Nonlocal Interaction,” Commun. Math. Phys.38, 11–28 (1974)

    1974

  56. [56]

    ¨Uber irreversible Strahlungsvorg¨ ange,

    M. Planck, “ ¨Uber irreversible Strahlungsvorg¨ ange,” Annalen der Physik306, no. 1, 69–122 (1900). doi:10.1002/andp.19003060105

    work page doi:10.1002/andp.19003060105 1900

  57. [57]

    ¨Uber irreversible Strahlungsvorg¨ ange. Erste Mittheilung,

    M. Planck, “ ¨Uber irreversible Strahlungsvorg¨ ange. Erste Mittheilung,” Sitzungsberichte der K¨ oniglich Preussischen Akademie der Wissenschaften zu Berlin, 57–68 (1897)

  58. [58]

    ¨Uber irreversible Strahlungsvorg¨ ange. Zweite Mittheilung,

    M. Planck, “ ¨Uber irreversible Strahlungsvorg¨ ange. Zweite Mittheilung,” Sitzungsberichte der K¨ oniglich Preussischen Akademie der Wissenschaften zu Berlin, 715–717 (1897)

  59. [59]

    ¨Uber irreversible Strahlungsvorg¨ ange. Dritte Mittheilung,

    M. Planck, “ ¨Uber irreversible Strahlungsvorg¨ ange. Dritte Mittheilung,” Sitzungsberichte der K¨ oniglich Preussischen Akademie der Wissenschaften zu Berlin, 1122–1145 (1897)

  60. [60]

    ¨Uber irreversible Strahlungsvorg¨ ange. Vierte Mittheilung,

    M. Planck, “ ¨Uber irreversible Strahlungsvorg¨ ange. Vierte Mittheilung,” Sitzungsberichte der K¨ oniglich Preussischen Akademie der Wissenschaften zu Berlin, 449–476 (1898)

  61. [61]

    ¨Uber irreversible Strahlungsvorg¨ ange. F¨ unfte Mittheilung,

    M. Planck, “ ¨Uber irreversible Strahlungsvorg¨ ange. F¨ unfte Mittheilung,” Sitzungsberichte der K¨ oniglich Preussischen Akademie der Wissenschaften zu Berlin, 440–480 (1899)

  62. [62]

    On the Physical Units of Nature,

    G. J. Stoney, “On the Physical Units of Nature,”Philo- sophical Magazine, Series 5,11, no. 69, 381–390 (1881). doi:10.1080/14786448108627031

    work page doi:10.1080/14786448108627031

  63. [63]

    Natural Systems of Units: To the Cen- tenary Anniversary of the Planck System,

    K. A. Tomilin, “Natural Systems of Units: To the Cen- tenary Anniversary of the Planck System,” inProceed- ings of the XXII Workshop on High Energy Physics and Field Theory, Protvino, Russia, 287–296 (1999)

    1999

  64. [64]

    Natural Units Before Planck,

    J. D. Barrow, “Natural Units Before Planck,”Quar- terly Journal of the Royal Astronomical Society24, 24–26 (1983). 21

    1983

  65. [65]

    On Absolute Units, I: Choices,

    F. Wilczek, “On Absolute Units, I: Choices,”Physics Today58, no. 10, 12–13 (2005). doi:10.1063/1.2138392

    work page doi:10.1063/1.2138392 2005

  66. [66]

    Planck-Scale Physics: Facts and Be- liefs,

    D. Meschini, “Planck-Scale Physics: Facts and Be- liefs,”Foundations of Science12, 277–294 (2007). doi:10.1007/s10699-006-9102-3

    work page doi:10.1007/s10699-006-9102-3 2007

  67. [67]

    Sulla teoria degli urti tra particelle ele- mentari,

    G. Wataghin, “Sulla teoria degli urti tra particelle ele- mentari,” Z. Phys.88, 92–98 (1934)

    1934

  68. [68]

    Quantum Theory of Non-Local Fields. Part I. Free Fields,

    H. Yukawa, “Quantum Theory of Non-Local Fields. Part I. Free Fields,” Phys. Rev.77, 219–226 (1950)

    1950

  69. [69]

    Causalit´ e et structure de la matriceS,

    E. C. G. Stueckelberg and D. Rivier, “Causalit´ e et structure de la matriceS,” Helv. Phys. Acta23, 215– 222 (1950)

    1950

  70. [70]

    On the Formulation of Quantum Field Theory with Nonlocal Interaction,

    D. I. Blokhintsev, “On the Formulation of Quantum Field Theory with Nonlocal Interaction,” Sov. Phys. JETP17, 116 (1947)

    1947

  71. [71]

    Quantization of Non-Local Field The- ory,

    G. V. Efimov, “Quantization of Non-Local Field The- ory,” Int. J. Theor. Phys.10, 19–37 (1974)

    1974

  72. [72]

    Nonlocal Gauge Theories,

    N. V. Krasnikov, “Nonlocal Gauge Theories,” Theor. Math. Phys.73, 1184–1190 (1987)

    1987

  73. [73]

    The Convergent Nonlocal Gravita- tion,

    Yu. V. Kuz’min, “The Convergent Nonlocal Gravita- tion,” Sov. J. Nucl. Phys.50, 1011–1014 (1989)

    1989

  74. [74]

    Finite Nonlocal Gauge Field Theory,

    J. W. Moffat, “Finite Nonlocal Gauge Field Theory,” Phys. Rev. D41, 1177–1184 (1990)

    1990

  75. [75]

    Nonlocal Yang-Mills,

    G. Kleppe and R. P. Woodard, “Nonlocal Yang-Mills,” Nucl. Phys. B388, 81–112 (1992)

    1992

  76. [76]

    Superrenormalizable Gauge and Gravitational Theories,

    E. T. Tomboulis, “Superrenormalizable Gauge and Gravitational Theories,” arXiv:hep-th/9702146

    Pith/arXiv arXiv

  77. [77]

    Bouncing Universes in String-Inspired Gravity,

    T. Biswas, A. Mazumdar, and W. Siegel, “Bouncing Universes in String-Inspired Gravity,” JCAP03, 009 (2006), arXiv:hep-th/0508194

    Pith/arXiv arXiv 2006

  78. [78]

    Non-Local SFT Tachyon and Cosmol- ogy,

    A. S. Koshelev, “Non-Local SFT Tachyon and Cosmol- ogy,” JHEP04, 029 (2007), arXiv:hep-th/0701103

    Pith/arXiv arXiv 2007

  79. [79]

    Towards Singularity- and Ghost-Free Theories of Gravity,

    T. Biswas, E. Gerwick, T. Koivisto, and A. Mazum- dar, “Towards Singularity- and Ghost-Free Theories of Gravity,” Phys. Rev. Lett.108, 031101 (2012), arXiv:1110.5249 [gr-qc]

    Pith/arXiv arXiv 2012

  80. [80]

    Super-Renormalizable Quantum Grav- ity,

    L. Modesto, “Super-Renormalizable Quantum Grav- ity,” Phys. Rev. D86, 044005 (2012), arXiv:1107.2403 [hep-th]

    Pith/arXiv arXiv 2012

Showing first 80 references.