Appendix A: Long Range Dynamics and Vacuum Seizing
235
According to Eq. (25.31), (in the proof of Theorem 25.3), the intermediate states
which contribute to the two point function < Q R A t > (briefly called Goldstone
excitations) have an energy spectrum at vanishing momentum, briefly called the
Goldstone spectrum, given by ˜
J (ω) and therefore a non-trivial t-dependence of J (t)
implies a departure from the standard Goldstone spectrum.
More detailed information on the energy spectrum are obtained if the boundary
counter terms, which define α
t
π according to Eq. (A.12), are local generators of a
covariance Lie group G of A l (at least in the factorial representation π), in the sense
of the covariance group of the dynamics discussed in Chap. 28.
14
This means that there is a Lie group G of automorphisms β
g , g ∈ G, of A l generated by local charges Q
i
R , i = 1, ..., n, n = dim G, such that the space translations
and α
t
π are subgroups of G and, ∀A ∈ A l ,
lim
R→∞
< [ ∂β
g
(ΔH R )/∂g i | g=0 , A ] >=
i lim
R→∞
n
k=1
c
i
k < [ Q
k
R , A ] >,
(A.16)
where ΔH R denotes the boundary counter terms in the sphere of radius R and
c
i
k , i, k = 1, ...n, are suitable coefficients in strict analogy with Eq. (28.3). Then,
Theorem 28.2 applies and one gets information on the energy spectrum of the Goldstone excitations, at vanishing momentum.
The validity of Eq. (A.16) is equivalent to the stability of the family of states Σ
under the effective dynamics α
t
π (see the above reference, especially Sect. 7).
Summarizing, in the presence of long range interactions, the failure of local
generation of continuous symmetries by local charges on the algebra stable under α
t is
explained by the fact that the time evolution of local variables involves non-invariant
variables at infinity on which local charges cannot act (by asymptotic abelianess).
Nevertheless, the Goldstone spectrum, related to symmetry breaking in a factorial
representation π, is under control if the time evolution α
t
= lim V α
t
V is essentially
local were it not for the occurrence of variables at infinity. This means that whenever such variables at infinity are frozen to their c-number values, in each factorial
representation π, the dynamics is described by a one-parameter group of the nonsymmetric effective dynamics α
t
π , which leaves stable an algebra A l of essential
localization. As discussed in Chap. 28, this allows to deduce the low momentum
energy spectrum of the related Goldstone excitations.
Clearly, the symmetries of the dynamics α
t (obtained through the removal of
the infrared or volume cutoff) are generically lost in the effective dynamics α
t
π and
therefore broken in π. Thus, long range interactions do not only invalidate one of the
14 For an extended analysis, see G. Morchio and F. Strocchi, Ann. Phys. 185, 241 (1988), esp. Sect. 6.
235
According to Eq. (25.31), (in the proof of Theorem 25.3), the intermediate states
which contribute to the two point function < Q R A t > (briefly called Goldstone
excitations) have an energy spectrum at vanishing momentum, briefly called the
Goldstone spectrum, given by ˜
J (ω) and therefore a non-trivial t-dependence of J (t)
implies a departure from the standard Goldstone spectrum.
More detailed information on the energy spectrum are obtained if the boundary
counter terms, which define α
t
π according to Eq. (A.12), are local generators of a
covariance Lie group G of A l (at least in the factorial representation π), in the sense
of the covariance group of the dynamics discussed in Chap. 28.
14
This means that there is a Lie group G of automorphisms β
g , g ∈ G, of A l generated by local charges Q
i
R , i = 1, ..., n, n = dim G, such that the space translations
and α
t
π are subgroups of G and, ∀A ∈ A l ,
lim
R→∞
< [ ∂β
g
(ΔH R )/∂g i | g=0 , A ] >=
i lim
R→∞
n
k=1
c
i
k < [ Q
k
R , A ] >,
(A.16)
where ΔH R denotes the boundary counter terms in the sphere of radius R and
c
i
k , i, k = 1, ...n, are suitable coefficients in strict analogy with Eq. (28.3). Then,
Theorem 28.2 applies and one gets information on the energy spectrum of the Goldstone excitations, at vanishing momentum.
The validity of Eq. (A.16) is equivalent to the stability of the family of states Σ
under the effective dynamics α
t
π (see the above reference, especially Sect. 7).
Summarizing, in the presence of long range interactions, the failure of local
generation of continuous symmetries by local charges on the algebra stable under α
t is
explained by the fact that the time evolution of local variables involves non-invariant
variables at infinity on which local charges cannot act (by asymptotic abelianess).
Nevertheless, the Goldstone spectrum, related to symmetry breaking in a factorial
representation π, is under control if the time evolution α
t
= lim V α
t
V is essentially
local were it not for the occurrence of variables at infinity. This means that whenever such variables at infinity are frozen to their c-number values, in each factorial
representation π, the dynamics is described by a one-parameter group of the nonsymmetric effective dynamics α
t
π , which leaves stable an algebra A l of essential
localization. As discussed in Chap. 28, this allows to deduce the low momentum
energy spectrum of the related Goldstone excitations.
Clearly, the symmetries of the dynamics α
t (obtained through the removal of
the infrared or volume cutoff) are generically lost in the effective dynamics α
t
π and
therefore broken in π. Thus, long range interactions do not only invalidate one of the
14 For an extended analysis, see G. Morchio and F. Strocchi, Ann. Phys. 185, 241 (1988), esp. Sect. 6.