Skip to main content

Introduction to the fiber-bundle approach to gauge theories

  • Chapter
  • First Online:
Book cover Fiber Bundle Techniques in Gauge Theories

Part of the book series: Lecture Notes in Physics ((LNP,volume 67))

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

Access this chapter

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  • Arnol'd, V. I.: Matematicheskie metody klassicheskoi mekhaniki (Mathematical methods of classical mechanics), Nauka, Moscow, 1974.

    Google Scholar 

  • Bourbaki, N: Variétés différentielles et analytiques, Hermann, Paris, §§ 1–7, 1967, §§ 8–15, 1971.

    Google Scholar 

  • Chern, S-S.: The Geometry of Characteristic Classes, in Proc. of the 13-th Biennial Seminar, Canadian Math. Congress, Montreal, pp. 1–40.

    Google Scholar 

  • Chevalley, C.: Theory of Lie Groups, vol. I, Princeton U. Press, Princeton, 1946.

    Google Scholar 

  • De Rham, G.: Variétés différentiables, Hermann, Paris, 1955.

    Google Scholar 

  • Dieudonné, J.: Treatise on Analysis, Academic Press, N. Y., vol III, 1972, vol. IV, 1974 (vols. V, VI in French).

    Google Scholar 

  • Greub, W., S. Halperin, and R. Vanstone: Connections, Curvature, and Cohomology, Academic Press, N. Y., Vol. I., 1972, Vol. II, 1973, Vol. 111, 1974.

    Google Scholar 

  • Hawking, S. W., and G. F. R. Ellis: The Large-Scale Structure of the Universe, Cambridge University Press, 1973.

    Google Scholar 

  • Hermann, R.: Vector Bundles in Mathematical Physics, Benjamin, N.Y. and a large number of other books by the same author.

    Google Scholar 

  • Hirsch, M.: Differential Topology, Springer Verlag, N. Y., 1976.

    Google Scholar 

  • Hirzebruch, F.: Topological Methods in Algebraic Geometry, 3-rd ed. Springer Verlag, Berlin, 1966.

    Google Scholar 

  • Husemoller, D.: Fibre Bundles, 2-nd ed., Springer Verlag, N. Y. 1975.

    Google Scholar 

  • Kobayasi, S., and K. Nomizu, Foundations of Differential Geometry, Wiley, N. Y., Vol. I, 1963, Vol, 11,1969.

    Google Scholar 

  • Landau, L. D., and E. M. Lifshitz: Teoriya polya (Classical Field Theory) 6-th ed., Nauka, Moscow, 1973; Engl. Transl, Pergamon Press.

    Google Scholar 

  • Lichnerowicz, A.: Theorie globale des connexions et des groupes d'holonomie, Ed. Cremonese, Roma, 1955.

    Google Scholar 

  • Lightman, A. P., W. H. Press, R. H. Price, and S. A. Teukolsky:Problem Book in Relativity and Gravitation, Princeton U. P.,1975.

    Google Scholar 

  • Mayer, M. E.: Cîmpuri cuantice si particlee elementare (Quantized fields and elementary particles), Ed. Tehnica, Bucharest, 1969.

    Google Scholar 

  • Misner, C. W., R. S. Thorne, and J. A. Wheeler, Gravitation, Freeman, San Francisco, 1973.

    Google Scholar 

  • Pauli, W.: Die allgemeinen Prinzipien der Quantenmechanik, Handb. d. Physik, 2-nd ed., Bd. V, T. 1, Springer Verlag, Berlin, 1958.

    Google Scholar 

  • Spivak, M.:A Comprehensive Introduction to Differential Geometry, 3 vols., Publish or Perish, Waltham, 1971. Also: Calculus on Manifolds, Benjamin, 1969.

    Google Scholar 

  • Steenrod, N.; The Topology of Fibre Bundles, Princeton Univ. Press, 1951.

    Google Scholar 

  • Sternberg, S: Lectures on Differential Geometry, Prentice-Hall, Englewood Cliffs, 1963.

    Google Scholar 

  • Sulanke, R., and P. Wintgen, Differentialgeometrie und Faserbündel, Birkhäuser Verlag, Basel 1972.

    Google Scholar 

  • Vranceanu, G.: Lecons de geometrie differentielle, 3 vols. Ed. de l'Academie de la R. P. Roumaine, Bucharest/Gauthier-Villars, Paris, 1964 (also available in German and the Romanian original).

    Google Scholar 

  • Yang, C. N.: Lecture notes on gauge theories and fiber bundles (approximate title), University of Hawaii, 1975.

    Google Scholar 

  • Milnor, J. W., and J. D. Stasheff: Characteristic Classes, Princeton Univ. Press, 1974.

    Google Scholar 

  • E. Abers and B. W. Lee, Gauge Theories, Phys. Reports, 9, No l, 1–141 (1973).

    Google Scholar 

  • W. Ambrose and I. M. Singer, A Theorem on Holonomy, Trans. Amer. Math. Soc. 75, 428–443 (1953).

    Google Scholar 

  • R. L. Arnowitt and S. I. Fickler, Phys. Rev. 127, 1821 (1962).

    Google Scholar 

  • A. A. Belavin, A. M. Polyakov, A. S. Schwartz and S. Tyupkin, Pseudoparticle Solutions of the Yang-Mills Equations, Phys. Lett. 59B, 85 (1976).

    Google Scholar 

  • J. Bernstein, Spontaneous Symmetry Breaking, Gauge Theories, Higgs Mechanism and All That, Rev. Mod. Phys. 46, 1 (1974).

    Google Scholar 

  • K. Bleuler, Helv. Phys. Acta 23, 567 (1950).

    Google Scholar 

  • R. Balian, J. Drouffe and C. Itzykson, Gauge Fields on a Lattice, Phys. Rev. D10, 3376 (1974); D11, 2098 (1975).

    Google Scholar 

  • R. Bott and S. S. Chern, Hermitian Vector Bundles and the Equidistribution of Zeroes of their Holomorphic Sections, Acta Mathem. 114, 71–112 (1965).

    Google Scholar 

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

    Google Scholar 

  • C. Callan, R. F. Dashen and D. J. Gross, The Structure of the Gauge Theory Vacuum, Phys. Lett. 63B, 334 (1976); A Mechanism for Quark Confinement, Princeton Preprint COO-2220-94, 1977.

    Google Scholar 

  • S. Coleman, Classical Lumps and their Quantum Descendants, Erice Lecture Notes, 1975.

    Google Scholar 

  • B. S. De Witt. Phys. Rev. 162, 1195, 1239 (1967).

    Google Scholar 

  • S. Doplicher, R. Haag and J. E. Roberts, Fields, Observables, and Gauge Transformations, I, II, Commun. Math. Phys. 13, 1–23 (1969); 15, 173–200 (1971). Local Observables and Particle Statistics, I, Ibid. 23, 199–230 (1971); S. Doplicher and J. E. Roberts, Fields, Statistics and Nonabelian Gauge Groups, Ibid. 28, 331–348 (1972). Lectures by S. Doplicher at various conferences.

    Google Scholar 

  • L. D. Faddeev, Lecture at the Mathematical Physics Conference, Moscow, 1972; published in the Trudy Matem. Inst. im. Steklova (in Russian) 1975.

    Google Scholar 

  • L. D. Faddeev and V. S. Popov, Feynman Diagrams for the Yang-Mills Field, Phys. Lett. 25B, 29 (1967); Kiev Preprint, 1967; L. D. Faddeev, Teor. Matem. Fiz, l, 3 (1969) [Theor. Math. Phys. 1, 1 (1969)].

    Google Scholar 

  • E. Fermi, Rev. Mod. Phys. 4, 87 (1932).

    Google Scholar 

  • R. P. Feynman, Acta Physica Polonica 26, 697 (1963).

    Google Scholar 

  • P. L. Garcia, Gauge Algebras, Curvature and Symplectic Structure to appear in J, Differ. Geom.; Reducibility of the Symplectic Structure of Classical Fields, Proc. of the Symposium on Differential-Geometrical Methods in Physics, Bonn, 1975, K. Bleuler and A. Reetz, eds., Springer Lecture Notes in Math., 1977.

    Google Scholar 

  • L. Gårding and A. S. Wightman, Fields as operator-Valued Distributions in Relativistic Quantum Field Theory, Arkiv för Fysik 28, 129–184 (1964).

    Google Scholar 

  • M. Gell-Mann and S. L. Glashow, Ann. Phys. (N. Y.) 15, 437 (1961).

    Google Scholar 

  • J. Glimm and A. Jaffe, Quark Trapping for Lattice U(1) Gauge Fields, Harvard Preprint, 1977. Functional Integral Methods in Quantum Field Theory, Cargese 1976 Lectures, to be published.

    Google Scholar 

  • S. N. Gupta, Proc. Phys. Soc. (Lond.) 61A, 68 (1950).

    Google Scholar 

  • G. S. Guralnik, C. R. Hagen and T. W. B. Kibble, Phys. Rev. Lett. 13, 585 (1964).

    Google Scholar 

  • S. W. Hawking, Gravitational Instantons, Phys. Lett. 60A, 81 (1977).

    Google Scholar 

  • P. W. Higgs, Phys. Rev. Lett. 12, 132 (1964); Phys. Rev. 145, 1156 (1966)

    Google Scholar 

  • R. Jackiw and C. Rebbi, Vacuum Periodicity in a Yang-Mills Theory, Phys. Rev. Lett. 37, 172 (1976). conformal Properties of a Yang-Mills Pseudoparticle, Phys. Rev. D 14, 517 (1976).

    Google Scholar 

  • H. Kerbrat-Lunc, Ann. Inst. H. Poincaré 13A, 295 (1970).

    Google Scholar 

  • R. Kerner, Ann. Inst. H. Poincaré 9A, 143 (1968).

    Google Scholar 

  • T. W. B. Kibble, J. Math. Phys. 2, 212 (1961).

    Google Scholar 

  • T. W. B. Kibble, Phys. Rev. 155, 1554 (1967).

    Google Scholar 

  • H. G. Loos, Internal Holonomy Groups of Yang-Mills Fields, J. Math. Phys. 8, 2114–2124 (1967); Phys. Rev. 188, 2342 (1969).

    Google Scholar 

  • Lu Qi-keng, Gauge Fields and Connections in Principal Bundles, Chin. J. Phys. 23, 153–161 (1975) [Wuli Xuebao 23, 249–263 (1974)]

    Google Scholar 

  • E. Lubkin, Ann. Phys. (N. Y.) 23, 233 (1963).

    Google Scholar 

  • V. Lugo, Holes and Integrality of the Curvature, UCLA Preprint, TEP/8, May 1976.

    Google Scholar 

  • W. Marciano and H. Pagels, Chiral Charge Conservation and Gauge Fields, Phys. Rev. (1976), idem and Z. Parsa, Multiply Charged Magnetic Monopoles, SU (3) Pseudoparticles and Gravitational Pseudoparticles, Rockefeller Preprint C00-2232B-108, 1976.

    Google Scholar 

  • M. E. Mayer, Thesis, Unpublished, Univ. of Bucharest, 1956; Extended Invariance Properties of Quantized Fields, I, Preprint JINR, Dubna 1958 and Nuovo Cimento 11, 760–770 (1959).

    Google Scholar 

  • M. E. Mayer, C*-Bundles and Symmetries in Algebraic Quantum Field Theories, Proc. of Conf. on Noncompact Groups in Physics, Y. Chow, ed., Milwaukee, 1965; W. A. Benjamin, N. Y. 1966. Fibrations, Connections and Gauge Theories (An Afterthought to the Talk by A. Trautman, Proc. of the International Symposium on New Mathematical Methods in Physics, Bonn 1973, K. Bleuler and A. Reetz, eds. Talk at the Intern. Congr. of Mathematicians (Abstract N4), Vancouver, B. C., 1974. Gauge Fields as Quantized Connection Forms, Proc. of the Symposium on Differential-Geometrical Methods in Physics, Bonn, 1975, K. Bleuler and A. Reetz, eds., Lecture Notes in Mathematics, Springer Verlag, Berlin, 1977.

    Google Scholar 

  • M. E. Mayer, Gauge Fields and Characteristic Classes, to be published.

    Google Scholar 

  • A. A. Migdal, Rekursionnye uravneniya v kalibrovochnykh teoriyakhpolya (Recursion Ecuations in Gauge Field Theories), Zh. Eksp. Teor. Fiz. 69, 810–822 (1975) [Sov. Phys. JETP 42, 413–418 (1976)].

    Google Scholar 

  • E. Noether, Invariante Variationsprobleme, Nachr. Ges. Göttingen (math.-phys. Klasse) 1918, 235–257.

    Google Scholar 

  • V. I. Ogievetskii and I. V. Polubarinov, Zh. Eksp. Teor. Fiz. 41, 247 (1961) [Sov. Phys. JETP 14, 179 (1962)].

    Google Scholar 

  • K. Osterwalder, Gauge Theories on the Lattice, Cargese Lectures, 1976, to be published.

    Google Scholar 

  • A. M. Polyakov, JETP Lett. 20, 194 (1974); Sov. Phys. JETP 41, 988 (1975); Phys. Lett. 59B, 80, 82 (1975). Nordita Preprint,1976.

    Google Scholar 

  • M. Prasad and C. Sommerfield, Phys. Rev. Lett. 35, 760 (1975).

    Google Scholar 

  • J. E. Roberts, Local Cohomology and Superselection, Preprint,1976.

    Google Scholar 

  • A. Salam and J. C. Ward, Nuovo Cimento 11, 568 (1959).

    Google Scholar 

  • A. Salam, Nobel Symposium, 1968, Almquist & Wiksell, Stockholm.

    Google Scholar 

  • J. Schwinger, Phys. Rev. 82, 914 (1951); 125, 1043; 127, 324 (1962); 91, 714 (1953).

    Google Scholar 

  • I. E. Segal, Proc. Nat. Acad. Sci. USA 41, 1103 (1955); 42, 670 (1956). Quantization of Nonlinear Systems, J. Math. Phys. 1, 468–488 (1960); Quantized Differential Forms, Topology 7, 147–171 (1968).

    Google Scholar 

  • F. Strocchi, The Existence of Local Solutions to the Equations ∂μFμν = jν and□φ = j in QFT, Princeton Seminar Notes 1971-72.

    Google Scholar 

  • F. Strocchi and A. S. Wightman, Proof of the Charge Superselection Rule in Local Relativistic Quantum Field Theory, J. Math. Phys. 15, 2198–2224 (1974).

    Google Scholar 

  • W. E. Thirring, Ann. Phys. (N. Y.) 16, 96 (1961).

    Google Scholar 

  • G. 't Hooft, Nucl. Phys. B33, 173; B35, 167 (1971).

    Google Scholar 

  • G. 't Hooft, Symmetry Breaking through Bell-Jackiw Anomalies, Phys. Rev. Lett. 37, 8 (1976). Computation of Quantum Effects due to a Four-Dimensional Pseudoparticle, Harvard Preprint, 1976. Cal-Tech Seminar, 1976.

    Google Scholar 

  • G.'t Hooft and M. Veltman, Nucl. Phys. B44, 189 (1973).

    Google Scholar 

  • A. Trautman, Infinitesimal Connections in Physics, Proc. Internat. Symposium on New Mathem. Methods in Physics, Bonn 1973, K. Bleuler and A. Reetz, eds., Bonn., 1973, and earlier work quoted there.

    Google Scholar 

  • R. Utiyama, Invariant Theoretical Interpretation of Interaction, Phys. Rev. 101, 1597 (1956).

    Google Scholar 

  • M. Veltman, Nucl. Phys. B21, 288 (1971).

    Google Scholar 

  • S. Weinberg, A Theory of Leptons, Phys. Rev. Lett. 19, 1264 (1967).

    Google Scholar 

  • S. Weinberg, Rev. Mod. Phys. 46, 255 (1974).

    Google Scholar 

  • H. Weyl, Gravitation and Elektrizität, Sber. Preuss. Akad. Wiss. 1918, 465–480; Z. Physik 56, 330 (1929).

    Google Scholar 

  • T. T. Wu and C. N. Yang, Concept of Nonintegrable Phase Factors and Global Formulation of Gauge Fields Phys. Rev. 12D, 3845 (1975).

    Google Scholar 

  • C. N. Yang and R. L. Mills, Conservation of Isotopic Spin and Isotopic Gauge Invariance, Phys. Rev. 96, 191 (1954).

    Google Scholar 

  • C. N. Yang, Integral Formalism for Gauge Fields, Phys. Rev. Lett. 33, 445–447 (1974).

    Google Scholar 

  • Albeverio, S. A., and Høegh-Krohn, R. J.: Mathematical Theory of Feynman Path Integrals, LNM 523, Springer Verl., 1976.

    Google Scholar 

  • Feynman, R. P., and Hibbs, A. R.: Quantum Mechanics and Path Integrals, McGraw-Hill, New York, 1965

    Google Scholar 

  • Palais, R. S.: Seminar on the Atiyah-Singer Index Theorem, Princeton Univ. Press, 1965.

    Google Scholar 

  • M. F. Atiyah, N. J. Hitchin, and I. M. Singer, Deformations of Instantons (Oxford-Berkeley-MIT Preprint, recd. May 1977).

    Google Scholar 

  • L. S. Brown, R. D. Carlitz, and C. Lee, Massless Excitations in Instanton Fields (U. of Washington Preprint, recd. May 1977).

    Google Scholar 

  • C. G. Callan,Jr., R. Dashen, and D. J. Gross, A Mechanism for Quark Confinement (IAS Preprint,C00-2220-94, 1977).

    Google Scholar 

  • V. De Alfaro, S. Fubini, and G. Furlan, A New Classical Solution for The Yang-Mills Equation, Phys. Lett. 65B, 163 (1976).

    Google Scholar 

  • S. Fubini, A New Approach to Conformal Invariant Field Theories, Nuovo Cimento 34A, 521 (1976). V. De Alfaro, S. Fubini,and G. Furlan, Conformal Invariance in Quantum Mechanics, Nuovo Cimento 34A, 569 (1976).

    Google Scholar 

  • R. Jackiw, C. Nohl, and C. Rebbi, Phys. Rev D (to appear), and R. Jackiw and C. Rebbi, Phys. Lett. B (to appear), both quoted in [65].

    Google Scholar 

Download references

Editor information

Wolfgang Drechsler M. E. Mayer A. Böhm J. D. Dollard

Rights and permissions

Reprints and permissions

Copyright information

© 1977 Springer-Verlag

About this chapter

Cite this chapter

(1977). Introduction to the fiber-bundle approach to gauge theories. In: Drechsler, W., Mayer, M.E., Böhm, A., Dollard, J.D. (eds) Fiber Bundle Techniques in Gauge Theories. Lecture Notes in Physics, vol 67. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-08350-2_5

Download citation

  • DOI: https://doi.org/10.1007/3-540-08350-2_5

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-08350-4

  • Online ISBN: 978-3-540-37289-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics