Chapter 3
Symmetries in Classical Field Theory
As the previous discussion indicates, it is impossible to realize the phenomenon of
(spontaneous) breaking of a continuous symmetry in classical mechanical systems
with a finite number of degrees of freedom described by canonical variables. We are
thus led to consider infinite-dimensional systems, like classical fields.
Our main purpose is to recognize the existence of disjoint “phases”, in the set of
solutions of the classical field equations, with the interpretation of possible disjoint
realizations of the system (Chap. 5). The phenomenon of spontaneous symmetry
breaking in a given “phase” will then be explained by its instability under the symmetry transformation.
To simplify the discussion, we will focus our attention to the standard case of the
non-linear equation
ϕ + U
(ϕ) = 0,
(3.1)
where ≡ (∂ t )
2
− , ϕ = ϕ(x, t), x ∈ R
s
, t ∈ R, is a field taking values in R
n ,
(an n-component field), U (ϕ) is the potential, which for the moment will be assumed
to be sufficiently regular, and U
denotes its derivative.
Equation (3.1) can be derived by the stationarity of the following action integral
A(ϕ, ˙
ϕ) =
d
s x dt
−
1
2
(∇ϕ)
2
+
1
2
˙
ϕ
2
− U (ϕ)
.
A typical prototype is given by
U (ϕ) =
1
4
λ(ϕ
2
− a
2
)
2
(3.2)
which is the infinite-dimensional version of the double well potential discussed in
Chap. 1.
© 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_3
11
Symmetries in Classical Field Theory
As the previous discussion indicates, it is impossible to realize the phenomenon of
(spontaneous) breaking of a continuous symmetry in classical mechanical systems
with a finite number of degrees of freedom described by canonical variables. We are
thus led to consider infinite-dimensional systems, like classical fields.
Our main purpose is to recognize the existence of disjoint “phases”, in the set of
solutions of the classical field equations, with the interpretation of possible disjoint
realizations of the system (Chap. 5). The phenomenon of spontaneous symmetry
breaking in a given “phase” will then be explained by its instability under the symmetry transformation.
To simplify the discussion, we will focus our attention to the standard case of the
non-linear equation
ϕ + U
(ϕ) = 0,
(3.1)
where ≡ (∂ t )
2
− , ϕ = ϕ(x, t), x ∈ R
s
, t ∈ R, is a field taking values in R
n ,
(an n-component field), U (ϕ) is the potential, which for the moment will be assumed
to be sufficiently regular, and U
denotes its derivative.
Equation (3.1) can be derived by the stationarity of the following action integral
A(ϕ, ˙
ϕ) =
d
s x dt
−
1
2
(∇ϕ)
2
+
1
2
˙
ϕ
2
− U (ϕ)
.
A typical prototype is given by
U (ϕ) =
1
4
λ(ϕ
2
− a
2
)
2
(3.2)
which is the infinite-dimensional version of the double well potential discussed in
Chap. 1.
© 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_3
11