The advantages and disadvantages of some pedagogical non-relativistic quantum-mechanical models, used to illustrate spontaneous symmetry breakdown, are discussed. A spinor on the line subject to a magnetostatic interaction is presented as a toy model of the spontaneous breakdown of an internal symmetry.
REFERENCES
1.
P. K.
Aravind
, “ A simple geometrical model of spontaneous symmetry breaking
,” Am. J. Phys.
55
(5
), 437
–439
(1987
). 2.
J.
van Wezel
and J.
van den Brink
, “ Spontaneous symmetry breaking in quantum mechanics
,” Am. J. Phys.
75(7)
, 635
–638
(2007
). 3.
Y.
Nambu
, “ Axial vector current conservation in weak interactions
,” Phys. Rev. Lett.
4
, 380
–382
(1960
). 4.
M.
Baker
and S. L.
Glashow
, “ Spontaneous breakdown of elementary particle symmetries
,” Phys. Rev.
128
, 2462
–2471
(1962
). 5.
Y.
Nambu
, “ Symmetry breakdown and small mass bosons
,” in Evolutionary Trends in Physical Sciences
, edited by S.
Suzuki
and R.
Kubo
(Springer
, Heidelberg
, 1991
). 6.
J.
Goldstone
“ Field theories with superconductor solutions
,” Nuovo Cimento
19
, 154
–164
(1961
). 7.
J.
Goldstone
, A.
Salam
, and S.
Weinberg
, “ Broken symmetries
,” Phys. Rev.
127
, 965
–970
(1962
). 8.
P. W.
Anderson
, “ Plasmons, gauge invariance, and mass
,” Phys. Rev.
130
, 439
–442
(1963
). 9.
P. W.
Higgs
, “ Broken symmetries and the masses of gauge bosons
,” Phys. Rev. Lett.
13
, 508
–509
(1964
). 10.
F.
Halzen
and A. D.
Martin
, Quarks and Leptons: An Introductory Course in Modern Particle Physics
(Wiley
, New York
, 1984
). 11.
O.
Nachtman
, Elementary Particle Physics: Concepts and Phenomena
(Springer
, Berlin
, 1990
). 12.
J.
Sivardière
, “ A simple mechanical model exhibiting spontaneous symmetry breaking
,” Am. J. Phys.
51(11)
, 1016
–1018
(1983
). 13.
J. R.
Drugowich de Felício
and O.
Hipolito
, “ Spontaneous symmetry breaking in a simple mechanical model
,” Am. J. Phys.
53
(7
), 690
–693
(1985
). 14.
15.
S.
Weinberg
, The Quantum Theory of Fields
(Cambridge U. P.
, Cambridge
, 1998
), Vol. II, pp. 164
–165
. 16.
We are leaving out of this discussion the anti-linear anti-unitary time-reversal operator , and the products and ( being the charge conjugation operator), which are very relevant for SSB in QFT.
17.
L. D.
Landau
and E. M.
Lifshitz
, Quantum Mechanics: Non-relativistic Theory
(Addison-Wesley
, Reading
, 1965
), p. 60
. 18.
K.
Heyde
, Basic Ideas and Concepts in Nuclear Physics: An Introductory Approach
(IOP
, Bristol
, 2004
), p. 439
. 19.
L. D.
Landau
and E. M.
Lifshitz
, Quantum Mechanics: Non-relativistic Theory
(Addison-Wesley
, Reading
, 1965
), Chap. XVI, paragraph 118. 20.
T.
Appelquist
, M. J.
Bowick
, E.
Cohler
, and A. I.
Hauser
, “ Breaking of isospin symmetry in theories with a dynamical Higgs mechanism
,” Phys. Rev. D
31
(7
), 1676
–1684
(1985
). 21.
B.-Q.
Ma
, “ Sea quark content of nucleons: Flavour distribution asymmetry or isospin symmetry breaking?
,” Phys. Lett. B
274
(1
), 111
–115
(1992
). 22.
M. J.
Savage
, “ Strong-Isospin Breaking in the Neutron-Proton Mass difference
,” in Proceedings of the XXIV International Symposium on Lattice Field Theory
(Proceedings of Science-SISSA, Trieste, 2006
). 23.
E.
Witten
, “ Dynamical breaking of supersymmetry
,” Nucl. Phys. B
185
(3
), 513
–554
(1981
). Although the toy model presented in this section is partially inspired by the quantum-mechanical supersymmetric models discussed by Witten, our system is clearly not supersymmetric. This is evident because the energy levels of our model are not degenerate above the ground-state level, while the excited levels of quantum-mechanical supersymmetric models are always degenerate, whether supersymmetry is broken or not. © 2012 American Association of Physics Teachers.
2012
American Association of Physics Teachers
AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.
