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4 Nambu–Goldstone Conjecture It is possible that both explicit and spontaneous symmetry breaking be present. 170) An example that illustrates all of these possibilities is the inﬁnite ferromagnet, where the symmetry in question is rotational invariance. In the paramagnetic phase at temperature T > Tc there is exact symmetry; in the ferromagnetic phase, T < Tc , there is spontaneous symmetry breaking. When an external magnetic ﬁeld is applied, this gives explicit symmetry breaking for both T > Tc and T < Tc .
The ﬁrst mechanism to be considered is explicit symmetry breaking, where one adds to the symmetric Lagrangian (L0 ) a piece (L1 ) that is noninvariant under the full symmetry group G, although L1 may be invariant under some subgroup G of G. 164) The explicit breaking is used traditionally for the breaking of ﬂavor groups SU(3) and SU(4) in hadron physics. The second mechanism is spontaneous symmetry breaking (perhaps more appropriately called hidden symmetry). 166) This is because as a consequence of the dynamics the vacuum state is degenerate, and the choice of one as the physical vacuum breaks the symmetry.
22 B. Renner, Current Algebras, Pergamon Press, Elmsford, New York, 1968. L. Adler and R. A. Benjamin, New York, 1969. N. Bogoliubov, J. Phys. ) 11, 23 (1947). 25 W. Heisenberg, Z. Phys. 49, 619 (1928). 26 J. Bardeen, L. Cooper, and J. Schrieffer, Phys. Rev. 106, 162 (1957). W. Anderson, Phys. Rev. 130, 439 (1963). W. Higgs, Phys. Lett. 12, 132 (1964). W. Higgs, Phys. Rev. Lett. 13, 508 (1964). W. Higgs, Phys. Rev. 145, 1156 (1966). B. Kibble, Phys. Rev. 155, 1554 (1967). 1 Introduction Here we study the question of quantizing Yang–Mills ﬁeld theories.