Chapter 8
Examples
1) Non-linear Scalar Field in One Space Dimension
The model describes the simplest non-linear field theory and it can be regarded as
a prototype of field theories in one space dimension (s = 1). The model can also
be interpreted as a non-linear generalization of the wave equation. The interest of
the model is that, even at the classical level, it has stable solutions with a possible
particle interpretation.
36
The model is defined by the potential
U = −
1
2
m
2
ϕ
2
+
1
4
λϕ
4
=
1
4
λ(ϕ
2
− μ
2
)
2
−
1
4
λμ
4
, μ
2
≡ m
2
/λ,
(8.1)
and therefore the equations of motion read
ϕ = −λϕ(ϕ
2
− μ
2
).
(8.2)
i) Vacuum state solutions
The simplest solutions are the ground state solutions, invariant under space and time
translations, i.e. ϕ = const. If the field ϕ takes values in R, there are only three
possibilities
ϕ
±
0 = ±μ, ϕ 0 = 0.
(8.3)
By the discussion of Chaps. 5–7, ϕ
±
0 define disjoint Hilbert space sectors H±,
for which an energy–momentum density can be defined and for which the energy is
bounded below. The other constant solution ϕ 0 = 0, corresponding to the so-called
trivial vacuum sector, still defines a Hilbert space sector with energy–momentum
density, but the energy is not bounded below and therefore in this case the sector is
36 J. Goldstone and R. Jackiw, Phys. Rev. D11, 1486 (1975). See also R. Rajaraman, Solitons and
Instantons, North-Holland 1982 and references therein.
© 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_8
41
Examples
1) Non-linear Scalar Field in One Space Dimension
The model describes the simplest non-linear field theory and it can be regarded as
a prototype of field theories in one space dimension (s = 1). The model can also
be interpreted as a non-linear generalization of the wave equation. The interest of
the model is that, even at the classical level, it has stable solutions with a possible
particle interpretation.
36
The model is defined by the potential
U = −
1
2
m
2
ϕ
2
+
1
4
λϕ
4
=
1
4
λ(ϕ
2
− μ
2
)
2
−
1
4
λμ
4
, μ
2
≡ m
2
/λ,
(8.1)
and therefore the equations of motion read
ϕ = −λϕ(ϕ
2
− μ
2
).
(8.2)
i) Vacuum state solutions
The simplest solutions are the ground state solutions, invariant under space and time
translations, i.e. ϕ = const. If the field ϕ takes values in R, there are only three
possibilities
ϕ
±
0 = ±μ, ϕ 0 = 0.
(8.3)
By the discussion of Chaps. 5–7, ϕ
±
0 define disjoint Hilbert space sectors H±,
for which an energy–momentum density can be defined and for which the energy is
bounded below. The other constant solution ϕ 0 = 0, corresponding to the so-called
trivial vacuum sector, still defines a Hilbert space sector with energy–momentum
density, but the energy is not bounded below and therefore in this case the sector is
36 J. Goldstone and R. Jackiw, Phys. Rev. D11, 1486 (1975). See also R. Rajaraman, Solitons and
Instantons, North-Holland 1982 and references therein.
© 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_8
41