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arxiv: 2506.09126 · v2 · pith:2UF6MJQLnew · submitted 2025-06-10 · ✦ hep-th · hep-ph· math-ph· math.DG· math.MP

Chiral interactions of fermions and massive gauge fields in Kaluza-Klein models

Joao Baptista This is my paper

Pith reviewed 2026-05-22 01:07 UTC · model grok-4.3

classification ✦ hep-th hep-phmath-phmath.DGmath.MP
keywords Kaluza-Klein theorychiral fermionsnon-Killing vector fieldsmassive gauge bosonsRiemannian submersionsspin geometryextra dimensionsweak interactions
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The pith

Gauge fields from non-Killing vector fields on the internal space yield chiral interactions with fermions in Kaluza-Klein models.

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

In Kaluza-Klein theory, gauge fields arise from the higher-dimensional metric on M4 times K. The traditional approach ties them to Killing vector fields on K, which limits chiral couplings. This paper shows that non-Killing vector fields produce massive bosons that can be arbitrarily light, mix fermions of different masses, and couple asymmetrically to left- and right-handed fermions. These features circumvent no-go arguments and offer a geometric source for chiral interactions like the weak force. A reader cares because it suggests higher-dimensional geometry can naturally generate the chirality needed for realistic particle models.

Core claim

Gauge fields linked to non-Killing vector fields on the internal space K produce 4D massive yet arbitrarily light bosons that mix fermions of different masses and couple asymmetrically to left- and right-handed fermions. These properties are not easily satisfied by gauge fields linked to internal isometries. The results hold for a general compact manifold K and are illustrated on the two-sphere and two-torus using the language of spin geometry and Riemannian submersions. This offers a geometric source of chiral interactions with fermions.

What carries the argument

The 4D gauge fields obtained from the higher-dimensional metric components corresponding to non-Killing vector fields on K, which modify the horizontal distribution in the Riemannian submersion.

If this is right

  • These gauge fields can have masses made arbitrarily small through suitable choice of the internal metric.
  • They allow mixing of fermions with different four-dimensional masses.
  • They produce asymmetric couplings to left- and right-handed fermions.
  • This provides a potential geometric realization of the weak force in Kaluza-Klein theory.

Where Pith is reading between the lines

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

  • This mechanism could extend to building Kaluza-Klein models that embed both gravity and chiral gauge sectors geometrically.
  • Similar reductions might apply to other compact spaces to resolve chirality problems in higher-dimensional theories.
  • The resulting light massive bosons suggest possible signatures in precision measurements if the internal geometry scale is suitable.

Load-bearing premise

The higher-dimensional metric on M4 times K can be chosen so that the horizontal distribution defined by the non-Killing fields remains compatible with a Riemannian submersion structure while preserving the spinor bundle and Dirac operator properties.

What would settle it

A specific example where including a non-Killing vector field leads to either heavy gauge bosons that cannot be made light or to purely vector-like fermion couplings would show the claim does not hold in general.

read the original abstract

In Kaluza-Klein theory, gauge fields on $M_4$ arise as components of a higher-dimensional metric defined on $M_4 \times K$. The traditional expectation is that all the gauge fields of the Standard Model are linked to exact Killing vector fields on the internal space. This paper questions that assumption and investigates the properties of 4D gauge fields linked to non-Killing fields on $K$. It is shown that they have massive yet arbitrarily light bosons; they can mix fermions with different masses; and they can have asymmetric couplings to left- and right-handed fermions. None of these properties is easily satisfied by gauge fields linked to internal isometries. So the massive gauge fields produced in this manner circumvent traditional no-go arguments and offer a geometric source of chiral interactions with fermions. This may help to model the weak force within the Kaluza-Klein framework. Technically, the paper uses the language of spin geometry and Riemannian submersions. It studies the higher-dimensional Dirac operator with non-trivial background metrics. The results are derived for a general $K$. They are illustrated explicitly in the simpler cases where $K$ is the two-sphere and the two-torus.

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

2 major / 2 minor

Summary. The paper claims that in Kaluza-Klein reductions on M4 × K, gauge fields arising from non-Killing vector fields on the internal space K yield 4D massive but arbitrarily light bosons. These fields mix fermions of different masses and couple asymmetrically to left- and right-handed fermions, circumventing traditional no-go theorems for chiral interactions. The results are obtained from the higher-dimensional Dirac operator on Riemannian submersions, derived for general K and illustrated on S2 and T2.

Significance. If the central derivation holds without hidden post-hoc choices in the metric ansatz, the work provides a geometric mechanism for chiral gauge interactions in KK theory that does not rely on internal isometries. This could be relevant for modeling the weak sector. The use of spin geometry and the general-K treatment are positive features; the explicit S2/T2 examples help make the claims concrete.

major comments (2)
  1. [General-K derivation and Riemannian submersion setup] The central claim that non-Killing fields produce asymmetric chiral couplings rests on the assumption that a background metric on M4 × K can always be chosen so the horizontal distribution remains orthogonal to the vertical fibers while preserving a Riemannian submersion structure and the required properties of the spinor bundle and Dirac operator. This is used throughout the general-K derivation; an explicit check that the spin connection does not restore symmetric couplings would strengthen the result.
  2. [Dirac operator reduction] In the reduction of the Dirac operator, the mass-mixing terms and left/right asymmetry are stated to follow from the geometry of non-Killing fields. It would be useful to see the precise decomposition of the horizontal lift and the resulting 4D interaction Lagrangian to confirm that no additional assumptions beyond the submersion metric are introduced.
minor comments (2)
  1. [Introduction] The abstract and introduction could more clearly distinguish the new mechanism from prior work on non-isometric reductions; a short comparison paragraph would help readers.
  2. [Examples on S2 and T2] Notation for the horizontal and vertical distributions is introduced without an explicit diagram or coordinate chart in the general case; adding one would improve readability of the S2/T2 examples.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment of our work and the constructive suggestions for strengthening the presentation. We address each major comment below and will revise the manuscript to incorporate the requested clarifications and explicit checks.

read point-by-point responses

  1. Referee: [General-K derivation and Riemannian submersion setup] The central claim that non-Killing fields produce asymmetric chiral couplings rests on the assumption that a background metric on M4 × K can always be chosen so the horizontal distribution remains orthogonal to the vertical fibers while preserving a Riemannian submersion structure and the required properties of the spinor bundle and Dirac operator. This is used throughout the general-K derivation; an explicit check that the spin connection does not restore symmetric couplings would strengthen the result.

    Authors: The Riemannian submersion structure, with orthogonal horizontal and vertical distributions, is the standard geometric ansatz for Kaluza-Klein reductions on M4 × K and is not an additional post-hoc choice; it is required for the higher-dimensional metric to induce a well-defined 4D metric and gauge fields. The spinor bundle and Dirac operator are constructed compatibly with this structure using the standard spin geometry of Riemannian submersions. The chiral asymmetry originates from the failure of the non-Killing vector fields to preserve the horizontal distribution under Lie transport. We agree that an explicit verification would improve clarity, and the revised manuscript will include a short calculation in the general-K section confirming that spin-connection contributions do not cancel the left/right asymmetry. revision: yes

  2. Referee: [Dirac operator reduction] In the reduction of the Dirac operator, the mass-mixing terms and left/right asymmetry are stated to follow from the geometry of non-Killing fields. It would be useful to see the precise decomposition of the horizontal lift and the resulting 4D interaction Lagrangian to confirm that no additional assumptions beyond the submersion metric are introduced.

    Authors: The reduction proceeds from the higher-dimensional Dirac operator on the Riemannian submersion, where the horizontal lift of the non-Killing vector fields enters the covariant derivative. The resulting mass-mixing and chiral-asymmetric terms arise directly from the non-vanishing vertical components of the Lie brackets and the associated connection forms; no further assumptions are imposed beyond the submersion metric. To make this transparent, the revised version will add an explicit decomposition of the horizontal lift together with the projected 4D interaction Lagrangian in an expanded section or appendix. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation follows from standard spin geometry and Riemannian submersion

full rationale

The paper begins with the conventional Kaluza-Klein metric ansatz on M4 × K and the Dirac operator on the product manifold. Properties of massive light bosons, fermion mass mixing, and asymmetric left/right couplings are obtained by analyzing the horizontal distribution induced by non-Killing vector fields while preserving the Riemannian submersion and spinor bundle structure. No equation reduces to its own input by definition, no fitted parameter is relabeled as a prediction, and no load-bearing step relies on a self-citation chain or imported uniqueness theorem. The general-K derivation and S2/T2 examples remain self-contained against the geometric assumptions stated in the setup.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claims rest on the existence of a Riemannian submersion structure compatible with the spinor bundle when the metric is deformed away from pure isometries. No new particles are postulated; the bosons emerge from the geometry. The scale that makes the bosons light is a free choice of the internal metric rather than a fitted parameter.

free parameters (1)
  • internal metric scale
    Chosen to make the resulting 4D boson mass arbitrarily small while remaining massive; appears as a continuous parameter in the general-K construction.
axioms (2)
  • domain assumption The higher-dimensional manifold admits a Riemannian submersion whose horizontal distribution is spanned by the non-Killing vector fields.
    Invoked to define the 4D gauge fields and to reduce the Dirac operator.
  • standard math The spin structure on M4 × K descends consistently to the 4D effective theory.
    Required for the chiral fermion couplings to be well-defined.

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Works this paper leans on

74 extracted references · 74 canonical work pages

  1. [1]

    Witten: Search for a realistic Kaluza-Klein theory, Nucl

    E. Witten: Search for a realistic Kaluza-Klein theory, Nucl. Phys. B186 (1981), 412–428

    work page 1981

  2. [2]

    M. Duff, B. Nilsson and C. Pope: Kaluza-Klein supergravity, Phys. Reports 130 (1986), 1–142

    work page 1986

  3. [3]

    Bailin and A

    D. Bailin and A. Love: Kaluza-Klein theories, Rep. Prog. Phys.50 (1987), 1087–1170

    work page 1987

  4. [4]

    Coquereaux and A

    R. Coquereaux and A. Jadczyk: Riemannian geometry, fiber bundles, Kaluza-Klein theories and all that.... , World Scientific Publishing, 1988

    work page 1988

  5. [5]

    Castellani, P

    L. Castellani, P. Fr´ e and R. D’Auria: Supergravity and superstrings: a geometric perspective, Vol. 2, Part five, World Scientific Publishing, 1991

    work page 1991

  6. [6]

    Overduin and P

    J. Overduin and P. Wesson: Kaluza-Klein gravity, Phys. Reports 283 (1997), 303– 380

    work page 1997

  7. [7]

    Englert and R

    F. Englert and R. Brout: Phys. Rev. Lett. 13 (1964), 321

    work page 1964

  8. [8]

    Guralnik, C

    G. Guralnik, C. Hagen and T. Kibble: Phys. Rev. Lett. 13 (1964), 585

    work page 1964

  9. [9]

    Higgs: Phys

    P. Higgs: Phys. Lett. 12 (1964), 132

    work page 1964

  10. [10]

    Baptista: Internal symmetries in Kaluza-Klein models, J

    J. Baptista: Internal symmetries in Kaluza-Klein models, J. High Energ. Phys. 2024 (2024), 178

    work page 2024

  11. [11]

    Besse: Einstein manifolds, Classics in Mathematics, Springer-Verlag, 1987

    A. Besse: Einstein manifolds, Classics in Mathematics, Springer-Verlag, 1987

    work page 1987

  12. [12]

    Manton: Fermions and parity violation in dimensional reduction schemes, Nucl

    N. Manton: Fermions and parity violation in dimensional reduction schemes, Nucl. Phys. B193 (1981), 502–516

    work page 1981

  13. [13]

    Chapline and R

    G. Chapline and R. Slansky: Dimensional reduction and flavor chirality, Nucl. Phys. B209 (1982), 461–483

    work page 1982

  14. [14]

    Witten: Fermion quantum numbers in Kaluza-Klein theory, in Shelter Island II, Proc

    E. Witten: Fermion quantum numbers in Kaluza-Klein theory, in Shelter Island II, Proc. of the 1983 Shelter Island conference , MIT Press, 1985

    work page 1983

  15. [15]

    Wetterich: Massless spinors in more than four dimensions, Nucl

    C. Wetterich: Massless spinors in more than four dimensions, Nucl. Phys. B211 (1983), 177–188

    work page 1983

  16. [16]

    Wetterich: Dimensional reduction of Weyl, Majorana and Majorana-Weyl spinors, Nucl

    C. Wetterich: Dimensional reduction of Weyl, Majorana and Majorana-Weyl spinors, Nucl. Phys. B222 (1983), 20–44. 44

    work page 1983

  17. [17]

    Atiyah and F

    M. Atiyah and F. Hirzebruch: Spin-manifolds and group actions, in Essays on To- pology and Related Subjects, Springer-Verlag, 1970, 18–28

    work page 1970

  18. [18]

    Chapline and N

    G. Chapline and N. Manton: The geometrical significance of certain Higgs potentials: an approach to grand unification, Nucl. Phys. B184 (1981), 391–405

    work page 1981

  19. [19]

    Randjbar-Daemi, A

    S. Randjbar-Daemi, A. Salam and J. Strathdee: Spontaneous compactification in six-dimensional Einstein-Maxwell theory, Nucl. Phys. B214 (1983), 491–512

    work page 1983

  20. [20]

    Frampton and K

    P. Frampton and K. Yamamoto: Unitary flavor unification through higher dimen- sionality, Phys. Rev. Lett. 52 (1984), 2016–2018

    work page 1984

  21. [21]

    Cremades, L

    D. Cremades, L. Ibanez and F. Marchesano: Computing Yukawa couplings from magnetized extra dimensions, J. High Energ. Phys. 05 (2004), 079

    work page 2004

  22. [22]

    Wetterich: Dimensional reduction of fermions in generalized gravity, Nucl

    C. Wetterich: Dimensional reduction of fermions in generalized gravity, Nucl. Phys. B242 (1984), 473–502

    work page 1984

  23. [23]

    Weinberg: Quasi-riemannian theories of gravitation in more than four dimensions, Phys

    S. Weinberg: Quasi-riemannian theories of gravitation in more than four dimensions, Phys. Letters 138B (1984), 47–51

    work page 1984

  24. [24]

    Moffat: Chiral fermions in non-Riemannian Kaluza–Klein theory, J

    J. Moffat: Chiral fermions in non-Riemannian Kaluza–Klein theory, J. Math. Phys. 26 (1985), 528–531

    work page 1985

  25. [25]

    Neville: Torsion and chiral fermions in Kaluza-Klein theories, Phys

    D. Neville: Torsion and chiral fermions in Kaluza-Klein theories, Phys. Rev. D33 (1986), 363–369

    work page 1986

  26. [26]

    Tchrakian: Comment on ‘Torsion and chiral fermions in Kaluza-Klein theories’, Phys

    D. Tchrakian: Comment on ‘Torsion and chiral fermions in Kaluza-Klein theories’, Phys. Rev. D34 (1986), 3930–3931

    work page 1986

  27. [27]

    Dixon, J

    L. Dixon, J. Harvey, C. Vafa and E. Witten: Strings on orbifolds, Nucl. Phys. B261 (1985), 678–686

    work page 1985

  28. [28]

    Dixon, J

    L. Dixon, J. Harvey, C. Vafa and E. Witten: Strings on orbifolds (II), Nucl. Phys. B274 (1986), 285–314

    work page 1986

  29. [29]

    Pomarol and M

    A. Pomarol and M. Quir´ os: The standard model from extra dimensions,Phys. Letters 438B (1998), 255–260

    work page 1998

  30. [30]

    Dienes, E

    K. Dienes, E. Dudas and T. Gherghetta: Grand unification at intermediate mass scales through extra dimensions, Nuclear Phys. B537 (1999), 47–108

    work page 1999

  31. [31]

    Cheng, B

    H. Cheng, B. Dobrescu and C. Hill: Electroweak symmetry breaking and extra di- mensions, Nuclear Phys. B589 (2000), 249–268. 45

    work page 2000

  32. [32]

    Georgi, A

    H. Georgi, A. Grant and G. Hailu: Chiral fermions, orbifolds, scalars and fat branes, Phys. Rev. D63 (2001), 064027

    work page 2001

  33. [33]

    Sundrum: TASI 2004 lectures: to the fifth dimension and back, in Physics in D >= 4, Proc

    R. Sundrum: TASI 2004 lectures: to the fifth dimension and back, in Physics in D >= 4, Proc. Theor. Adv. St. Inst. Elementary Particle Physics (2005), 585–630

    work page 2004

  34. [34]

    Cs´ aki, J

    C. Cs´ aki, J. Hubisz and P. Meade: TASI lectures on electroweak symmetry breaking from extra dimensions, in Physics inD >= 4, Proc. Theor. Adv. St. Inst. Elementary Particle Physics (2005), 703–776

    work page 2005

  35. [35]

    Bourguignon: A mathematician’s visit to Kaluza-Klein theory, Rend

    J. Bourguignon: A mathematician’s visit to Kaluza-Klein theory, Rend. Sem. Mat. Univ. Politec. Torino (1989), 143–163

    work page 1989

  36. [36]

    Kaluza: Zum Unitt¨ asproblem in der Physik, Sitzungsber

    T. Kaluza: Zum Unitt¨ asproblem in der Physik, Sitzungsber. Preuss. Akad. Wiss. Berlin Math. Phys. K1 (1921), 966–972

    work page 1921

  37. [37]

    Klein: Quantentheorie und f¨ unfdimensionale Relativit¨ atstheorie,Zeitschrift Phys

    O. Klein: Quantentheorie und f¨ unfdimensionale Relativit¨ atstheorie,Zeitschrift Phys. 37 (1926), 895–906

    work page 1926

  38. [38]

    Einstein and P

    A. Einstein and P. Bergmann: On a generalization of Kaluza’s theory of electricity, Annals Math. 39 (1938), 683–701

    work page 1938

  39. [39]

    Jordan: Relativistische Gravitationstheorie mit variabler Gravitationskonstante, Naturwissenschaften 33 (1946), 250–251

    P. Jordan: Relativistische Gravitationstheorie mit variabler Gravitationskonstante, Naturwissenschaften 33 (1946), 250–251

    work page 1946

  40. [40]

    Thiry: Les ´ equations de la th´ eorie unitaire de Kaluza, Comptes Rendus Acad

    Y. Thiry: Les ´ equations de la th´ eorie unitaire de Kaluza, Comptes Rendus Acad. Sci. Paris 226 (1948), 216–218

    work page 1948

  41. [41]

    DeWitt: Dynamical theory of groups and fields, in Lectures at 1963 Les Houches School, Gordon and Breach, 1964, 585–820

    B. DeWitt: Dynamical theory of groups and fields, in Lectures at 1963 Les Houches School, Gordon and Breach, 1964, 585–820

    work page 1963

  42. [42]

    Kerner: Generalization of the Kaluza-Klein theory for an arbitrary non-abelian gauge group, Ann

    R. Kerner: Generalization of the Kaluza-Klein theory for an arbitrary non-abelian gauge group, Ann. Inst. H. Poincar´ e9 (1968), 143–152

    work page 1968

  43. [43]

    Cho: Higher-dimensional unifications of gravitation and gauge theories, J

    Y. Cho: Higher-dimensional unifications of gravitation and gauge theories, J. Math. Phys. 16 (1975), 2029–2035

    work page 1975

  44. [44]

    Baptista: Test particles in Kaluza-Klein models, Class

    J. Baptista: Test particles in Kaluza-Klein models, Class. Quantum Grav. 42 (2025), 045007

    work page 2025

  45. [45]

    O’Neill: The fundamental equations of a submersion,Michigan Math

    B. O’Neill: The fundamental equations of a submersion,Michigan Math. J. 13 (1966), 459–469. 46

    work page 1966

  46. [46]

    Ehresmann: Les connexions infinit´ esimales dans un espace fibr´ e diff´ erentiable, Colloque de Topologie (1950), 29–55

    C. Ehresmann: Les connexions infinit´ esimales dans un espace fibr´ e diff´ erentiable, Colloque de Topologie (1950), 29–55

    work page 1950

  47. [47]

    Hermann: A sufficient condition that a mapping of Riemannian manifolds be a fibre bundle, Proc

    R. Hermann: A sufficient condition that a mapping of Riemannian manifolds be a fibre bundle, Proc. Amer. Math. Soc. 11 (1960), 236–242

    work page 1960

  48. [48]

    Falcitelli, S

    M. Falcitelli, S. Ianus and A. Pastore: Riemannian submersions and related topics , World Scientific Publishing, 2004

    work page 2004

  49. [49]

    Moroianu, Op´ erateur de Dirac et submersions riemanniennes

    A. Moroianu, Op´ erateur de Dirac et submersions riemanniennes. Ph.D. thesis. ´Ecole Polytechnique, Paris, 1996

    work page 1996

  50. [50]

    Loubeau and R

    E. Loubeau and R. Slobodeanu: A characterization of Dirac morphisms, Commun. Math. Phys. 288 (2009), 1089–1102

    work page 2009

  51. [51]

    Reynolds, On conformal submersions and manifolds with exceptional structure groups, PhD Thesis, University of Edinburgh, 2011

    P. Reynolds, On conformal submersions and manifolds with exceptional structure groups, PhD Thesis, University of Edinburgh, 2011

    work page 2011

  52. [52]

    Kosmann: D´ eriv´ ees de Lie des spineurs,Ann

    Y. Kosmann: D´ eriv´ ees de Lie des spineurs,Ann. di Matematica Pura ed Applicata 91 (1971), 317–395

    work page 1971

  53. [53]

    Benn and R

    I. Benn and R. Tucker: An introduction to spinors and geometry with applications in physics, IOP Publishing, 1987

    work page 1987

  54. [54]

    Friedrich: Zur Abh¨ angigkeit des Dirac-Operators von der Spin-Struktur, Coll

    T. Friedrich: Zur Abh¨ angigkeit des Dirac-Operators von der Spin-Struktur, Coll. Math. 48 (1984), 57–62

    work page 1984

  55. [55]

    Ginoux: The Dirac spectrum, Springer, 2009

    N. Ginoux: The Dirac spectrum, Springer, 2009

    work page 2009

  56. [56]

    Lawson and M

    H. Lawson and M. Michelsohn: Spin geometry, Princeton Univ. Press, 1989

    work page 1989

  57. [57]

    Bourguignon, O

    J. Bourguignon, O. Hijazi, J. Milhorat, A. Moroianu and S. Moroianu: A spinorial approach to Riemannian and conformal geometry , European Mathematical Society, 2015

    work page 2015

  58. [58]

    Dolan: On the elimination of Pauli couplings in Kaluza-Klein theories using tor- sion, Phys

    B. Dolan: On the elimination of Pauli couplings in Kaluza-Klein theories using tor- sion, Phys. Letters 159B (1985), 279–283

    work page 1985

  59. [59]

    Lichnerowicz: Spineurs harmoniques, Comptes rendus Acad

    A. Lichnerowicz: Spineurs harmoniques, Comptes rendus Acad. Sc. Paris, groupe 1 257 (1963), 7–9

    work page 1963

  60. [60]

    Baum: Spin-Strukturen und Dirac-Operatoren ¨ uber pseudoriemannschen Mannig- faltigkeiten, B

    H. Baum: Spin-Strukturen und Dirac-Operatoren ¨ uber pseudoriemannschen Mannig- faltigkeiten, B. G. Teubner, 1981. 47

    work page 1981

  61. [61]

    Hamilton: Mathematical gauge theory: with applications to the Standard Model of particle physics , Universitext, Springer International Publishing, 2017

    M. Hamilton: Mathematical gauge theory: with applications to the Standard Model of particle physics , Universitext, Springer International Publishing, 2017

    work page 2017

  62. [62]

    Baez and J

    J. Baez and J. Huerta: The algebra of grand unified theories, Bull. Amer. Math. Soc. 47 (2010), 483–552

    work page 2010

  63. [63]

    Imayoshi and M

    Y. Imayoshi and M. Taniguchi: An introduction to Teichm¨ uller spaces, Springer- Verlag, 1992

    work page 1992

  64. [64]

    B¨ ar: The Dirac operator on space forms of positive curvature,J

    C. B¨ ar: The Dirac operator on space forms of positive curvature,J. Math. Soc. Japan 48 (1996), 69–83

    work page 1996

  65. [65]

    Gracia-Bond´ ıa, J

    J. Gracia-Bond´ ıa, J. V´ arilly and H. Figueroa:Elements of noncommutative geometry, Birkh¨ auser, 2001

    work page 2001

  66. [66]

    Silva: Lectures in symplectic geometry, Springer, 2008

    A. Silva: Lectures in symplectic geometry, Springer, 2008

    work page 2008

  67. [67]

    Jackson: Classical electrodynamics, 3rd ed., John Wiley & Sons, 1998

    J. Jackson: Classical electrodynamics, 3rd ed., John Wiley & Sons, 1998

    work page 1998

  68. [68]

    Rios and E

    P. Rios and E. Straume: Symbol correspondences for spin systems, Birkh¨ auser, 2014

    work page 2014

  69. [69]

    Freidel and K

    L. Freidel and K. Krasnov: The fuzzy sphere ⋆-product and spin networks, J. Math. Phys. 43 (2002), 1737–1754

    work page 2002

  70. [70]

    Torres del Castillo: 3-D Spinors, spin-weighted functions and their applications , Springer, 2003

    G. Torres del Castillo: 3-D Spinors, spin-weighted functions and their applications , Springer, 2003

    work page 2003

  71. [71]

    Newman and R

    E. Newman and R. Penrose: Note on the Bondi-Metzner-Sachs group, J. Math. Phys. 7 (1966), 863–870

    work page 1966

  72. [72]

    Goldberg, A

    J. Goldberg, A. Macfarlane, E. Newman, F. Rohrlich and E. Sudarshan: Spin- s spherical harmonics and ð, J. Math. Phys. 8 (1967), 2155–2161

    work page 1967

  73. [73]

    Benn and J

    I. Benn and J. Kress: First-order Dirac symmetry operators, Class. Quantum Grav. 21 (2004), 427–431

    work page 2004

  74. [74]

    Cariglia, P

    M. Cariglia, P. Krtouˇ s and D. Kubizˇ n´ ak: Commuting symmetry operators of the Dirac equation, Killing-Yano and Schouten-Nijenhuis brackets, Phys. Rev. D84 (2011), 024004. 48

    work page 2011