Appendix B: Breaking of the Galilei Group and Plasmon Energy Spectrum
241
On the quasi-local algebra A 0 the (local) generators of G are determined up to
space divergences and may be taken, respectively, as
G R = m
d
3 x f R (x) x ρ(x), N R =
d
3 x f R (x) ρ(x),
P R =
1
2 i
d
3 x f R (x) [(∇ψ
∗
)ψ − ψ
∗
∇ψ](x) = m
d
3 x f R (x)j(x).
(B.7)
The only non-trivial commutator is
[ G i , P k ] = m δ ik N .
Then, in a factorial representation defined by a translationally invariant state ω
with < A >≡ ω(A), one has (all operators at time t = 0)
< ∂β
v
( j k (y))/∂v i > | v=0 = i lim
R→∞
< [ G i,R , j k (y) ] >= δ ik < ρ >,
(B.8)
and a non-zero density < ρ >= ρ B implies the breaking of the Galilei group. This
is actually a general feature of non-relativistic systems with the algebraic symmetry
of the Galilei group and non-zero density.
21
iv) Plasmon as Goldstone excitations with energy gap
As anticipated, for the breaking of the Galilei group, one expects that the long range
Coulomb interaction allows to evade the conclusions of the Goldstone theorem, with
an energy gap in the Goldstone spectrum in a factorial representation defined by a
ground state invariant under space translations and rotations.
Proposition B.1 In a factorial representation defined by a ground state invariant
under space translations and rotations, with expectations denoted by < >, one has
i lim
R→∞
< [ G i,R (t), j i (0) ] >= ρ B cos(ω p t), ω
2
p ≡ 4πe
2
ρ B /m,
(B.9)
and therefore the Goldstone spectrum consists of quasi-particle excitations with
infinite lifetime in the limit k → 0 and with energy ω(k) → ω p as k → 0 (plasmons
as Goldstone excitations with energy gap).
Proof. As discussed before, the time evolution is defined by removing the infrared
cutoff, i.e. G i,R (t) = α
t
(G i,R ) = w − lim L→∞ α
t
L (G i,R ), before the limit R → ∞.
We divide the proof in a few steps. For the breaking of the Galilei group, the Goldstone
energy spectrum at k → 0 is given by the Fourier transform of
J i (t) ≡ i lim
R→∞
< [G i,R , α
t
( j i ) ] >,
21 J. Swieca, in “Cargese Lectures in Physics”, Vol. 4, D. Kastler ed., Gordon and Breach, New
York, 1970; Comm. Math. Phys. 4, 1 (1967).
241
On the quasi-local algebra A 0 the (local) generators of G are determined up to
space divergences and may be taken, respectively, as
G R = m
d
3 x f R (x) x ρ(x), N R =
d
3 x f R (x) ρ(x),
P R =
1
2 i
d
3 x f R (x) [(∇ψ
∗
)ψ − ψ
∗
∇ψ](x) = m
d
3 x f R (x)j(x).
(B.7)
The only non-trivial commutator is
[ G i , P k ] = m δ ik N .
Then, in a factorial representation defined by a translationally invariant state ω
with < A >≡ ω(A), one has (all operators at time t = 0)
< ∂β
v
( j k (y))/∂v i > | v=0 = i lim
R→∞
< [ G i,R , j k (y) ] >= δ ik < ρ >,
(B.8)
and a non-zero density < ρ >= ρ B implies the breaking of the Galilei group. This
is actually a general feature of non-relativistic systems with the algebraic symmetry
of the Galilei group and non-zero density.
21
iv) Plasmon as Goldstone excitations with energy gap
As anticipated, for the breaking of the Galilei group, one expects that the long range
Coulomb interaction allows to evade the conclusions of the Goldstone theorem, with
an energy gap in the Goldstone spectrum in a factorial representation defined by a
ground state invariant under space translations and rotations.
Proposition B.1 In a factorial representation defined by a ground state invariant
under space translations and rotations, with expectations denoted by < >, one has
i lim
R→∞
< [ G i,R (t), j i (0) ] >= ρ B cos(ω p t), ω
2
p ≡ 4πe
2
ρ B /m,
(B.9)
and therefore the Goldstone spectrum consists of quasi-particle excitations with
infinite lifetime in the limit k → 0 and with energy ω(k) → ω p as k → 0 (plasmons
as Goldstone excitations with energy gap).
Proof. As discussed before, the time evolution is defined by removing the infrared
cutoff, i.e. G i,R (t) = α
t
(G i,R ) = w − lim L→∞ α
t
L (G i,R ), before the limit R → ∞.
We divide the proof in a few steps. For the breaking of the Galilei group, the Goldstone
energy spectrum at k → 0 is given by the Fourier transform of
J i (t) ≡ i lim
R→∞
< [G i,R , α
t
( j i ) ] >,
21 J. Swieca, in “Cargese Lectures in Physics”, Vol. 4, D. Kastler ed., Gordon and Breach, New
York, 1970; Comm. Math. Phys. 4, 1 (1967).