170
25 Breaking of Continuous Symmetries: Goldstone’s Theorem
aim is to show in a simple way the connection between symmetry breaking of a
continuous symmetry and absence of energy gap.
The idea is that if the ground state ω of an extended system is not symmetric
under a continuous symmetry β
λ , λ ∈ R, leaving the Hamiltonian invariant, then the
states ω β R , obtained from ω by applying a symmetry transformation β R localized in
a region of radius R, have the same energy of the ground state except for boundary
terms. Since the symmetry is continuous one can smooth the transition region so that
the boundary terms vanish when R → ∞, and so does the energy of the states ω β R .
To formalize the idea, one abstracts from the Lagrangian (or Hamiltonian) formulation the information (Noether’s theorem) that the invariance under a continuous
symmetry β
λ implies the existence of a conserved current, whose charge density
generates the symmetry transformation, namely, ∀A ∈ A L (= the local algebra), the
infinitesimal variation under β
λ is given by
146
δ A = dβ
λ
(A)/dλ| λ=0 = i lim
R→∞
[Q R , A],
(25.1)
Q R =
|x|≤R
d
s x j 0 (x, 0),
(25.2)
∂ t j 0 (x, t) + div j(x, t) = 0.
(25.3)
A relevant point is that the above infinitesimal generation of the symmetry holds
for a subalgebra A 0 of A, containing A L , stable under time translations. The above
equations encode the essential features of a continuous symmetry without relying on
the definition of the Lagrangian.
For symmetries which commute with space and time translations, the current is
assumed to transform covariantly under space and time translations
U (a, τ ) j μ (x, t)U (a, τ )
−1
= j μ (x + a, t + τ ), μ = 0, 1, . . .
(25.4)
Briefly, an internal continuous symmetry β
λ satisfying the above properties is
said to be locally generated by a charge density associated with a conserved current.
25.1 The Goldstone Theorem
The (heuristic) version of the Goldstone theorem,
147 which does not use manifest
relativistic covariance, says ( A = A
∗ , j 0 = j
∗
0 covers the general case since any B
is = B 1 + i B 2 , B i = B
∗
i ; in fact, < [ j 0 , B] > = 0 implies that at least one of the
two < [ j 0 , B i ] > = 0 and then for it one may similarly decompose j 0 = j
1
0 + i j
2
0 ,
j
i
0 = ( j
i
0 )
∗
):
146 Equation (25.1) may be understood to hold as a bilinear form on a dense set of states, in each
relevant representation; actually, all what is needed is its validity on the ground state.
147 The non-relativistic version has been discussed in particular by R.V. Lange, Phys. Rev. Lett. 14,
3 (1965); Phys. Rev. 146, 301 (1966) and by J.A. Swieca, Comm. Math. Phys. 4, 1 (1967).
25 Breaking of Continuous Symmetries: Goldstone’s Theorem
aim is to show in a simple way the connection between symmetry breaking of a
continuous symmetry and absence of energy gap.
The idea is that if the ground state ω of an extended system is not symmetric
under a continuous symmetry β
λ , λ ∈ R, leaving the Hamiltonian invariant, then the
states ω β R , obtained from ω by applying a symmetry transformation β R localized in
a region of radius R, have the same energy of the ground state except for boundary
terms. Since the symmetry is continuous one can smooth the transition region so that
the boundary terms vanish when R → ∞, and so does the energy of the states ω β R .
To formalize the idea, one abstracts from the Lagrangian (or Hamiltonian) formulation the information (Noether’s theorem) that the invariance under a continuous
symmetry β
λ implies the existence of a conserved current, whose charge density
generates the symmetry transformation, namely, ∀A ∈ A L (= the local algebra), the
infinitesimal variation under β
λ is given by
146
δ A = dβ
λ
(A)/dλ| λ=0 = i lim
R→∞
[Q R , A],
(25.1)
Q R =
|x|≤R
d
s x j 0 (x, 0),
(25.2)
∂ t j 0 (x, t) + div j(x, t) = 0.
(25.3)
A relevant point is that the above infinitesimal generation of the symmetry holds
for a subalgebra A 0 of A, containing A L , stable under time translations. The above
equations encode the essential features of a continuous symmetry without relying on
the definition of the Lagrangian.
For symmetries which commute with space and time translations, the current is
assumed to transform covariantly under space and time translations
U (a, τ ) j μ (x, t)U (a, τ )
−1
= j μ (x + a, t + τ ), μ = 0, 1, . . .
(25.4)
Briefly, an internal continuous symmetry β
λ satisfying the above properties is
said to be locally generated by a charge density associated with a conserved current.
25.1 The Goldstone Theorem
The (heuristic) version of the Goldstone theorem,
147 which does not use manifest
relativistic covariance, says ( A = A
∗ , j 0 = j
∗
0 covers the general case since any B
is = B 1 + i B 2 , B i = B
∗
i ; in fact, < [ j 0 , B] > = 0 implies that at least one of the
two < [ j 0 , B i ] > = 0 and then for it one may similarly decompose j 0 = j
1
0 + i j
2
0 ,
j
i
0 = ( j
i
0 )
∗
):
146 Equation (25.1) may be understood to hold as a bilinear form on a dense set of states, in each
relevant representation; actually, all what is needed is its validity on the ground state.
147 The non-relativistic version has been discussed in particular by R.V. Lange, Phys. Rev. Lett. 14,
3 (1965); Phys. Rev. 146, 301 (1966) and by J.A. Swieca, Comm. Math. Phys. 4, 1 (1967).