Chapter 7
Noether Theorem and Symmetry
Breaking
The existence of sectors, i.e. of “disjoint physical worlds” in the set of solutions of
the non-linear equation (4.6), provides the mathematical and physical basis for the
mechanism of spontaneous symmetry breaking briefly discussed in Chap. 2. We can
now understand the relation between the Noether theorem, the existence of conserved
currents and the occurrence of spontaneous symmetry breaking, which, among other
things, implies the lack of existence of the corresponding generators. As shown by
the following Propositions, the mechanism of spontaneous symmetry breaking is
related to the instability of a physical world under a symmetry operation.
Proposition 7.1
30 Let G denote the group of internal symmetries of (4.6). Then
1) G maps sectors into sectors and HSS into HSS
G : H ϕ → H g(ϕ) , ∀g ∈ G,
giving rise to orbits of sectors and of HSS.
2) Each HSS H ϕ determines a subgroup G ϕ of G such that
G ϕ : H ϕ → H ϕ .
G ϕ is called the stability group of H ϕ and H ϕ is the carrier of a representation
of its stability group.
3) A necessary and sufficient condition for G ϕ being the stability group of H ϕ is
that there exists one element ¯
ϕ ∈ H ϕ such that
G ϕ ¯
ϕ ∈ H ϕ .
(7.1)
30 Ref. II.
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer
Nature Switzerland AG 2021
F. Strocchi, Symmetry Breaking, Theoretical and Mathematical Physics,
https://doi.org/10.1007/978-3-662-62166-0_7
35
Noether Theorem and Symmetry
Breaking
The existence of sectors, i.e. of “disjoint physical worlds” in the set of solutions of
the non-linear equation (4.6), provides the mathematical and physical basis for the
mechanism of spontaneous symmetry breaking briefly discussed in Chap. 2. We can
now understand the relation between the Noether theorem, the existence of conserved
currents and the occurrence of spontaneous symmetry breaking, which, among other
things, implies the lack of existence of the corresponding generators. As shown by
the following Propositions, the mechanism of spontaneous symmetry breaking is
related to the instability of a physical world under a symmetry operation.
Proposition 7.1
30 Let G denote the group of internal symmetries of (4.6). Then
1) G maps sectors into sectors and HSS into HSS
G : H ϕ → H g(ϕ) , ∀g ∈ G,
giving rise to orbits of sectors and of HSS.
2) Each HSS H ϕ determines a subgroup G ϕ of G such that
G ϕ : H ϕ → H ϕ .
G ϕ is called the stability group of H ϕ and H ϕ is the carrier of a representation
of its stability group.
3) A necessary and sufficient condition for G ϕ being the stability group of H ϕ is
that there exists one element ¯
ϕ ∈ H ϕ such that
G ϕ ¯
ϕ ∈ H ϕ .
(7.1)
30 Ref. II.
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer
Nature Switzerland AG 2021
F. Strocchi, Symmetry Breaking, Theoretical and Mathematical Physics,
https://doi.org/10.1007/978-3-662-62166-0_7
35