Skip to main content
SpringerLink
Log in
Book cover

A Primer for Chiral Perturbation Theory pp 49–64Cite as

Spontaneous Symmetry Breaking and the Goldstone Theorem

  • Chapter
  • First Online:

  • 2762 Accesses

Part of the book series: Lecture Notes in Physics ((LNP,volume 830))

Abstract

So far we have concentrated on the chiral symmetry of the QCD Hamiltonian and the explicit symmetry breaking by the quark masses. We have discussed the importance of chiral symmetry for the properties of Green functions with particular emphasis on the relations among different Green functions as expressed through the chiral Ward identities. Now it is time to address a second aspect which, for the low-energy structure of QCD, is equally important, namely, the concept of spontaneous symmetry breaking. A (continuous) symmetry is said to be spontaneously broken or hidden if the ground state of the system is no longer invariant under the full symmetry group of the Hamiltonian. In this chapter we will first illustrate this by means of a discrete symmetry and then turn to the case of a spontaneously broken continuous global symmetry.

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   79.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   99.99
Price excludes VAT (USA)
  • Compact, lightweight 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.

    The case of a quantum field theory with an infinite volume \(V\) has to be distinguished from, say, a nonrelativistic particle in a one-dimensional potential of a shape similar to the function of Fig. 2.2. For example, in the case of a symmetric double-well potential, the solutions with positive parity always have lower energy eigenvalues than those with negative parity (see, e.g., Ref. [11]).

  2. 2.

    The field \(\Upphi'\) instead of \(\Upphi\) is assumed to vanish at infinity.

  3. 3.

    For continuous symmetry groups one may have a non-countably infinite number of ground states.

  4. 4.

    The linear sigma model [6, 7, 17] is constructed in terms of the O(4) multiplet \((\sigma,\pi_1,\pi_2,\pi_3).\) Since the group O(4) is locally isomorphic to SU(2) \(\times\) SU(2), the linear sigma model is a popular framework for illustrating the spontaneous symmetry breaking in two-flavor QCD.

  5. 5.

    In the beginning, the discussion of spontaneous symmetry breaking in field theories [9, 13–15] was driven by an analogy with the theory of superconductivity [1, 2, 4, 5].

  6. 6.

    The Lagrangian is invariant under the full group O(3) which can be decomposed into its two components: the proper rotations connected to the identity, SO(3), and the rotation-reflections. For our purposes it is sufficient to discuss SO(3).

  7. 7.

    We say, somewhat loosely, that \(T_1\;\hbox{and}\;T_2\) do not annihilate the ground state or, equivalently, finite group elements generated by \(T_1\;\hbox{and}\;T_2\) do not leave the ground state invariant. This should become clearer later on.

  8. 8.

    The restriction to compact groups allows for a complete decomposition into finite-dimensional irreducible unitary representations.

  9. 9.

    Using the replacements \(Q_k\to \hat l_k\;\hbox{and}\;\Upphi_l\to\hat x_l,\) note the analogy with \(i[\hat l_k,\hat x_l]=-\varepsilon_{klm}\hat x_m.\)

  10. 10.

    The abbreviation \(\sum\!\!\!\!\!\!\!\!\int_n|n\rangle\langle n|\) includes an integral over the total momentum \(\vec{p}\) as well as all other quantum numbers necessary to fully specify the states.

References

  1. Bardeen, J., Cooper, L.N., Schrieffer, J.R.: Phys. Rev 106, 162 (1957)

    Article  MathSciNet  ADS  Google Scholar 

  2. Bardeen, J., Cooper, L.N., Schrieffer, J.R.: Phys. Rev. 108, 1175 (1957)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  3. Bernstein, J.: Rev. Mod. Phys 46, 7 (1974) (Erratum, ibid. 47, 259 (1975))

    Article  ADS  Google Scholar 

  4. Bogoliubov, N.N.: Sov. Phys. JETP 7, 41 (1958)

    MathSciNet  Google Scholar 

  5. Bogoliubov, N.N.: Zh. Eksp. Teor. Fiz 34, 58 (1958)

    Google Scholar 

  6. Cheng, T.P., Li, L.F.: Gauge Theory of Elementary Particle Physics (Chap. 5.3). Clarendon, Oxford (1984)

    Google Scholar 

  7. Gell-Mann, M., Lévy, M.: Nuovo. Cim. 16, 705 (1960)

    Article  MATH  Google Scholar 

  8. Georgi, H.: Weak Interactions and Modern Particle Theory (Chaps. 2.4–2.6). Benjamin/Cummings, Menlo Park (1984)

    Google Scholar 

  9. Goldstone, J.: Nuovo. Cim. 19, 154 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  10. Goldstone, J., Salam, A., Weinberg, S.: Phys. Rev. 127, 965 (1962)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  11. Greiner, W.: Theoretical Physics. Quantum Theory, vol. 4a (in German). Deutsch, Thun (1985)

    Google Scholar 

  12. Li, L.F., Pagels, H.: Phys. Rev. Lett. 26, 1204 (1971)

    Article  ADS  Google Scholar 

  13. Nambu, Y.: Phys. Rev. Lett. 4, 380 (1960)

    Article  ADS  Google Scholar 

  14. Nambu, Y., Jona-Lasinio, G.: Phys. Rev. 122, 345 (1961)

    Article  ADS  Google Scholar 

  15. Nambu, Y., Jona-Lasinio, G.: Phys. Rev. 124, 246 (1961)

    Article  ADS  Google Scholar 

  16. Ryder, L.H.: Quantum Field Theory (Chap. 8). Cambridge University Press, Cambridge (1985)

    Google Scholar 

  17. Schwinger, J.S.: Ann. Phys. 2, 407 (1957)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  18. Weinberg, S.: The Quantum Theory of Fields. Modern Applications, vol. 2 (Chap. 19). Cambridge University Press, Cambridge (1996)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Johannes Gutenberg-Universität Mainz, Institut für Kernphysik, Johann-Joachim-Becher-Weg 45, 55099, Mainz, Germany

    Dr. Stefan Scherer

  2. Department of Physics and Astronomy, University of South Carolina, 712 Main Street, Columbia, SC, 29208, USA

    Dr. Matthias R. Schindler

  3. Department of Physics, The George Washington University, Washington, DC, USA

    Dr. Matthias R. Schindler

Authors
  1. Dr. Stefan Scherer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dr. Matthias R. Schindler
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Stefan Scherer .

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Scherer, S., Schindler, M.R. (2011). Spontaneous Symmetry Breaking and the Goldstone Theorem. In: A Primer for Chiral Perturbation Theory. Lecture Notes in Physics, vol 830. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-19254-8_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-19254-8_2

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-19253-1

  • Online ISBN: 978-3-642-19254-8

  • eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)

Publish with us

Policies and ethics

Search

Navigation