Chapter 26
∗ The Goldstone Theorem at Non-zero
Temperature
The proof of the Goldstone theorem can be extended to the case of non-zero temperature T = 1/β, i.e. to representations defined by KMS states. In this case, the interest
of the theorem is in the prediction of the Goldstone quasi-particles, which crucially
depends on the integrability of the charge density commutators. The absence of an
energy gap (as in Theorem 25.1) is not very significant, since it is already implied
by general properties (like time-like clustering) of the KMS states.
164
As we shall see, in the non-zero temperature case the fine mathematical points
discussed in Sect. 25.3, e.g. the distributional properties of J (x, t), become more
relevant also in view of some puzzling statements that have appeared in the literature.
Theorem 26.1 (Non-relativistic Goldstone Theorem for T = 0)
Under the assumptions I, II, III of Theorem 25.3, with π a representation defined
by a translationallly invariant KMS state , the same conclusions hold (existence
Goldstone quasi-particles).
Moreover, if < > denotes the expectation on , the Fourier transform ˜
W(k, ω)
of the two-point function
W(x, t) ≡< j 0 ( f x ) α −t (A) >,
164 R. Haag, D. Kastler and E.B. Trych-Pohlmeyer, Comm. Math. Phys. 38,137 (1974), Prop. 3. As a
consequence of this result, several papers have been devoted to a version of the Goldstone theorem
which relates symmetry breaking to poor clustering, rather than to the absence of energy gap:
L. Landau, J. Fernando Perez and W.F. Wreszinski, J. Stat. Phys. 26, 755 (1981); Ph. Martin, Nuovo
Cim. 68, 302 (1982); M. Fannes, J.V. Pulé and A. Verbeure, Lett. Math. Phys. 6, 385 (1982) and the
reviews Ch. Gruber and P.A. Martin, Goldstone theorem in Statistical mechanics, in Mathematical
Problems in Theoretical Physics, (Berlin Conference 1981), R. Schrader et al. eds., Springer 1982,
p. 25; W.F. Wreszinski, Fortschr. Phys. 35, 379 (1987).
© 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_26
187
∗ The Goldstone Theorem at Non-zero
Temperature
The proof of the Goldstone theorem can be extended to the case of non-zero temperature T = 1/β, i.e. to representations defined by KMS states. In this case, the interest
of the theorem is in the prediction of the Goldstone quasi-particles, which crucially
depends on the integrability of the charge density commutators. The absence of an
energy gap (as in Theorem 25.1) is not very significant, since it is already implied
by general properties (like time-like clustering) of the KMS states.
164
As we shall see, in the non-zero temperature case the fine mathematical points
discussed in Sect. 25.3, e.g. the distributional properties of J (x, t), become more
relevant also in view of some puzzling statements that have appeared in the literature.
Theorem 26.1 (Non-relativistic Goldstone Theorem for T = 0)
Under the assumptions I, II, III of Theorem 25.3, with π a representation defined
by a translationallly invariant KMS state , the same conclusions hold (existence
Goldstone quasi-particles).
Moreover, if < > denotes the expectation on , the Fourier transform ˜
W(k, ω)
of the two-point function
W(x, t) ≡< j 0 ( f x ) α −t (A) >,
164 R. Haag, D. Kastler and E.B. Trych-Pohlmeyer, Comm. Math. Phys. 38,137 (1974), Prop. 3. As a
consequence of this result, several papers have been devoted to a version of the Goldstone theorem
which relates symmetry breaking to poor clustering, rather than to the absence of energy gap:
L. Landau, J. Fernando Perez and W.F. Wreszinski, J. Stat. Phys. 26, 755 (1981); Ph. Martin, Nuovo
Cim. 68, 302 (1982); M. Fannes, J.V. Pulé and A. Verbeure, Lett. Math. Phys. 6, 385 (1982) and the
reviews Ch. Gruber and P.A. Martin, Goldstone theorem in Statistical mechanics, in Mathematical
Problems in Theoretical Physics, (Berlin Conference 1981), R. Schrader et al. eds., Springer 1982,
p. 25; W.F. Wreszinski, Fortschr. Phys. 35, 379 (1987).
© 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_26
187