Skip to main content
SpringerLink
Log in

Lattice Gauge Theories

  • Chapter
Statistical Approach to Quantum Field Theory

Part of the book series: Lecture Notes in Physics ((LNP,volume 100))

  • 5319 Accesses

Abstract

According to present day knowledge all fundamental interactions in nature are described by gauge theories. In the first part we summarized concepts and properties of continuum gauge theories which are needed in the remaining chapters of this book. These include local gauge transformations, gauge potential and field strength, covariant derivative, parallel transport and gauge invariant Lagrangians for Euclidean Higgs models. Following Wilson we discretize gauge theories on a space-time lattice by replacing the Lie algebra valued continuum gauge field by Lie group valued variables on the links. Thereby the functional integral becomes a finite-dimensional integral which can be estimated by stochastic simulation techniques. We study the weak and strong coupling limits of lattice Higgs models, calculate the mean field approximations and discuss the expected phase diagram at zero temperature. In the second part we prove Elitzur’s theorem, sketch the strong coupling expansion, introduce the string tension and show in detail how to extract glueball masses. Towards the end we comment on gauge theories at finite temperature and in particular the breaking of center symmetry in pure lattice gauge theories.

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 69.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Gravity is an exception.

References

  1. D. Ivanenko, G. Sardanashvily, The gauge treatment of gravity. Phys. Rep. 94, 1 (1983)

    Article  ADS  MathSciNet  Google Scholar 

  2. F.W. Hehl, J.D. McCrea, E.W. Mielke, Y. Ne’eman, Metric affine gauge theory of gravity: field equations, Noether identities, world spinors, and breaking of dilation invariance. Phys. Rep. 258, 1 (1995)

    Article  ADS  MathSciNet  Google Scholar 

  3. S. Pokorski, Gauge Field Theories (Cambridge University Press, Cambridge, 2000)

    Book  MATH  Google Scholar 

  4. K. Huang, Quarks, Leptons and Gauge Fields (World Scientific, Singapore, 1992)

    MATH  Google Scholar 

  5. L. O’Raifeartaigh, Group Structure of Gauge Theories (Cambridge University Press, Cambridge, 1986)

    Book  MATH  Google Scholar 

  6. A. Das, Lectures on Quantum Field Theory (World Scientific, Singapore, 2008)

    Book  MATH  Google Scholar 

  7. F.J. Wegner, Duality in generalized Ising models and phase transitions without local order parameters. J. Math. Phys. 10, 2259 (1971)

    Article  ADS  MathSciNet  Google Scholar 

  8. K.G. Wilson, Confinement of quarks. Phys. Rev. D 10, 2445 (1974)

    Article  ADS  Google Scholar 

  9. M. Creutz, L. Jacobs, C. Rebbi, Experiments with a gauge invariant Ising system. Phys. Rev. Lett. 42, 1390 (1979)

    Article  ADS  Google Scholar 

  10. M. Creutz, Confinement and the critical dimensionality of spacetime. Phys. Rev. Lett. 43, 553 (1979)

    Article  ADS  Google Scholar 

  11. M. Creutz, Monte Carlo simulations in lattice gauge theories. Phys. Rep. 95, 201 (1983)

    Article  ADS  Google Scholar 

  12. I. Montvay, G. Münster, Quantum Fields on a Lattice (Cambridge University Press, Cambridge, 1994)

    Book  Google Scholar 

  13. H.J. Rothe, Lattice Gauge Theories: An Introduction (World Scientific, Singapore, 2012)

    Book  Google Scholar 

  14. T. DeGrand, C. DeTar, Lattice Methods for Quantum Chromodynamics (World Scientific, Singapore, 2006)

    Book  MATH  Google Scholar 

  15. C. Gattringer, C. Lang, in Quantum Chromodynamics on the Lattice. Lect. Notes Phys., vol. 788, (2010)

    Chapter  Google Scholar 

  16. L.P. Kadanoff, The application of renormalization group techniques to quarks and strings. Rev. Mod. Phys. 49, 267 (1977)

    Article  ADS  MathSciNet  Google Scholar 

  17. J.B. Kogut, An introduction to lattice gauge theory and spin systems. Rev. Mod. Phys. 51, 659 (1979)

    Article  ADS  MathSciNet  Google Scholar 

  18. J.B. Kogut, The lattice gauge theory approach to quantum chromodynamics. Rev. Mod. Phys. 55, 775 (1983)

    Article  ADS  Google Scholar 

  19. P. de Forcrand, O. Jahn, Comparison of SO(3) and SU(2) lattice gauge theory. Nucl. Phys. B 651, 125 (2003)

    Article  ADS  MATH  Google Scholar 

  20. R.L. Karp, F. Mansouri, J.S. Rho, Product integral formalism and non-Abelian Stokes theorem. J. Math. Phys. 40, 6033 (1999)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  21. R.L. Karp, F. Mansouri, J.S. Rho, Product integral representations of Wilson lines and Wilson loops, and non-Abelian Stokes theorem. Turk. J. Phys. 24, 365 (2000)

    Google Scholar 

  22. R. Giles, Reconstruction of gauge potentials from Wilson loops. Phys. Rev. D 24, 2160 (1981)

    Article  ADS  MathSciNet  Google Scholar 

  23. J. Fröhlich, G. Morchio, F. Strocchi, Higgs phenomenon without symmetry breaking order parameter. Nucl. Phys. B 190, 553 (1981)

    Article  ADS  Google Scholar 

  24. F. Hausdorff, Die symbolische Exponentialformel in der Gruppentheorie. Ber. Verh. Saechs. Akad. Wiss. Leipz. 58, 19 (1906)

    Google Scholar 

  25. K. Wilson, in Recent Developments of Gauge Theories, ed. by G. ’t Hofft, et al. (Plenum, New York, 1980)

    Google Scholar 

  26. K. Symanzik, Continuum limit and improved action in lattice theories. 1. Principles and ϕ 4 theory. Nucl. Phys. B 226, 187 (1983)

    Article  ADS  MathSciNet  Google Scholar 

  27. M. Luscher, P. Weisz, Computation of the action for on-shell improved lattice gauge theories at weak coupling. Phys. Lett. B 158, 250 (1985)

    Article  ADS  Google Scholar 

  28. K. Langfeld, Improved actions and asymptotic scaling in lattice Yang–Mills theory. Phys. Rev. D 76, 094502 (2007)

    Article  ADS  Google Scholar 

  29. J.M. Drouffe, J.B. Zuber, Strong coupling and mean field methods in lattice gauge theories. Phys. Rep. 102, 1 (1983)

    Article  ADS  MathSciNet  Google Scholar 

  30. G. Arnold, B. Bunk, T. Lippert, K. Schilling, Compact QED under scrutiny: its first order. Nucl. Phys. B, Proc. Suppl. 119, 864 (2003)

    Article  ADS  MATH  Google Scholar 

  31. K. Langfeld, B. Lucini, A. Rago, The density of states in gauge theories. Phys. Rev. Lett. 109, 111601 (2012)

    Article  ADS  Google Scholar 

  32. J. Carlsson, B. McKellar, SU(N) glueblall masses in 2+1 dimensions. Phys. Rev. D 68, 074502 (2003)

    Article  ADS  Google Scholar 

  33. S. Uhlmann, R. Meinel, A. Wipf, Ward identities for invariant group integrals. J. Phys. A 40, 4367 (2007)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  34. K. Osterwalder, E. Seiler, Gauge field theories on a lattice. Ann. Phys. 10, 440 (1978)

    Article  ADS  MathSciNet  Google Scholar 

  35. E. Fradkin, S. Shenker, Phase diagrams of lattice gauge theories with Higgs fields. Phys. Rev. D 19, 3682 (1979)

    Article  ADS  Google Scholar 

  36. C. Bonati, G. Cossu, M. D’Elia, A. Di Giacomo, Phase diagram of the lattice SU(2) Higgs model. Nucl. Phys. B 828, 390 (2010)

    Article  ADS  MATH  Google Scholar 

  37. J. Greensite, An Introduction to the Confinement Problem. Lecture Notes in Physics (Springer, Berlin, 2011)

    Book  MATH  Google Scholar 

  38. S. Elitzur, Impossibility of spontaneously breaking local symmetries. Phys. Rev. D 12, 3978 (1975)

    Article  ADS  Google Scholar 

  39. Y. Blum, P.K. Coyle, S. Elitzur, E. Rabinovici, S. Solomon, H. Rubinstein, Investigation of the critical behavior of the critical point of the Z2 gauge lattice. Nucl. Phys. B 535, 731 (1998)

    Article  ADS  Google Scholar 

  40. C. Itzikson, J.M. Drouffe, Statistical Field Theory, vol. I. Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, 1989)

    Book  Google Scholar 

  41. B. Wellegehausen, A. Wipf, C. Wozar, Casimir scaling and string breaking in G2 gluodynamics. Phys. Rev. D 83, 016001 (2011)

    Article  ADS  Google Scholar 

  42. G.S. Bali, K. Schilling, C. Schlichter, Observing long color flux tubes in SU(2) lattice gauge theory. Phys. Rev. D 51, 5165 (1995)

    Article  ADS  Google Scholar 

  43. E. Seiler, Upper bound on the color-confining potential. Phys. Rev. D 18, 482 (1978)

    Article  ADS  Google Scholar 

  44. C. Bachas, Convexity of the quarkonium potential. Phys. Rev. D 33, 2723 (1986)

    Article  ADS  Google Scholar 

  45. M. Lüscher, K. Symanzik, P. Weisz, Anomalies of the free loop wave equation in WKB approximation. Nucl. Phys. B 173, 365 (1980)

    Article  ADS  Google Scholar 

  46. M. Lax, Symmetry Principles in Solid State and Molecular Physics (Wiley, New York, 1974)

    Google Scholar 

  47. J.S. Lomount, Applications of Finite Groups (Academic Press, New York, 1959)

    Google Scholar 

  48. Y. Chen et al., Glueball spectrum and matrix elements on anisotropic lattices. Phys. Rev. D 73, 014516 (2006)

    Article  ADS  Google Scholar 

  49. M. Teper, An improved method for lattice glueball calculations. Phys. Lett. B 183, 345 (1986)

    Article  ADS  Google Scholar 

  50. J.I. Kapusta, C. Gale, Finite-Temperature Field Theory: Principles and Applications (Cambridge University Press, Cambridge, 2006)

    Book  Google Scholar 

  51. F. Karsch, Lattice QCD at high temperature and density. Lect. Notes Phys. 583, 209 (2002)

    Article  ADS  Google Scholar 

  52. G. Boyd, J. Engels, F. Karsch, E. Laermann, C. Legeland, M. Lütgemeier, B. Petersson, Equation of state for the SU(3) gauge theory. Phys. Rev. Lett. 75, 4169 (1995)

    Article  ADS  Google Scholar 

  53. B. Wellegehausen, Effektive Polyakov-Loop Modelle für SU(N)- und G2-Eichtheorien (Effective Polyakov loop models for SU(N) and G2 gauge theories). Diploma Thesis, Jena (2008)

    Google Scholar 

  54. B. Lucini, M. Teper, U. Wenger, The high temperature phase transition in SU(N) gauge theories. J. High Energy Phys. 0401, 061 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  55. A.M. Polyakov, Quark confinement and topology of gauge groups. Nucl. Phys. B 120, 429 (1977)

    Article  ADS  MathSciNet  Google Scholar 

  56. B. Svetitsky, L.G. Yaffe, Critical behavior at finite temperature confinement transitions. Nucl. Phys. B 210, 423 (1982)

    Article  ADS  Google Scholar 

  57. K. Holland, P. Minkowski, M. Pepe, U.J. Wiese, Exceptional confinement in G(2) gauge theory. Nucl. Phys. B 668, 207 (2003)

    Article  ADS  MATH  Google Scholar 

  58. B. Wellegehausen, A. Wipf, C. Wozar, Phase diagram of the lattice G2 Higgs Model. Phys. Rev. D 83, 114502 (2011)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Theoretisch-Physikalisches-Institut, Friedrich-Schiller-Universität Jena, Jena, Germany

    Andreas Wipf

Authors
  1. Andreas Wipf
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Wipf, A. (2013). Lattice Gauge Theories. In: Statistical Approach to Quantum Field Theory. Lecture Notes in Physics, vol 100. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33105-3_13

Download citation

Publish with us

Policies and ethics

Search

Navigation