Chapter 25
Breaking of Continuous Symmetries:
Goldstone’s Theorem
For a long time, the mechanism of spontaneous breaking of continuous symmetries
has been recognized to be at the basis of many collective phenomena and, in particular,
of phase transitions in statistical mechanics; recently, it has played a crucial role in
the developments of theoretical physics, both at the level of many-body physics and
for the unification of elementary particle interactions.
For relativistic systems and more generally for systems with short range dynamics, the clarification of the mechanism has been achieved to a high level of rigour
and formalized in the so-called Goldstone’s theorem.
145 The result is that the conditions for the applicability of the conclusions, a subject of discussions in the early
developments, are now out of question.
The important point is that the Goldstone theorem provides non-perturbative
exact information on the excitation spectrum, since it predicts the low momentum
behaviour of the energy, ω(k) → 0, as k → 0, of the elementary excitations (Goldstone bosons) associated with the broken symmetry generators. The examples are
many and they appear in different branches of physics, like the spin waves in the
theory of ferromagnetism, the Landau phonons in the theory of superfluidity, the
phonon excitations in crystals, the pions as Goldstone particles of chiral symmetry
breaking, etc.
In this chapter, we first give the simple “heuristic proof” of the Goldstone theorem
(in the zero temperature case) without caring about subtle mathematical points; the
145 J. Goldstone, A. Salam and S. Weinberg, Phys. Rev. 127, 965 (1962); D. Kastler, D.W. Robinson
and J.A. Swieca, Comm. Math. Phys. 2, 108 (1966); D. Kastler, Broken Symmetries and the Goldstone Theorem in Axiomatic Field theory, in Proceedings of the 1967 International Conference on
Particles and Fields, C.R. Hagen et al. eds., Interscience 1967; J.A. Swieca, Goldstone theorem
and related topics, in Cargése Lectures in Physics, Vol. 4, D. Kastler ed., Gordon and Breach 1970;
R.F. Streater, Spontaneously broken symmetries, in Many degrees of freedom in Field Theory, L.
Streit ed., Plenum Press 1978.
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer
Nature Switzerland AG 2021
F. Strocchi, Symmetry Breaking, Theoretical and Mathematical Physics,
https://doi.org/10.1007/978-3-662-62166-0_25
169
Breaking of Continuous Symmetries:
Goldstone’s Theorem
For a long time, the mechanism of spontaneous breaking of continuous symmetries
has been recognized to be at the basis of many collective phenomena and, in particular,
of phase transitions in statistical mechanics; recently, it has played a crucial role in
the developments of theoretical physics, both at the level of many-body physics and
for the unification of elementary particle interactions.
For relativistic systems and more generally for systems with short range dynamics, the clarification of the mechanism has been achieved to a high level of rigour
and formalized in the so-called Goldstone’s theorem.
145 The result is that the conditions for the applicability of the conclusions, a subject of discussions in the early
developments, are now out of question.
The important point is that the Goldstone theorem provides non-perturbative
exact information on the excitation spectrum, since it predicts the low momentum
behaviour of the energy, ω(k) → 0, as k → 0, of the elementary excitations (Goldstone bosons) associated with the broken symmetry generators. The examples are
many and they appear in different branches of physics, like the spin waves in the
theory of ferromagnetism, the Landau phonons in the theory of superfluidity, the
phonon excitations in crystals, the pions as Goldstone particles of chiral symmetry
breaking, etc.
In this chapter, we first give the simple “heuristic proof” of the Goldstone theorem
(in the zero temperature case) without caring about subtle mathematical points; the
145 J. Goldstone, A. Salam and S. Weinberg, Phys. Rev. 127, 965 (1962); D. Kastler, D.W. Robinson
and J.A. Swieca, Comm. Math. Phys. 2, 108 (1966); D. Kastler, Broken Symmetries and the Goldstone Theorem in Axiomatic Field theory, in Proceedings of the 1967 International Conference on
Particles and Fields, C.R. Hagen et al. eds., Interscience 1967; J.A. Swieca, Goldstone theorem
and related topics, in Cargése Lectures in Physics, Vol. 4, D. Kastler ed., Gordon and Breach 1970;
R.F. Streater, Spontaneously broken symmetries, in Many degrees of freedom in Field Theory, L.
Streit ed., Plenum Press 1978.
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer
Nature Switzerland AG 2021
F. Strocchi, Symmetry Breaking, Theoretical and Mathematical Physics,
https://doi.org/10.1007/978-3-662-62166-0_25
169