Skip to main content
SpringerLink
Log in
Book cover

Geometrodynamics of Gauge Fields pp 227–259Cite as

Spinor Bundles

  • Chapter
  • First Online:

  • 1498 Accesses

Part of the book series: Mathematical Physics Studies ((MPST))

Abstract

The intended unification of electromagnetic, weak, strong, and gravitational interactions within the geometric concept of gauge invariance is still flawed by one essential shortcoming that cannot be disregarded.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    (Weyl 1924, p. 609), “eine dynamische Theorie der Materie am aussichtsreichsten: die Materie ein felderregendes Agens, das Feld ein extensives Medium, das die Wirkungen von Körper zu Körper überträgt.”

  2. 2.

    Fermions with a higher half-integer spin are not to be considered here.

  3. 3.

    In order to make possible a generalization of the Dirac equation in the higher-dimensional spaces of the Kaluza–Klein theory, the formalism is again developed for arbitrary dimensions (Brauer & Weyl 1935). For this purpose, the work of Kerner (1980) is rather useful.

  4. 4.

    This restriction is to be explained later on.

  5. 5.

    In compliance with our previous conventions, we denote the covariant derivative in a vector bundle by the same symbol D, since it is always possible to infer from the field in question which representation of the structure group or its covering group is meant.

  6. 6.

    A corresponding conservation law is also valid for the axial vector current \(j_{n+1}:=i\overline{\psi }{}^{*}\gamma \gamma ^{n+1}\psi \) of a Dirac field without mass. However, a quantum-theoretic treatment by means of a functional integral, due to an anomaly, results in (cf., for instance, Jackiw 1977)

    $$ \langle - \infty | D \tau _{A}| \infty \rangle ^{\overset{(+)}{-}} = \frac{1}{(4.) 8\pi ^{2}} \mathrm {Tr} (\varOmega ^{g} \wedge \varOmega ^{g(*)}) . $$

    Thus in quantum field theory, the conservation of the axial current is violated by topological contributions of the Pontryagin class, and this in dependence on the asymptotic helicity states concerning \(\gamma ^{n+1}\). This is the physical content of the celebrated Atiyah–Singer index theorem (Römer 1981a, b; Eguchi et al. 1980).

  7. 7.

    At an earlier stage, this was described as a “mixed” theory, since the affine connection and the tetrad field occur as independent variables in the variation procedure.

  8. 8.

    The Heisenberg–Pauli–Weyl spinor equation within two dimensions serves as the starting point for the quantum-field-theoretic Thirring model (1958). This equation has solutions not only exhibiting quark confinement (Chang et al. 1975; Hortaçsu 1977), but also with particle characteristics (Yamamoto 1977; Rañada & Rañada 1984).

  9. 9.

    As far as we know, this term occurs for the first time in Kaluza (1921), although in a differing context. Theoretically, it was coined by Born & Infeld (1934, 1935) and, as far as a quantum-theoretic model is concerned, by Finkelstein  (1949).

  10. 10.

    According to the highly speculative notions of Jehle  (1977, 1981), this should be asserted almost literally.

  11. 11.

    ...ferner der Grund, warum wir an der Energie oder trägen Masse eines zusammengesetzten Körpers die nicht auflösbare Energie seiner letzten materiellen Elementarbestandteile der auflösbaren Energie ihrer wechselseitigen Bindung gegenüberstellen.” (Weyl 1924, p. 592).

  12. 12.

    Hermann Weyl’s still valid admonitory statement is to be remembered here:

    The formulation of Dirac’s theory of the electron in the frame of general relativity has to its credit one feature that should be appreciated even by the atomic physicist who feels safe in ignoring the role of gravitation in the building up of the elementary particles: its explanation of the quantum-mechanical principle of “gauge invariance” that connects Dirac’s \(\psi \) with the electromagnetic potentials

    (Weyl 1950).

  13. 13.

    This parameter appears as fundamental length \(\ell \) in both (11.4.3) and the Ansatz (11.4.15).

References

  • Alvarez A, Soler M (1983) Energetic stability criterion for a nonlinear spinorial model. Phys Rev Lett 50(17):1230

    Article  ADS  Google Scholar 

  • Anderson D, Brown E Jr, Peterson FP (1966) Spin cobordism. Bull Am Math Soc 72(2):256–260

    Article  MathSciNet  MATH  Google Scholar 

  • Anderson DL (1971) Stability of time-dependent particlelike solutions in nonlinear field theories, II. J Math Phys 12:945–952

    Google Scholar 

  • Asselmeyer-Maluga T, Brans CH (2015) How to include fermions into general relativity by exotic smoothness. Gen Relativ Gravit 47(3):30

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Atiyah MF, Bott R, Shapiro A (1964) Clifford modules. Topology 3:3–38

    Article  MathSciNet  MATH  Google Scholar 

  • Avis S, Isham C (1980) Generalized spin structures on four dimensional space-times. Commun Math Phys 72(2):103–118

    Article  ADS  MathSciNet  Google Scholar 

  • Bargmann V (1932) Bemerkungen zur allgemein-relativistischen Fassung der Quantentheorie, Sitzungsber preuss. Akad Wiss Phys Math Kl.XXV, 346

    Google Scholar 

  • Barut A (1980) Stable particles as building blocks of matter. Surv High Energy Phys 1(2):113–140

    Article  ADS  Google Scholar 

  • Barut A, Kraus J (1977) Form-factor corrections to superpositronium and short-distance behavior of the magnetic moment of the electron. Phys Rev D 16(1):161

    Article  ADS  Google Scholar 

  • Bjorken JD, Drell SD (1964) Relativistic quantum mechanics. McGraw-Hill, San Francisco

    MATH  Google Scholar 

  • Born M (1934) On the quantum theory of the electromagnetic field. In: Proceedings of the Royal Society of London. Series A, vol 143, pp 410–437

    Google Scholar 

  • Born M, Infeld L (1934) Foundations of the new field theory. In: Proceedings of the Royal Society of London. Series A, vol 144(852), pp 425–451

    Google Scholar 

  • Born M, Infeld L (1935) On the quantization of the new field theory, II. In: Proceedings of the Royal Society of London. Series A, mathematical and physical sciences, vol 150(869), pp 141–166

    Google Scholar 

  • Börner G, Dürr H (1970) Nonlinear spinor theory in de Sitter space, Il. Nuovo Cimento A 66(1):185–201

    Article  ADS  MathSciNet  Google Scholar 

  • Brauer R, Weyl H (1935) Spinors in n dimensions. Am J Math 57(2):425–449

    Article  MathSciNet  MATH  Google Scholar 

  • Braunss G (1964) Nichtlineare Spinorgleichungen und affiner Zusammenhang. Z für Naturforschung A 19(7–8):825–827

    ADS  MathSciNet  Google Scholar 

  • Braunss G (1965) On the rôle of the group O(4) of local complex orthogonal transformations in a nonlinear theory of elementary particles. Z für Naturforschung A 20(5):649–655

    Article  ADS  MATH  Google Scholar 

  • Brill DR, Wheeler JA (1957) Interaction of neutrinos and gravitational fields. Rev Mod Phys 29(3):465

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Callan CG, Gross DJ (1975) Quantum perturbation theory of solitons. Nucl Phys B 93(1):29–55

    Article  ADS  Google Scholar 

  • Cartan É (1966) The theory of spinors. Hermann, Paris

    Google Scholar 

  • Chandrasekhar S (1976) The solution of Dirac’s equation in Kerr geometry. In: Proceedings of the Royal Society of London a: mathematical, physical and engineering sciences, vol 349, pp 571–575

    Google Scholar 

  • Chang S-J, Ellis S, Lee B (1975) Chiral confinement: an exact solution of the massive Thirring model. Phys Rev D 11(12):3572

    Article  ADS  Google Scholar 

  • Chodos A, Jaffe R, Johnson K, Thorn CB, Weisskopf V (1974) New extended model of hadrons. Phys Rev D 9(12):3471

    Article  ADS  MathSciNet  Google Scholar 

  • Datta B (1971) Spinor fields in general relativity: Noether’s theorem and the conservation laws in Riemann–Cartan space, II: generalized field equations and application to the Dirac field. Il Nuovo Cimento B Ser 11(6b(1)):1–16

    Google Scholar 

  • Deppert W, Mielke EW (1979) Localized solutions of the nonlinear Heisenberg–Klein–Gordon equation: in flat and exterior Schwarzschild space-time. Phys Rev D 20(6):1303

    Google Scholar 

  • Derrick G, Kay-Kong W (1968) Particle motion and interaction in nonlinear field theories. J Math Phys 9(2):232–240

    Article  ADS  Google Scholar 

  • De Broglie L (1960) Non-linear Wave Mechanics, Elsevier, Amsterdam

    Google Scholar 

  • Dirac PAM (1928) The quantum theory of the electron. Part I, II. In: Proceedings of the Royal Society of London A: mathematical, physical and engineering sciences, vol 118, pp 351–361

    Google Scholar 

  • Dirac PAM (1962) An extensible model of the electron. In: Proceedings of the Royal Society of London A: mathematical, physical and engineering sciences, vol. 268, pp 57–67

    Google Scholar 

  • Dürr H (1973) Poincaré gauge invariant spinor theory and the gravitational field, II. Gen Relat Gravit 4(1):29–52

    Article  ADS  Google Scholar 

  • Eguchi T (1977) Collective phenomena and renormalization of nonlinear spinor theories. In: Stump DR, Weingarten DH (eds) Quark confinement and field theory. Wiley, New York, p 13

    Google Scholar 

  • Eguchi T, Gilkey PB, Hanson AJ (1980) Gravitation, gauge theories and differential geometry. Phys Rep 66(6):213–393

    Article  ADS  MathSciNet  Google Scholar 

  • Einstein A (1919) Spielen Gravitationsfelder im Aufbau der materiellen Elementarteilchen eine wesentliche Rolle?, Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften (Berlin), 1:349–356

    Google Scholar 

  • Einstein A (1928) Riemann-Geometrie mit Aufrechterhaltung des Begriffs des Fernparallelismus; Neue Möglichkeit für eine einheitliche Feldtheorie von Gravitation und Elektrizität. Sitzungsber. Preuss. Akad. Wiss. (Berlin), Phys. math. Kl. 217: 224

    Google Scholar 

  • Einstein S, Finkelstein R (1977) A class of solutions of the Dirac equation in the Kerr–Newman space. J Math Phys 18(4):664–671

    Article  ADS  Google Scholar 

  • Finkelstein D, Rubinstein J (1968) Connection between spin, statistics, and kinks. J Math Phys 9(11):1762–1779

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Finkelstein R (1949) On the quantization of a unitary field theory. Phys Rev 75(7):1079

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Finkelstein R (1960) Spacetime of the elementary particles. J Math Phys 1(5):440–451

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Finkelstein R (1961a) Elementary interactions in spaces with torsion. Ann Phys 15(2):223–249

    Article  ADS  MATH  Google Scholar 

  • Finkelstein R (1961b) Spinor fields in spaces with torsion. Ann Phys 12(2):200–221

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Finkelstein R, Fronsdal C, Kaus P (1956) Nonlinear spinor field. Phys Rev 103(5):1571

    Article  ADS  MATH  Google Scholar 

  • Finkelstein R, LeLevier R, Ruderman M (1951) Nonlinear spinor fields. Phys Rev 83(2):326

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Finkelstein R, Ramsay W (1962) The strong couplings in a space with torsion. Ann Phys 17(3):379–403

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Flügge S (1971) Practical quantum mechanics. Springer, Heidelberg

    Book  MATH  Google Scholar 

  • Fock V (1929) Geometrisierung der Diracschen Theorie des Elektrons. Zeitschrift für Physik 57(3–4):261–277

    Google Scholar 

  • Friedberg R, Lee T (1977) Fermion-field nontopological solitons. Phys Rev D 15(6):1694

    Article  ADS  Google Scholar 

  • Garcia L, Rañada AF (1980) A classical model of the nucleon. Prog Theor Phys 64(2):671–693

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Gel’fand IM, Minlos RA, Shapiro ZY (1963) Representations of the rotation and Lorentz groups and their applications. Oxford, New York

    MATH  Google Scholar 

  • Gell-Mann M, Ne’eman Y (1964) The eightfold way, Benjamin, New York

    Google Scholar 

  • Geroch R (1968) Spinor structure of space-times in general relativity, I. J Math Phys 9(11):1739–1744

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Gürsey F (1957) Relativistic kinematics of a classical point particle in spinor form. Il Nuovo Cimento 5:784

    Google Scholar 

  • Hamilton MJ, Das A (1977) On the combined Dirac–Einstein–Maxwell field equations. J Math Phys 18(10):2026–2030

    Article  ADS  Google Scholar 

  • Hehl F, Datta B (1971) Nonlinear spinor equation and asymmetric connection in general relativity. J Math Phys 12(7):1334–1339

    Article  ADS  MathSciNet  Google Scholar 

  • Hehl FW (1974) Spin and torsion in general relativity II: geometry and field equations. Gen Relativ Gravit 5(5):491–516

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Hehl FW, Lord EA, Ne’eman Y (1978) Hyper momentum in hadron dynamics and in gravitation. Phys. Rev. D 17:428

    Google Scholar 

  • Hehl FW, Von der Heyde P, Kerlick GD, Nester JM (1976) General relativity with spin and torsion: Foundations and prospects. Rev Mod Phys 48(3):393

    Article  ADS  MathSciNet  Google Scholar 

  • Heisenberg W (1932) Über den Bau der Atomkerne, I, II. Zeitschr f Phys 77(1):1–11

    Article  ADS  MATH  Google Scholar 

  • Heisenberg W (1957) Quantum theory of fields and elementary particles. Reviews of Modern Physics 29(3):269

    Google Scholar 

  • Heisenberg W (1967) Einführung in die einheitliche Feldtheorie der Elementarteilchen. Hirzel, Stuttgart

    Google Scholar 

  • Heisenberg W (1974) The unified field theory of elementary particles: some recent advances. Naturwissenschaften 61(1):1–5

    Article  ADS  Google Scholar 

  • Heisenberg W, Euler H (1936) Folgerungen aus der Diracschen Theorie des Positrons. Zeitschrift für Physik 98(11–12):714–732

    Article  ADS  MATH  Google Scholar 

  • Helgason S (2001) Differential geometry and symmetric spaces, vol 341, American Mathematical Society

    Google Scholar 

  • Hortaçsu M (1977) Confined solutions of the Thirring model coupled to a Schwinger field. Phys Rev D 15(10):3003

    Article  ADS  Google Scholar 

  • Inomata A (1978) Effect of the self-induced torsion of the Dirac sources on gravitational singularities. Phys Rev D 18(10):3552

    Article  ADS  Google Scholar 

  • Isham CJ (1978) Spinor fields in four dimensional space-time. In: Proceedings of the Royal Society of London A: mathematical, physical and engineering sciences, vol 364, pp 591–599

    Google Scholar 

  • Ivanenko D (1938) Bemerkungen zur Theorie der Wechselwirkung. Physikalische Zeitschrift der Sowjetunion 13(2)

    Google Scholar 

  • Ivanenko D (1957) Non-linear generalizations of the field theory and the constant of minimal length. Il Nuovo Cimento Ser 10 6(1):349–355

    Article  MathSciNet  Google Scholar 

  • Ivanenko DD (1958) Bemerkungen zu einer einheitlichen nichtlinearen Theorie der Materie. In: Max-Planck-Festschrift, Hrsg B, Kockel MW, Papapetrou A, Deutscher Verlag der Wissenschaften, Berlin

    Google Scholar 

  • Jackiw R (1977) Quantum meaning of classical field theory. Rev Mod Phys 49(3):681

    Article  ADS  MathSciNet  Google Scholar 

  • Jehle H (1977) Electron-muon puzzle and the electromagnetic coupling constant. Phys Rev D 15(12):3727

    Article  ADS  Google Scholar 

  • Jehle H (1981) Flux quantization and electromagnetic properties of particles. Phys Lett B 104(3):203–206

    Article  ADS  MathSciNet  Google Scholar 

  • Joos H (1979) Introduction to quark confinement in QCD. In: Quarks and leptons as fundamental particles. Springer, pp 407–483

    Google Scholar 

  • Jost R (1965) The general theory of quantized fields. Am Math Soc Providence, Rhode Island

    Google Scholar 

  • Joutei HB, Chakrabarti A (1979) Kerr–Schild geometry, spinors, and instantons. Phys Rev D 19(2):457

    Article  ADS  MathSciNet  Google Scholar 

  • Kalinowski MW (1984) Spinor fields in non-abelian Klein–Kaluza theories. Int J Theor. Phys. 23(2):131–146

    Article  Google Scholar 

  • Kaluza T (1921) Zum Unitätsproblem der Physik, Sitzungsber. Preuss Akad Wiss Berlin (Math Phys) (966972), 45

    Google Scholar 

  • Kazmierczak M (2010) Nontrivial realization of the space-time translations in the theory of quantum fields. Preprint arXiv:1009.3042 [hep-th]

  • Kerner R (1980) Spinors on fibre bundles and their use in invariant models. In: Differential geometrical methods in Mathematical Physics. Springer, pp 349–358

    Google Scholar 

  • Kerr RP (1963) Gravitational field of a spinning mass as an example of algebraically special metrics. Phys Rev Lett 11(5):237

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Klauder J, Wheeler JA (1957) On the question of a neutrino analog to electric charge. Rev Mod Phys 29(3):516

    Article  ADS  MathSciNet  Google Scholar 

  • Kokkedee JJ (1969) The quark model. Benjamin, New York

    Google Scholar 

  • Korteweg DJ, De Vries G (1895) On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Philosophical Magazine, 39(240):422–443

    Google Scholar 

  • Kumar A, Nisichenko V, Rybakov YP (1979) Stability of charged solitons. Int J Theor Phys 18(6):425–432

    Article  MathSciNet  Google Scholar 

  • Luehr C, Rosenbaum M (1974) Spinor connections in general relativity. J Math Phys 15(7):1120–1137

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Luehr C, Rosenbaum M (1984) Unified gauge theory for electromagnetism and gravitation based on twistor bundles. J Math Phys 25(2):380–387

    Article  ADS  MathSciNet  Google Scholar 

  • Mackey G (1968) Induced representations of groups and quantum. Benjamin, New York

    MATH  Google Scholar 

  • Makhankov V (1978) Dynamics of classical solitons (in non-integrable systems). Phys Rep 35(1):1–128

    Article  ADS  MathSciNet  Google Scholar 

  • Marciano W, Pagels H (1978) Quantum chromodynamics. Phys Rep 36(3):137–276

    Article  ADS  Google Scholar 

  • Mathieu P, Morris T (1985) Energetic instability of classical nontopological solitons with spinorial conserved charge. Phys Lett B 155(3):156–158

    Article  ADS  MathSciNet  Google Scholar 

  • Menius A Jr, Rosen N (1942) Calculations on classical field theory. Phys Rev 62(9–10):436

    Article  ADS  Google Scholar 

  • Mie G (1912) Grundlagen einer Theorie der Materie (Erste und zweite Mitteilung). Ann Physik (Leipzig) 344(11):1–40

    Google Scholar 

  • Mie G (1913) Grundlagen einer Theorie der Materie (dritte Mitteilung). Ann Phys 345(1):1–66

    Article  MATH  Google Scholar 

  • Mielke E (1979) Mass formula for solitons with quantized charge, Lettere al Nuovo Cimento 25(14), 424–428

    Google Scholar 

  • Mielke EW (1977a) Conformal changes of metrics and the initial-value problem of general relativity. Gen Relativ Gravit 8(5):321–345

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Mielke EW (1977b) Outline of a new geometrodynamical model of extended baryons. Phys Rev Lett 39(9):530

    Article  ADS  MathSciNet  Google Scholar 

  • Mielke EW (1977c) Quantenfeldtheorie im de Sitter–Raum. Fortschr Phys 25(1–12):401–457

    Article  MathSciNet  Google Scholar 

  • Mielke EW (1978) Note on localized solutions of a nonlinear Klein–Gordon equation related to Riemannian geometry. Phys Rev D 18(12):4525

    Article  ADS  MathSciNet  Google Scholar 

  • Mielke EW (1979) Mass formular for solitons with quantized charge, Lettere al Nuovo Cimento 25:424

    Google Scholar 

  • Mielke EW (1980) On exact solutions of the nonlinear Heisenberg-Klein-Gordon equation in a space-time of constant spacelike curvature. J Math Phys 21(3):543–546

    Article  ADS  Google Scholar 

  • Mielke EW (1981a) Gauge-theoretical foundation of color geometrodynamics. In: Doebner HD (ed) Differential geometric methods in mathematical physics, proceedings of the 1978 Clausthal meeting Lecture Notes in Physics, vol 139. Springer, Berlin, pp 135–151

    Google Scholar 

  • Mielke EW (1981b) Toward exact solutions of the nonlinear Heisenberg–Pauli–Weyl spinor equation. J Math Phys 22(9):2034–2039

    Article  ADS  Google Scholar 

  • Mielke EW (1981c) Outline of a nonlinear, relativistic quantum mechanics of extended particles. Fortschr. Phys. 30:551

    Google Scholar 

  • Mielke EW, Scherzer R (1981) Geon-type solutions of the nonlinear Heisenberg–Klein–Gordon equation. Phys Rev D 24(8):2111

    Article  ADS  Google Scholar 

  • Milnor J (1963) Spin structures on manifolds. L’Enseigment Mathematique 9:198

    ADS  MathSciNet  MATH  Google Scholar 

  • Milnor J (1965) On the Stiefel–Whitney numbers of complex manifolds and of spin manifolds. Topology 3(3):223–230

    Article  MathSciNet  MATH  Google Scholar 

  • Milnor JW, Stasheff JD (1974) Characteristic classes, vol 76. Princeton University Press, Princeton

    Book  MATH  Google Scholar 

  • Morris T (1980) Is the electron a soliton. Hadronic J 3(5):1360–1392

    MathSciNet  Google Scholar 

  • Nester JM (1977) Effective equivalence of the Einstein–Cartan and Einstein theories of gravity. Phys Rev D 16(8):2395

    Article  ADS  MathSciNet  Google Scholar 

  • Newman ET, Couch E, Chinnapared K, Exton A, Prakash A, Torrence R (1965) Metric of a rotating, charged mass. J Math Phys 6(6):918–919

    Article  ADS  MathSciNet  Google Scholar 

  • Pati JC, Salam A (1973) Unified lepton-hadron symmetry and a gauge theory of the basic interactions. Phys Rev D 8(4):1240

    Article  ADS  Google Scholar 

  • Pati JC, Salam A (1974) Lepton number as the fourth “color”. Phys Rev D 10(1):275

    Article  ADS  Google Scholar 

  • Peres A (1962) Spinor fields in generally covariant theories. Il Nuovo Cimento 24:389–452

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Rañada A (1978) S wave kinks of the Dirac–Weyl equation. J Phys A Math Gen 11(2):341

    Article  ADS  MathSciNet  Google Scholar 

  • Rañada AF (1985) A geometrical interpretation of the Pauli exclusion principle in classical field theory. Found Phys 15(1):89–100

    Article  ADS  MathSciNet  Google Scholar 

  • Rañada AF (1986) Baryonlike and mesonlike solitons in a one-dimensional Dirac model of extended particles. Phys. Rev. D 33: 1714

    Google Scholar 

  • Rañada AF, Rañada MF (1984) Nonlinear model of c-number confined Dirac quarks. Phys Rev D 29(5):985

    Article  ADS  MathSciNet  Google Scholar 

  • Rañada AF, Rañada MF (1983) A model of extended nonlinear Dirac quarks Physica D 9:251

    Google Scholar 

  • Rodičhev V (1961) Twisted space and nonlinear field equations. Sov Phys JETP 13(5)

    Google Scholar 

  • Römer H (1981a) Atiyah-Singer index theorem and quantum field theory. In: Doebner HD (ed) Differential geometric methods in mathematical physics. Proceedings of the 1978 Clausthal meeting, Lecture Notes in Physics, Vol 139 Springer, Berlin, p 167–211

    Google Scholar 

  • Römer H (1981b) A universality property of axial anomalies. Phys Lett B 101(1):55–60

    Article  ADS  MathSciNet  Google Scholar 

  • Rose M (1961) Relativistic Electron Theory, Wiley, New York

    Google Scholar 

  • Rose ME, Furry W (1961) Relativistic electron theory. Am J Phys 29(12):866–866

    Article  ADS  Google Scholar 

  • Rosen G (1965) Particlelike solutions to nonlinear scalar wave theories. J Math Phys 6(8):1269–1272

    Article  ADS  MathSciNet  Google Scholar 

  • Rosen N (1939) A field theory of elementary particles. Phys Rev 55(1):94

    Article  ADS  MATH  Google Scholar 

  • Rosen N, Rosenstock HB (1952) The force between particles in a nonlinear field theory. Phys Rev 85(2):257

    Article  ADS  MATH  Google Scholar 

  • Salam A (1977) Gauge unification of basic forces, particularly of gravitation with strong interactions. Ann N Y Acad Sci 294(1):12–36

    Article  ADS  Google Scholar 

  • Schrödinger E (1932) Diracsches Elektron im Schwerefeld I. Sitzber Preuss Akad Wiss Phys Math, Klasse XI 105

    Google Scholar 

  • Schunck FE, Mielke EW (2003) Topical Review: General relativistic boson stars. Class Quantum Grav 20(20):R301–R356

    Google Scholar 

  • Report on waves. Report of the British Association for the Advancement of Science, pp 311–390

    Google Scholar 

  • Soler M (1970) Classical, stable, nonlinear spinor field with positive rest energy. Phys Rev D 1(10):2766

    Article  ADS  Google Scholar 

  • Stumpf H (1980) Functional quantum theory of the nonlinear spinor field as a lepton–hadron model with quark-confinement. Z für Naturforschung A 35(10):1104–1107

    Google Scholar 

  • Stumpf H (1981) Confinement and radical unification in functional quantum theory of the nonlinear spinorfield. Z für Naturforschung A 36(8):785–788

    Article  ADS  Google Scholar 

  • Takahashi K (1979a) Color vanishing solitons from nonlinear Dirac equation. Prog Theor Phys 61(4):1251–1254

    Article  ADS  Google Scholar 

  • Takahashi K (1979b) Soliton solutions of nonlinear Dirac equations. J Math Phys 20(6):1232–1238

    Article  ADS  MATH  Google Scholar 

  • Taylor JC (1979) Gauge theories of weak interactions. Cambridge University Press, Cambridge

    Google Scholar 

  • Tetrode H (1928) Allgemein-relativistische Quantentheorie des Elektrons. Z für Physik 50(5–6):336–346

    Article  ADS  MATH  Google Scholar 

  • Teukolsky SA (1972) Rotating black holes: separable wave equations for gravitational and electromagnetic perturbations. Phys Rev Lett 29(16):1114

    Article  ADS  Google Scholar 

  • Teukolsky SA (1973) Perturbations of a rotating black hole, I. Fundamental equations for gravitational, electromagnetic, and neutrino-field perturbations. Astrophys J 185:635–648

    Article  ADS  Google Scholar 

  • Thirring W (1958) On interacting spinor fields in one dimension. Il Nuovo Cimento 9:1007

    Google Scholar 

  • Trautman A (1970) Fibre bundles associated with space-time. Rep Math Phys 1(1):29–62

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Trautman A (1972) On the Einstein–Cartan equations I-IV. Bull Acad Pol Sci Ser Sci Math Astron Phys, 20, 185, 503, 895, 21, 345

    Google Scholar 

  • Trautman A (2008) Connections and the Dirac operator on spinor bundles. J Geom Phys 58(2):238–252

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Van Der Merwe PDT (1979) Two-point function of nonlinear spinor theory. Phys Rev D 19:1746–1756

    Article  ADS  Google Scholar 

  • Vázquez L (1977) Localised solutions of a non-linear spinor field. J Phys A Math Gen 10(8):1361

    Article  ADS  MathSciNet  Google Scholar 

  • Werle J (1977) Dirac spinor solitons or bags. Phys Lett B 71(2):357–359

    Article  ADS  MathSciNet  Google Scholar 

  • Weyl H (1924) Was ist Materie? Naturwissenschaften 12(30):604–611

    Article  ADS  MATH  Google Scholar 

  • Weyl H (1929a) Elektron und Gravitation, I. Z für Physik 56(5):330–352

    Article  ADS  MATH  Google Scholar 

  • Weyl H (1929b) Gravitation and the electron. Proc Natl Acad Sci 15(4):323–334

    Article  ADS  MATH  Google Scholar 

  • Weyl H (1950) A remark on the coupling of gravitation and electron. Phys Rev 77(5):699

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Wheeler JA (1955) Geons. Phys Rev 97(2):511

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Wheeler JA (1962) Geometrodynamics. Academic Press, New York

    Google Scholar 

  • Wheeler J (1968) Superspace and the nature of quantum geometrodynamics. In: Zabusky Norman J (ed) Topics in Nonlinear Physics. Springer, New York, pp 615–724

    Google Scholar 

  • Wigner E (1929) Eine Bemerkung zu Einsteins neuer Formulierung des allgemeinen Relativitätsprinzips, Zeitschr f Physik 53:592

    Google Scholar 

  • Yamamoto H (1977) Spinor soliton as an elementary particle. Prog Theor Phys 58(3):1014–1023

    Article  ADS  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Física, CBI Iztapalapa, Universidad Autónoma Metropolitana, Mexico City, México

    Eckehard W. Mielke

Authors
  1. Eckehard W. Mielke
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eckehard W. Mielke .

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Mielke, E.W. (2017). Spinor Bundles. In: Geometrodynamics of Gauge Fields. Mathematical Physics Studies. Springer, Cham. https://doi.org/10.1007/978-3-319-29734-7_11

Download citation

Publish with us

Policies and ethics

Search

Navigation