Chapter 10
Appendix
10.1 Properties of the Free Wave Propagator
a) W (t) maps S × S into S × S
If u ∈ S(R
s
) × S(R
s
) (S(R
s
) is the Schwartz space of C
∞ test functions decreasing
at infinity faster than any inverse polynomial), then the solution of the free wave
equation is easily obtained by Fourier transform and one has
W (t)
ϕ 0 (k)
ψ 0 (k)
=
cos|k|t (sin|k|t)/|k|
−|k|sin|k|t cos|k|t
ϕ 0 (k)
ψ 0 (k)
.
(10.1)
cos |k|t, (sin |k|t)/|k| etc. are multipliers of S ≡ S(R
s
) continuous in t and
d
dt
W (t)| t=0 =
0 1
|k|
2 0
= K .
(10.2)
The group property is easily checked.
b) Hyperbolic character of W (t). Huygens’ principle.
Let Ω R−t be concentric spheres in R
s of radius R − t, 0 ≤ t ≤ R − δ, δ > 0, for
simplicity centered at the origin, then
W (t)u 0 Ω R−t ≤ e
|t|/2
u 0 Ω R .
(10.3)
This is a mathematical formulation of Huygens’ principle: the norm of u(t) in Ω R−t
depends only on the norm of u(0) in Ω R (influence domain). We start by proving
(10.3) for u ∈ S × S. The free wave equation implies
1
2
d
dt
[(∇ϕ)
2
+ ψ
2
] − ∇ · (ψ∇ϕ) = 0
(10.4)
© 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_10
53
Appendix
10.1 Properties of the Free Wave Propagator
a) W (t) maps S × S into S × S
If u ∈ S(R
s
) × S(R
s
) (S(R
s
) is the Schwartz space of C
∞ test functions decreasing
at infinity faster than any inverse polynomial), then the solution of the free wave
equation is easily obtained by Fourier transform and one has
W (t)
ϕ 0 (k)
ψ 0 (k)
=
cos|k|t (sin|k|t)/|k|
−|k|sin|k|t cos|k|t
ϕ 0 (k)
ψ 0 (k)
.
(10.1)
cos |k|t, (sin |k|t)/|k| etc. are multipliers of S ≡ S(R
s
) continuous in t and
d
dt
W (t)| t=0 =
0 1
|k|
2 0
= K .
(10.2)
The group property is easily checked.
b) Hyperbolic character of W (t). Huygens’ principle.
Let Ω R−t be concentric spheres in R
s of radius R − t, 0 ≤ t ≤ R − δ, δ > 0, for
simplicity centered at the origin, then
W (t)u 0 Ω R−t ≤ e
|t|/2
u 0 Ω R .
(10.3)
This is a mathematical formulation of Huygens’ principle: the norm of u(t) in Ω R−t
depends only on the norm of u(0) in Ω R (influence domain). We start by proving
(10.3) for u ∈ S × S. The free wave equation implies
1
2
d
dt
[(∇ϕ)
2
+ ψ
2
] − ∇ · (ψ∇ϕ) = 0
(10.4)
© 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_10
53