234
Appendix A: Long Range Dynamics and Vacuum Seizing
A.3 Local Generation of Continuous Symmetries
As stressed in Chap. 25, a crucial condition for the derivation of the Goldstone
theorem is that the continuous symmetry β
λ
, λ ∈ R is generated by a local (spacetime covariant) charge on an algebra stable under time evolution, at least in the
expectation of the ground state, Eq. (25.29).
It was further argued that the Coulomb delocalization induced by the dynamics
might explain the failure of such a local generation. We can now add a sharper
conclusion by exploiting the fact that, as a consequence of the long range interaction,
the symmetric time evolution α
t may involve variables at infinity which are not
pointwise invariant under β
λ .
Proposition A.2 Let A be the variable which defines the symmetry breaking order
parameter, < δ A > = 0, if its time evolution involves a non-symmetric function of a
variable at infinity, say B ∞ , such that in the ground/equilibrium state expectations
< β
λ
(B ∞ ) > =< B ∞ >,
(A.13)
then, β
λ is not generated by a local charge on α
t
(A).
Proof. By asymptotic abelianess, for any local charge Q R , [ Q R , B ∞ ] = 0 and
therefore < δ B ∞ > = 0 cannot be obtained through the limit of commutators of
local charges. Hence,
i lim
R→∞
< [ Q R , α
t
(A) ] > =< δα
t
(A) > .
(A.14)
In a factorial representation π which leads to an effective localization of the time
evolution, (at the expense of a non-symmetric dynamics α
t
π ), a continuous symmetry
β
λ may be generated by a local charge on the algebra A l of essential localization,
stable under the non-symmetric dynamics α
t
π . Indeed, in the applications, in particular
for the implications of symmetry breaking on the energy spectrum, it is enough that
the order parameter A belongs to an algebra stable under α
t
π , on which the continuous
symmetry β
λ is locally generated.
Then, one has
J (t) ≡ i lim
R→∞
< [ Q R , α
t
(A) ] >= i lim
R→∞
< [ Q R , α
t
π (A) ] >=
=
d
dλ
< β
λ
α
t
π (A) > | λ=0 =
d
dλ
< β
λ
α
t
(A) > | λ=0 =< δ A > .
(A.15)
Hence, a non-trivial time dependence of J (t) is possible compatibly with β
λ
α
t
=
α
t
β
λ , because the symmetry is locally generated on α
t
π (A) and not on α
t
(A).
Appendix A: Long Range Dynamics and Vacuum Seizing
A.3 Local Generation of Continuous Symmetries
As stressed in Chap. 25, a crucial condition for the derivation of the Goldstone
theorem is that the continuous symmetry β
λ
, λ ∈ R is generated by a local (spacetime covariant) charge on an algebra stable under time evolution, at least in the
expectation of the ground state, Eq. (25.29).
It was further argued that the Coulomb delocalization induced by the dynamics
might explain the failure of such a local generation. We can now add a sharper
conclusion by exploiting the fact that, as a consequence of the long range interaction,
the symmetric time evolution α
t may involve variables at infinity which are not
pointwise invariant under β
λ .
Proposition A.2 Let A be the variable which defines the symmetry breaking order
parameter, < δ A > = 0, if its time evolution involves a non-symmetric function of a
variable at infinity, say B ∞ , such that in the ground/equilibrium state expectations
< β
λ
(B ∞ ) > =< B ∞ >,
(A.13)
then, β
λ is not generated by a local charge on α
t
(A).
Proof. By asymptotic abelianess, for any local charge Q R , [ Q R , B ∞ ] = 0 and
therefore < δ B ∞ > = 0 cannot be obtained through the limit of commutators of
local charges. Hence,
i lim
R→∞
< [ Q R , α
t
(A) ] > =< δα
t
(A) > .
(A.14)
In a factorial representation π which leads to an effective localization of the time
evolution, (at the expense of a non-symmetric dynamics α
t
π ), a continuous symmetry
β
λ may be generated by a local charge on the algebra A l of essential localization,
stable under the non-symmetric dynamics α
t
π . Indeed, in the applications, in particular
for the implications of symmetry breaking on the energy spectrum, it is enough that
the order parameter A belongs to an algebra stable under α
t
π , on which the continuous
symmetry β
λ is locally generated.
Then, one has
J (t) ≡ i lim
R→∞
< [ Q R , α
t
(A) ] >= i lim
R→∞
< [ Q R , α
t
π (A) ] >=
=
d
dλ
< β
λ
α
t
π (A) > | λ=0 =
d
dλ
< β
λ
α
t
(A) > | λ=0 =< δ A > .
(A.15)
Hence, a non-trivial time dependence of J (t) is possible compatibly with β
λ
α
t
=
α
t
β
λ , because the symmetry is locally generated on α
t
π (A) and not on α
t
(A).