36
7 Noether Theorem and Symmetry Breaking
Proof. By the characterization of internal symmetries given by (3.6), (3.7), u
(t) −
u(t) ∈ C
0
(H
1
⊕ L
2
, R) implies
g(ϕ
(t)) − g(ϕ(t)) = A g (ϕ
(t) − ϕ(t)) ∈ C
0
(H
1
⊕ L
2
, R),
so that sectors are mapped into sectors.
Furthermore, if u 0 = {ϕ 0 , ψ 0 }, with ϕ 0 ∈ L
∞
(R
s
), ψ 0 ∈ L
2
(R
s
), satisfies condition
b) of Theorem 5.2, it follows that A g ϕ 0 + a g ∈ L
∞
(R
s
), A g ψ 0 ∈ L
2
(R
s
) and, by
(3.6), (3.10),
Δg(ϕ 0 ) − U
(g(ϕ 0 )) = A g (Δϕ 0 − U
(ϕ 0 )) ∈ H
−1
(R
s
),
i.e. g maps HSS into HSS.
Finally, for any element ϕ of H ϕ 0 , putting χ = ϕ − ϕ 0 , one has
g(ϕ) − ϕ 0 = A g χ + g(ϕ 0 ) − ϕ 0
(7.2)
and since for any g ∈ G ϕ 0 , g(ϕ 0 ) − ϕ 0 ∈ H
1
(R
s
) ⊕ L
2
(R
s
), by (7.2) the mapping
g induces an affine mapping on H
1
⊕ L
2 to which H ϕ 0 is naturally identified, by
Theorem 5.2.
Conversely, by arguing as for (7.2), if ∃ ¯
ϕ ∈ H ϕ 0 such that g( ¯
ϕ) − ¯
ϕ ∈ H
1
⊕ L
2 so
does g(ϕ) − ϕ 0 , ∀ϕ ∈ H ϕ 0 , i.e. g ∈ G ϕ 0 .
Since, as discussed before, different HSSs define “disjoint physical worlds”, an
internal symmetry of the field equation (4.6) gives rise to a symmetry of the physical
world described by the Hilbert sector H ϕ only if it maps H ϕ into H ϕ . Otherwise the
symmetry is spontaneously broken.
As discussed in the Introduction, if H ϕ is not stable under G, its elements cannot
be classified in terms of irreducible representations of G. It is now clear what distinguishes the infinite-dimensional case with respect to the finite-dimensional one.
In the latter case, degenerate ground states related by a continuous symmetry, cannot be separated by potential barriers and one can move from one to the other by
physically realizable operations. In the infinite-dimensional case, degenerate ground
states characterize different large distance behaviours of the field configurations, so
that, even if they are related by a continuous symmetry, they cannot be related by
physically realizable operations, since the latter ones must both involve finite energy
and be essentially localized.
When the field equations can be derived by a Lagrangian, the link between the
invariance group of the Lagrangian and the existence of conservation laws is provided
by the classical Noether’s theorem. The existence of a continuity equation or a local
conservation law, however, does not in general imply the existence of a conserved
charge, since, first of all, the integral which defines the charge
Q
i
=
d
3 x J
i
0 (x)
may not converge.
7 Noether Theorem and Symmetry Breaking
Proof. By the characterization of internal symmetries given by (3.6), (3.7), u
(t) −
u(t) ∈ C
0
(H
1
⊕ L
2
, R) implies
g(ϕ
(t)) − g(ϕ(t)) = A g (ϕ
(t) − ϕ(t)) ∈ C
0
(H
1
⊕ L
2
, R),
so that sectors are mapped into sectors.
Furthermore, if u 0 = {ϕ 0 , ψ 0 }, with ϕ 0 ∈ L
∞
(R
s
), ψ 0 ∈ L
2
(R
s
), satisfies condition
b) of Theorem 5.2, it follows that A g ϕ 0 + a g ∈ L
∞
(R
s
), A g ψ 0 ∈ L
2
(R
s
) and, by
(3.6), (3.10),
Δg(ϕ 0 ) − U
(g(ϕ 0 )) = A g (Δϕ 0 − U
(ϕ 0 )) ∈ H
−1
(R
s
),
i.e. g maps HSS into HSS.
Finally, for any element ϕ of H ϕ 0 , putting χ = ϕ − ϕ 0 , one has
g(ϕ) − ϕ 0 = A g χ + g(ϕ 0 ) − ϕ 0
(7.2)
and since for any g ∈ G ϕ 0 , g(ϕ 0 ) − ϕ 0 ∈ H
1
(R
s
) ⊕ L
2
(R
s
), by (7.2) the mapping
g induces an affine mapping on H
1
⊕ L
2 to which H ϕ 0 is naturally identified, by
Theorem 5.2.
Conversely, by arguing as for (7.2), if ∃ ¯
ϕ ∈ H ϕ 0 such that g( ¯
ϕ) − ¯
ϕ ∈ H
1
⊕ L
2 so
does g(ϕ) − ϕ 0 , ∀ϕ ∈ H ϕ 0 , i.e. g ∈ G ϕ 0 .
Since, as discussed before, different HSSs define “disjoint physical worlds”, an
internal symmetry of the field equation (4.6) gives rise to a symmetry of the physical
world described by the Hilbert sector H ϕ only if it maps H ϕ into H ϕ . Otherwise the
symmetry is spontaneously broken.
As discussed in the Introduction, if H ϕ is not stable under G, its elements cannot
be classified in terms of irreducible representations of G. It is now clear what distinguishes the infinite-dimensional case with respect to the finite-dimensional one.
In the latter case, degenerate ground states related by a continuous symmetry, cannot be separated by potential barriers and one can move from one to the other by
physically realizable operations. In the infinite-dimensional case, degenerate ground
states characterize different large distance behaviours of the field configurations, so
that, even if they are related by a continuous symmetry, they cannot be related by
physically realizable operations, since the latter ones must both involve finite energy
and be essentially localized.
When the field equations can be derived by a Lagrangian, the link between the
invariance group of the Lagrangian and the existence of conservation laws is provided
by the classical Noether’s theorem. The existence of a continuity equation or a local
conservation law, however, does not in general imply the existence of a conserved
charge, since, first of all, the integral which defines the charge
Q
i
=
d
3 x J
i
0 (x)
may not converge.