184
25 Breaking of Continuous Symmetries: Goldstone’s Theorem
Fourier transform is proportional to δ(ω); one has to prove that the point ω = 0 arises
from states orthogonal to the ground state and it is the limit of the energy spectrum
when k → 0. This is essentially guaranteed by the charge integrability condition
which ensures that ˜
J (k, t) is a continuous function of k, so that the limit R → ∞,
which corresponds to the limit k → 0 of ˜
J (k, t), is related to the continuous limit of
the energy spectral support on real symmetric test functions ˜
g(ω) = ˜
g(ω) = ˜
g(−ω)
lim
k→0
˜
J (k, ω) = −2(2π)
2 lim
k→0
Im < j 0 ( f )Ψ 0 , d E(ω) d E(k) AΨ 0 > .
The charge integrability (condition) settles the problem of the possible non-continuity
of ˜
J (k, ω) at k → 0, raised by Klein and Lee
162 as a mechanism for evading the
Goldstone theorem and accounting for an energy gap associated with symmetry
breaking. The recourse to (approximate) locality to guarantee analyticity in k, as
advocated by Kibble and collaborators,
163 isolates a much too strong condition,
which in particular is not satisfied by systems with long range dynamics, whereas,
as discussed in Sect. 25.2, the integrability of the charge density commutators seems
general enough.
Proof. By the spectral theorem applied to U (x) U (−t), one has that ˜
J f (k, ω) is a
measure in k and ω, and by the regularity of f it is a finite measure in k. Furthermore,
by the absolute integrability (in x) of J f (x, t), one has that, as a distribution in ω,
˜
J f (k, ω) is continuous in k as k → 0. Thus
˜
J (ω) = lim
k→0
˜
J (k, ω) = ˜
J (0, ω).
By definition J (t) is real, so that ˜
J (ω) = ˜
J (−ω) vanishes on test functions ˜
g(ω) =
− ˜
g(−ω), whereas if ˜
g(ω) = ˜
g(−ω) one has
˜
J ( ˜
g) = (2π)
2 lim
k→0
i
dω ˜
g(ω)[< j 0 ( f )Ψ 0 , d E(−ω) d E(k) AΨ 0 >
−< j o ( f )Ψ 0 , d E(ω) d E(−k) AΨ 0 >]
= −2(2π)
2 Im
dω ˜
g(ω) < j 0 ( f )Ψ 0 , d E(−ω) d E(k = 0) AΨ 0 > .
Thus, as a distribution on real symmetric test functions
˜
J (ω) = −2(2π)
2 Im < j 0 ( f )Ψ 0 , d E(−ω) d E(k = 0) AΨ 0 > .
(25.31)
162 A. Klein and B.W Lee, Phys. Rev. Lett. 12, 266 (1964).
163 T.W.B. Kibble, Broken Symmetries, in Proc. Oxford Internat. Conf. on Elementary Particles,
Oxford 1965, p.19; G.S. Guralnik, C.R. Hagen and T.W.B. Kibble, Broken Symmetries and the
Goldstone Theorem, in Advances in Particle Physics, Vol 2., R.L. Cool, R.E. Marshak eds., Interscience 1968.
25 Breaking of Continuous Symmetries: Goldstone’s Theorem
Fourier transform is proportional to δ(ω); one has to prove that the point ω = 0 arises
from states orthogonal to the ground state and it is the limit of the energy spectrum
when k → 0. This is essentially guaranteed by the charge integrability condition
which ensures that ˜
J (k, t) is a continuous function of k, so that the limit R → ∞,
which corresponds to the limit k → 0 of ˜
J (k, t), is related to the continuous limit of
the energy spectral support on real symmetric test functions ˜
g(ω) = ˜
g(ω) = ˜
g(−ω)
lim
k→0
˜
J (k, ω) = −2(2π)
2 lim
k→0
Im < j 0 ( f )Ψ 0 , d E(ω) d E(k) AΨ 0 > .
The charge integrability (condition) settles the problem of the possible non-continuity
of ˜
J (k, ω) at k → 0, raised by Klein and Lee
162 as a mechanism for evading the
Goldstone theorem and accounting for an energy gap associated with symmetry
breaking. The recourse to (approximate) locality to guarantee analyticity in k, as
advocated by Kibble and collaborators,
163 isolates a much too strong condition,
which in particular is not satisfied by systems with long range dynamics, whereas,
as discussed in Sect. 25.2, the integrability of the charge density commutators seems
general enough.
Proof. By the spectral theorem applied to U (x) U (−t), one has that ˜
J f (k, ω) is a
measure in k and ω, and by the regularity of f it is a finite measure in k. Furthermore,
by the absolute integrability (in x) of J f (x, t), one has that, as a distribution in ω,
˜
J f (k, ω) is continuous in k as k → 0. Thus
˜
J (ω) = lim
k→0
˜
J (k, ω) = ˜
J (0, ω).
By definition J (t) is real, so that ˜
J (ω) = ˜
J (−ω) vanishes on test functions ˜
g(ω) =
− ˜
g(−ω), whereas if ˜
g(ω) = ˜
g(−ω) one has
˜
J ( ˜
g) = (2π)
2 lim
k→0
i
dω ˜
g(ω)[< j 0 ( f )Ψ 0 , d E(−ω) d E(k) AΨ 0 >
−< j o ( f )Ψ 0 , d E(ω) d E(−k) AΨ 0 >]
= −2(2π)
2 Im
dω ˜
g(ω) < j 0 ( f )Ψ 0 , d E(−ω) d E(k = 0) AΨ 0 > .
Thus, as a distribution on real symmetric test functions
˜
J (ω) = −2(2π)
2 Im < j 0 ( f )Ψ 0 , d E(−ω) d E(k = 0) AΨ 0 > .
(25.31)
162 A. Klein and B.W Lee, Phys. Rev. Lett. 12, 266 (1964).
163 T.W.B. Kibble, Broken Symmetries, in Proc. Oxford Internat. Conf. on Elementary Particles,
Oxford 1965, p.19; G.S. Guralnik, C.R. Hagen and T.W.B. Kibble, Broken Symmetries and the
Goldstone Theorem, in Advances in Particle Physics, Vol 2., R.L. Cool, R.E. Marshak eds., Interscience 1968.