Skip to main content
SpringerLink
Log in

Maxwell and Yang–Mills Theory

  • Chapter
  • First Online:
Geometrodynamics of Gauge Fields

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

  • 1500 Accesses

Abstract

The Lagrangian formalism has to be seen as the point of departure for classical field theories. This approach is also providing the scaffold for (canonical) quantum field theories. Instead of using a local notation, our representation of the formalism will be based upon differential forms which are globally defined on a pseudo-Riemannian manifold of dimension n.

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

Access this chapter

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

Institutional subscriptions

Notes

  1. 1.

    Mack (1981) interpreted this postulate physically as a result of a “Naheinformationsprinzip”, regarding it as a generalization of the principle of action-at-close-distances.

  2. 2.

    More generally, one could imagine the existence of topological modified Lagrangians \({\buildrel \theta \over L} =-(\cos \theta \,F\wedge \,^* F+\sin \theta \,F\wedge F)/2\), where \(\theta \) is the’vacuum’ angle of duality rotations or higher order functionals of the two quadratic invariants.

  3. 3.

    The displacement current \(\partial \mathfrak {D}/\partial t\), where \(\mathfrak {D}=\varepsilon \mathbf {E}\), was anticipated 1839 by James Mac Cullagh, cf. Darrigol (2010). It later on turned out to be a necessary ingredient for rendering electromagnetism relativistic-invariant.

  4. 4.

    In four dimensions, B resembles the two-form potential for the gauge-invariant field strength or excitation \(H=dB\), the Kalb–Ramond axion three-form.

  5. 5.

    According to De Alfaro (1979a, b), the so-called meron solutions with half-integer topological charge are supposed to be dominant instead.

References

  • Alvarez O (1985) Topological quantization and cohomology. Commun Math Phys 100(2):279–309

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Atiyah MF (1979) Geometry of Yang-Mills fields. Lezioni Fermiane, Accademia Nazionale dei Lincei Scuola Normale Superione, Pisa

    MATH  Google Scholar 

  • Atiyah MF, Hitchin NJ, Singer IM (1978) Self-duality in four-dimensional Riemannian geometry. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 362:425–461

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Babelon O, Viallet CM (1979) The geometrical interpretation of the Faddeev-Popov determinant. Phys Lett B 85(2):246–248

    Article  ADS  MathSciNet  Google Scholar 

  • Babelon O, Viallet CM (1981) The Riemannian geometry of the configuration space of gauge theories. Commun Math Phys 81(4):515–525

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Baekler P (1981) A spherically symmetric vacuum solution of the quadratic Poincaré gauge field theory of gravitation with Newtonian and confinement potentials. Phys Lett B 99(4):329–332

    Google Scholar 

  • Batelaan H, Tonomura A (2009) The Aharonov-Bohm effects: variations on a subtle theme. Phys Today 62(9):38

    Article  Google Scholar 

  • Belavin AA, Polyakov AM, Schwartz AS, Tyupkin YS (1975) Pseudoparticle solutions of the Yang-Mills equations. Phys Lett B 59(1):85–87

    Article  ADS  MathSciNet  Google Scholar 

  • Bernstein J (1974) Spontaneous symmetry breaking, gauge theories, the Higgs mechanism and all that. Rev Mod Phys 46(1):7

    Article  ADS  MathSciNet  Google Scholar 

  • Bogomol’nyi EB (1976) The stability of classical solutions. Sov J Nucl Phys 24(4) (Engl. Transl.)

    Google Scholar 

  • Born M, Infeld L (1934) Foundations of the new field theory. Proc R Soc Lond Ser A 144(852):425–451

    Google Scholar 

  • Borowiec A, Francaviglia M (2005) Three-Dimensional Chern-Simons and BF theories. In: AIP Conference Proceedings vol 751, pp 165–167

    Google Scholar 

  • Bourguignon JP, Lawson HB (1981) Stability and isolation phenomena for Yang-Mills fields. Commun Math Phys 79(2):189–230

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Broda B (2005) BF system-Encyclopedic entry, pp 53–54. arXiv preprint. hep-th/0502045

  • Cattaneo AS, Cotta-Ramusino P, Fucito F, Martellini M, Rinaldi M, Tanzini A, Zeni M (1998) Four-dimensional Yang-Mills theory as a deformation of topological BF theory. Commun Math Phys 197(3):571–621

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Cattaneo AS, Rossi CA (2005) Wilson surfaces and higher dimensional knot invariants. Commun Math Phys 256(3):513–537

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Cheng TP, Li LF (1984) Gauge Theory of Elementary Particle Physics. Clarendon Press, Oxford

    Google Scholar 

  • Daniel M, Viallet CM (1980) The geometrical setting of gauge theories of the Yang-Mills type. Rev Mod Phys 52(1):175

    Article  ADS  MathSciNet  Google Scholar 

  • Darrigol O (2010) James MacCullagh’s ether: An optical route to Maxwell’s equations?. Eur Phys J H, 35: 133–172

    Google Scholar 

  • de Alfaro V, Fubini S, Furlan G (1979a) Gauge theories and strong gravity. Il Nuovo Cimento A 50(4):523–554

    Google Scholar 

  • de Alfaro V, Fubini S, Furlan G (1979b) A new approach to the theory of gravitation. Nuovo Cimento B 57:227–252

    Article  ADS  MathSciNet  Google Scholar 

  • Deser S, Drechsler W (1979) Generalized gauge field copies. Phys Lett B 86(2):189–192

    Article  ADS  MathSciNet  Google Scholar 

  • Donaldson S (2006) Gauge theory: Mathematical applications. In: J.-PFLNS Tsun (ed) Encyclopedia of Mathematical Physics. Academic Press, Oxford, pp 468–481

    Google Scholar 

  • Eddington AS (1924) The Mathematical Theory of Relativity, 2nd edn. Chelsea Publ, New York

    MATH  Google Scholar 

  • Faddeev LD, Slavnov AA (1980) Gauge Fields: Introduction to Quantum Theory. Benjamin/Cummings Publishing Company Inc, Reading, MA

    MATH  Google Scholar 

  • Felsager B, Leinaas JM (1980) Geometric interpretation of magnetic fields and the motion of charged particles. Nucl Phys B 166(1):162–188

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Franz W (1939) Elektroneninterferenzen im Magnetfeld. Verhandlungen der Deutschen Physikalischen Gesellschaft 2:65

    Google Scholar 

  • Gell-Mann M (1956) The interpretation of the new particles as displaced charge multiplets, Il Nuovo Cimento 4(2):848–866

    Google Scholar 

  • Gell-Mann M, Neeman Y (1964) The Eightfold Way. Benjamin, New York

    Google Scholar 

  • Gu CH (1981) On the minima of Yang-Mills functionals. J Differ Geom 16(1):147–151

    MathSciNet  MATH  Google Scholar 

  • Gu CH, Hu HS, Shen CL, Yang CN (1978) Geometrical interpretation of instanton solutions in Euclidean space. Sci Sin 21(6):767–772

    Google Scholar 

  • Hehl FW (1981) Zur Eichfeldtheorie der Gravitation. Grundlagenprobleme der modernen Physik, BI-Wissenschaftsverlag, Mannheim, p 103

    Google Scholar 

  • Hehl FW, Lemke J, Mielke EW (1991) Two lectures on fermions and gravity. Geometry and theoretical physics. Springer, Heidelberg, pp 56–140

    MATH  Google Scholar 

  • Hehl FW, Macías A, Mielke EW, Obukhov YN (1999) On the structure of the energy-momentum and the spin currents in Dirac’s electron theory. In: Harvey A (ed) On Einstein’s Path. Festschrift for E. Schucking on the occasion of his 70th birthday, Springer, New York, pp 257–274

    Google Scholar 

  • Hehl FW, McCrea JD, Mielke EW, Ne’eman Y (1995) Metric-affine gauge theory of gravity: field equations. Noether identities, world spinors, and breaking of dilation invariance. Phys Rep 258(1):1–171

    Article  ADS  MathSciNet  Google Scholar 

  • Hennig J, Nitsch J (1981) Gravity as an internal Yang-Mills gauge field theory of the Poincaré group. Gen Relat Gravit 13(10):947–962

    Article  ADS  MATH  Google Scholar 

  • Higgs PW (1964a) Broken symmetries and the masses of gauge bosons. Phys Rev Lett 13(16):508

    Article  ADS  MathSciNet  Google Scholar 

  • Higgs PW (1964b) Broken symmetries, massless particles and gauge fields. Phys Lett 12(2):132–133

    Article  ADS  Google Scholar 

  • Higgs PW (1966) Spontaneous symmetry breakdown without massless bosons. Phys Rev 145(4):1156

    Article  ADS  MathSciNet  Google Scholar 

  • Horndeski GW (1978) Gauge invariance and charge conservation. Tensor 32:131–139

    MathSciNet  Google Scholar 

  • Horndeski GW (1980) Gauge invariance and charge conservation in non-Abelian gauge theories. Arch Ration Mech Anal 75:211–227

    MathSciNet  MATH  Google Scholar 

  • Horowitz GT (1989) Exactly soluble diffeomorphism invariant theories. Commun Math Phys 125(3):417–437

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  • Jackiw R (1980) Introduction to the Yang-Mills quantum theory. Rev Mod Phys 52(4):661

    Article  ADS  MathSciNet  Google Scholar 

  • Jackiw R, Nohl C, Rebbi C (1977) Conformal properties of pseudoparticle configurations. Phys Rev D 15(6):1642

    Article  ADS  Google Scholar 

  • Jost J, Peng X, Wang G (1996) Variational aspects of the Seiberg-Witten functional. Calc Var, 4:205–218

    Google Scholar 

  • Kibble TWB (1967) Symmetry breaking in non-Abelian gauge theories. Phys Rev 155(5):1554

    Article  ADS  Google Scholar 

  • Kobayashi S, Nomizu K (1963). Foundations of Differential Geometry, Vol I (Interscience, New York)

    Google Scholar 

  • Lucchesi C, Piguet O, Sorella SP (1993) Renormalization and finiteness of topological BF theories. Nuclear Phys B 395(1):325–353

    Article  ADS  MathSciNet  Google Scholar 

  • Mack G (1981) Physical principles, geometrical aspects, and locality properties of gauge field theories. Fortschritte der Physik 29(4):135–185

    Article  ADS  MathSciNet  Google Scholar 

  • Madore J (1977) A classification of SU(3) magnetic monopoles. Commun Math Phys 56(2):115–123

    Google Scholar 

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

    Google Scholar 

  • Maxwell JC (1892) A Treatise on Electricity and Magnetism, vol 1, 3rd edn. Clarendon Press, Oxford

    Google Scholar 

  • Mielke EW (1977a) Knot wormholes in geometrodynamics? Gen Relativ Gravitation 8(3):175–196

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Mielke EW (1984) Reduction of the Poincaré gauge field equations by means of duality rotations. J Math Phys 25(3):663–668

    Article  ADS  MathSciNet  Google Scholar 

  • Mielke EW, Maggiolo AAR (2007) S-duality in 3D gravity with torsion. Ann Phys 322(2):341–362

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Mills R (1979) Model of confinement for gauge theories. Phys Rev Lett 43(8):549

    Article  ADS  Google Scholar 

  • Misner CW, Wheeler JA (1957) Classical physics as geometry. Ann Phys 2:525–603

    Google Scholar 

  • Montonen C, Olive D (1977) Magnetic monopoles as gauge particles? Phys Lett B 72(1):117–120

    Article  ADS  Google Scholar 

  • Percacci R (1981) Global definition of nonlinear sigma model and some consequences. J Math Phys 22(9):1892–1895

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Plebański JF (1977) On the separation of Einsteinian substructures. J Math Phys 18(12):2511–2520

    Article  ADS  MATH  Google Scholar 

  • Polyakov AM (1974) Particle spectrum in quantum field theory. Sov J Exp Theoret Phys Lett 20:194

    ADS  Google Scholar 

  • Prasad MK, Rossi P (1981a) Construction of exact multimonopole solutions. Phys Rev D 24(8):2182

    Article  ADS  MathSciNet  Google Scholar 

  • Prasad MK, Rossi P (1981b) Construction of exact Yang-Mills-Higgs multimonopoles of arbitrary charge. Phys Rev Lett 46(13):806

    Google Scholar 

  • Prasad MK, Sommerfield CM (1975) Exact classical solution for the ’t Hooft monopole and the Julia-Zee dyon. Phys Rev Lett 35(12):760

    Google Scholar 

  • Quirós M, Mittelbrunn JR, Rodriguez E (1982) On the topological meaning of magnetic charge quantization. J Math Phys 23(5):873–877

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Rainich GY (1925) Electrodynamics in the general relativity theory. Trans Am Math Soc 27(1):106–136

    Google Scholar 

  • Rose-Innes AC, Rhoderick EH (1969) Introduction to Superconductivity. Pergamon Publishing Company, Elmsford

    Google Scholar 

  • Rund H, Lovelock D (1972) Variational principles in the general theory of relativity. Jahresbericht der Deutschen Mathematiker-Vereinigung 74:1–65

    MATH  Google Scholar 

  • Saçlıo\({\tilde{\rm {g}}}\)lu C, (1999) Monopole condensates in Seiberg-Witten theory. Phys Lett B 454(3):315–318

    Google Scholar 

  • Salam A (1980) Gauge unification of fundamental forces. Rev Mod Phys 52(3):525

    Article  ADS  MathSciNet  Google Scholar 

  • Schwinger J (1962) Non-Abelian gauge fields commutation relations. Phys Rev 125(3):1043

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Seiberg N, Witten E (1994) Electric-magnetic duality, monopole condensation, and confinement in N= 2 supersymmetric Yang-Mills theory. Nuclear Phys B 426(1):19–52

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Simms DJ, Woodhouse NMJ (1976) Lectures on geometric quantization, vol 53. Springer, Heidelberg

    MATH  Google Scholar 

  • Singer IM (1978) Some remarks on the Gribov ambiguity. Commun Math Phys 60(1):7–12

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Slavnov AA (2009) Lorentz-invariant quantization of the Yang-Mills theory free of the Gribov ambiguity. Theoret Math Phys 161(2):1497–1502

    Article  MathSciNet  MATH  Google Scholar 

  • Sommerfeld A (1910) Zur Relativitätstheorie.I. Vierdimensionale Vektoralgebra, Annalen der Physik, vol 32, footnote p 768

    Google Scholar 

  • Strazhev VI, Tomil’chik LM (1973) Current state of the Dirac monopole problem. Technical report, Institute of Physics, Academy of Sciences of the Belorussian SSR, Minsk

    Google Scholar 

  • ’t Hooft G (1974) Magnetic monopoles in unified gauge theories. Nuclear Phys: B 79(2):276–284

    Google Scholar 

  • ’t Hooft G (1977) Gauge theory for strong interactions. New Phenom Subnucl. Phys: Part A, Plenum Press, New York, pp 261–295

    Google Scholar 

  • Taylor JC (1976) Gauge theories of weak interactions. Cambridge University Press, Cambridge

    Google Scholar 

  • Tchrakian DH (1981) A derivation of the Prasad-Sommerfield solution. J Phys A: Math Gen 14(6):1439

    Article  ADS  MathSciNet  Google Scholar 

  • Tchrakian DH (1983) Some geometrical models with torsion for monopole-like solutions 55:7

    Google Scholar 

  • Thorne KS, Lee DL, Lightman AP (1973) Foundations for a theory of gravitation theories. Phys Rev D 7(12):3563

    Article  ADS  Google Scholar 

  • Trautman A (1977) Solutions of the Maxwell and Yang-Mills equations associated with Hopf fibrings. Int J Theor Phys 16(8):561–565

    Article  Google Scholar 

  • Weyl H (1918) Gravitation und Elektrizität. Abhand Preuss Akad Wiss 25:465–480

    Google Scholar 

  • Weyl H (1924) Was ist Materie?. Springer, Heidelberg, pp 1–59

    Book  MATH  Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  • Wheeler JA (1970) Superspace. In: Gilbert RP, Newton RC (eds) Analytic Methods in Mathematical Physics. Gordon and Breach, New York, p 335

    Google Scholar 

  • Wightman AS (1973) Relativistic wave equations as singular hyperbolic systems in partial differential equations. In: Proceedings of Symposia in Pure Mathematics, vol 23

    Google Scholar 

  • Witten E (1977) Some exact multipseudoparticle solutions of classical Yang-Mills theory. Phys Rev Lett 38(3):121

    Article  ADS  Google Scholar 

  • Wu TT, Yang CN (1975) Concept of nonintegrable phase factors and global formulation of gauge fields. Phys Rev D 12(12):3845

    Article  ADS  MathSciNet  Google Scholar 

  • Yang CN, Mills RL (1954) Conservation of isotopic spin and isotopic gauge invariance. Phys Rev 96(1):191

    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). Maxwell and Yang–Mills Theory. In: Geometrodynamics of Gauge Fields. Mathematical Physics Studies. Springer, Cham. https://doi.org/10.1007/978-3-319-29734-7_3

Download citation

Publish with us

Policies and ethics

Search

Navigation