202
28 An Extension of Goldstone Theorem to Non-symmetric Hamiltonians
H L = H inv,L (s) + h
|i|≤L
s
3
i ,
(28.6)
where H inv,L (s) is a rotationally invariant spin Hamiltonian with finite range interactions, having a translationally invariant ground state.
The rotations and the dynamics generate a Lie group G, as the covariance group
of the dynamics. As a consequence of the finite range, the time evolution induces a
delocalization of fast decrease
189 ; then the commutators of
S
a
R (t) ≡
|i|≤R
s
a
i (t), α = 1, 2, 3,
with a local A are absolutely summable in norm (as distributions in t) and the same
property holds for the algebra A 0 generated by the time evolved elements of A L .
Under general technical conditions one can also prove that (28.3) hold on A 0 .
190
The presence of the external magnetic field implies the breaking of the symmetries
generated by S
1
R , S
2
R and the matrix ˜
c is given by
˜
c ii = 0, i = 1, 2, ˜
c 12 = −ih = ˜
c 21 .
Then, Theorem 18.1 implies that there are Goldstone quasi-particles with energy
ω(k) satisfying
lim
k→0
ω(k) = h.
189 By (17.16), if A ∈ A(V 0 ), there are suitable positive constants C, v such that for |t| < v −1 |x|,
||[s
a
i , α t (A)]|| = ||[α −t (s
a
i ), A]|| ≤ Ce
−dist(i,V A )/2 ,
(28.7)
where dist (i, V A ) is the distance between the lattice point i and the localization region V A of A.
This implies that a fast decrease of the delocalization induced by the dynamics holds for all A of
the form A = α τ (B), τ ∈ R, B ∈ A L and therefore for the algebra A 0 generated by them.
190 G. Morchio and F. Strocchi, Ann. Phys. 185, 241 (1988).
28 An Extension of Goldstone Theorem to Non-symmetric Hamiltonians
H L = H inv,L (s) + h
|i|≤L
s
3
i ,
(28.6)
where H inv,L (s) is a rotationally invariant spin Hamiltonian with finite range interactions, having a translationally invariant ground state.
The rotations and the dynamics generate a Lie group G, as the covariance group
of the dynamics. As a consequence of the finite range, the time evolution induces a
delocalization of fast decrease
189 ; then the commutators of
S
a
R (t) ≡
|i|≤R
s
a
i (t), α = 1, 2, 3,
with a local A are absolutely summable in norm (as distributions in t) and the same
property holds for the algebra A 0 generated by the time evolved elements of A L .
Under general technical conditions one can also prove that (28.3) hold on A 0 .
190
The presence of the external magnetic field implies the breaking of the symmetries
generated by S
1
R , S
2
R and the matrix ˜
c is given by
˜
c ii = 0, i = 1, 2, ˜
c 12 = −ih = ˜
c 21 .
Then, Theorem 18.1 implies that there are Goldstone quasi-particles with energy
ω(k) satisfying
lim
k→0
ω(k) = h.
189 By (17.16), if A ∈ A(V 0 ), there are suitable positive constants C, v such that for |t| < v −1 |x|,
||[s
a
i , α t (A)]|| = ||[α −t (s
a
i ), A]|| ≤ Ce
−dist(i,V A )/2 ,
(28.7)
where dist (i, V A ) is the distance between the lattice point i and the localization region V A of A.
This implies that a fast decrease of the delocalization induced by the dynamics holds for all A of
the form A = α τ (B), τ ∈ R, B ∈ A L and therefore for the algebra A 0 generated by them.
190 G. Morchio and F. Strocchi, Ann. Phys. 185, 241 (1988).