42
8 Examples
not energetically stable under external perturbations (see Chap. 6). This would be
the only vacuum state solution available in Segal’s approach.
If the Field ϕ takes values in R
n
, n > 1, the internal symmetry group is the
continuous group G of transformations (3.6), (3.7) with λ = 1, a = 0. In this case,
besides the trivial vacuum solution ϕ 0 = 0, the non-trivial vacuum solutions are
given by the points of the orbit
{ϕ
g
0 ≡ A g ¯
ϕ 0 , g ∈ G, ¯
ϕ
2
0 = μ
2
}.
(8.4)
For n = 1, the internal symmetry group is the discrete group
Z 2 : ϕ → −ϕ.
Clearly, in all cases, the internal symmetry group is unbroken in the trivial vacuum
sector H 0 , but it is spontaneously broken in each “pure phase” H g , defined by ϕ
g
0 .
ii) Time-independent solutions defining physical Hilbert space sectors. Kinks
Another interesting class are the time-independent solutions, which satisfy
(∂ x )
2
ϕ = λϕ(ϕ
2
− μ
2
).
(8.5)
This implies
∂ x (
1
2
ϕ
2
x −
1
4
λ(ϕ
2
− μ
2
)
2
) = 0, ϕ x ≡ ∂ x ϕ,
i.e.
1
2
ϕ
2
x =
1
4
λ(ϕ
2
− μ
2
)
2
+ C, C = constant.
(8.6)
For simplicity, we consider the case in which ϕ takes values in R, leaving the straightforward generalization as an exercise.
The discussion of the solutions of (8.5), as given in the literature, (see, e.g. the
references in the previous footnote), is done under the condition that they have finite
energy when the potential is so renormalized that it vanishes at its absolute minimum.
This means that
1
2
(∇ϕ)
2
+
1
4
λ(ϕ
2
− μ
2
)
2
∈ L
1
.
By the discussion of Chap. 5, this appears as too restrictive, since it does not consider
the possibility of energy renormalization, (6.4), and in particular it crucially depends
on the overall scale of the potential (it also excludes the trivial vacuum solution
ϕ 0 = 0!). For these reasons we prefer to leave open the energy renormalization.
To simplify the discussion we will only assume that ϕ has (bounded) limits
ϕ(±∞), when x → ±∞ (regularity at infinity). Then, quite generally, since U is
by assumption of class C
2 , also U
(ϕ) has bounded limits as x → ±∞ and (8.5)
implies that d
2
ϕ/dx
2 also does. On the other hand, for any test function f of compact
8 Examples
not energetically stable under external perturbations (see Chap. 6). This would be
the only vacuum state solution available in Segal’s approach.
If the Field ϕ takes values in R
n
, n > 1, the internal symmetry group is the
continuous group G of transformations (3.6), (3.7) with λ = 1, a = 0. In this case,
besides the trivial vacuum solution ϕ 0 = 0, the non-trivial vacuum solutions are
given by the points of the orbit
{ϕ
g
0 ≡ A g ¯
ϕ 0 , g ∈ G, ¯
ϕ
2
0 = μ
2
}.
(8.4)
For n = 1, the internal symmetry group is the discrete group
Z 2 : ϕ → −ϕ.
Clearly, in all cases, the internal symmetry group is unbroken in the trivial vacuum
sector H 0 , but it is spontaneously broken in each “pure phase” H g , defined by ϕ
g
0 .
ii) Time-independent solutions defining physical Hilbert space sectors. Kinks
Another interesting class are the time-independent solutions, which satisfy
(∂ x )
2
ϕ = λϕ(ϕ
2
− μ
2
).
(8.5)
This implies
∂ x (
1
2
ϕ
2
x −
1
4
λ(ϕ
2
− μ
2
)
2
) = 0, ϕ x ≡ ∂ x ϕ,
i.e.
1
2
ϕ
2
x =
1
4
λ(ϕ
2
− μ
2
)
2
+ C, C = constant.
(8.6)
For simplicity, we consider the case in which ϕ takes values in R, leaving the straightforward generalization as an exercise.
The discussion of the solutions of (8.5), as given in the literature, (see, e.g. the
references in the previous footnote), is done under the condition that they have finite
energy when the potential is so renormalized that it vanishes at its absolute minimum.
This means that
1
2
(∇ϕ)
2
+
1
4
λ(ϕ
2
− μ
2
)
2
∈ L
1
.
By the discussion of Chap. 5, this appears as too restrictive, since it does not consider
the possibility of energy renormalization, (6.4), and in particular it crucially depends
on the overall scale of the potential (it also excludes the trivial vacuum solution
ϕ 0 = 0!). For these reasons we prefer to leave open the energy renormalization.
To simplify the discussion we will only assume that ϕ has (bounded) limits
ϕ(±∞), when x → ±∞ (regularity at infinity). Then, quite generally, since U is
by assumption of class C
2 , also U
(ϕ) has bounded limits as x → ±∞ and (8.5)
implies that d
2
ϕ/dx
2 also does. On the other hand, for any test function f of compact