238
Appendix B: Breaking of the Galilei Group and Plasmon Energy Spectrum
where ψ
∗
, ψ are the electron creation and annihilation operators, and U L (x) =
(e
2
/|x − y|) f L (x − y),( f L defined as in Sect. 25.2, Eq. (25.12)) is the infrared regularized Coulomb potential.
The Hamiltonian H L generates the infrared cutoff dynamics α
t
L of the fermion
fields ψ
∗
, ψ and of the (quasi-) local algebra A 0 generated by them at t = 0; then,
according to the general discussion in the previous Appendix, the first problem is the
removal of the infrared cutoff.
To this purpose, as a family Σ of infrared regular states we consider those ω
such that a) ∀A, B ∈ A 0 , the correlation functions |x|
−2
ω(A (ρ(x, t) − ρ B ) B) ≡
|x|
−2
ω AB (ρ(x, t) − ρ B )are absolutely integrable in x.
16
By exploiting this condition, one can control the removal of the infrared cutoff at
each order of the Taylor expansion in t.
The family Σ is also assumed to be invariant under space translations.
ii) Time evolution and variables at infinity
To display the occurrence of variables at infinity in the time evolution of local variables, we consider the equation of motion for the local variable Δψ at t = 0:
dΔψ
dt
| t=0 =
i
2m
Δ
2
ψ − 2i(∂ i U L ∗ ˜
ρ) ∂ i ψ − i(U L ∗ ˜
ρ) Δψ − i(ΔU L ∗ ˜
ρ) ψ,
(B.2)
where ˜
ρ ≡ ρ − ρ B , ρ ≡ ψ
∗
ψ and ∗ denotes the (spacial) convolution product.
Now, as a consequence of condition a), with A, B polynomials of ψ
∗
, ψ, putting
ρ V ≡ V
−1
V d
3 x ρ(x), one has that lim V →∞ ρ V ≡ ρ ∞ exists with respect to the
family Σ and that ρ ∞ = ρ B .
17 Furthermore, one has
ΔU L (x) = −4π e
2
δ(x) + σ L (x),
where, by the (compact) support of f L , supp σ L ⊂ [ L < |x| < L(1 + ε) ], and
d
3 x σ L (x) = 4π e
2 , since
d
3 x Δ(U f L ) = 0.
Then, the last term on the right-hand side of Eq. (B.2) becomes i [ 4πe
2
ρ(x) −
σ L ∗ ρ ]ψ(x) ≡ T L (x)ψ(x).
16 The existence of the correlation functions of the electric field
E(x, t) = −
d 3 y ∇U L (x − y) (ρ(y, t) − ρ B ) in the limit L → ∞, i.e. the weak convergence
of α t
L (E), is guaranteed by the (infrared regularity) condition that the correlation functions
|x| −1 ω AB (ρ(x) − ρ B ), with A, B ∈ A 0 , are absolutely integrable in x. This means that at large
distances, the electron density approaches the background density ρ B faster than |x| −2 .
To get the group law for the dynamics in the limit L → ∞, one should actually have the ultrastrong convergence of α t
L (e.g. for the two-point function of the electric field one should have the
integrability in x, y of the functions |x| −2 |y| −2 ω AB ((ρ(x, t) − ρ B ) (ρ(y, t) − ρ B ))). This is the
reason for the above condition a).
17 In fact, one has
|ω(ρ V − ρ B )| ≤ sup x∈V |x| 2 V −1
V d 3 x |x| −2 |ω(ρ(x) − ρ B )| ≤ sup x∈V |x| 2 V −1 C → V →∞ 0.
Appendix B: Breaking of the Galilei Group and Plasmon Energy Spectrum
where ψ
∗
, ψ are the electron creation and annihilation operators, and U L (x) =
(e
2
/|x − y|) f L (x − y),( f L defined as in Sect. 25.2, Eq. (25.12)) is the infrared regularized Coulomb potential.
The Hamiltonian H L generates the infrared cutoff dynamics α
t
L of the fermion
fields ψ
∗
, ψ and of the (quasi-) local algebra A 0 generated by them at t = 0; then,
according to the general discussion in the previous Appendix, the first problem is the
removal of the infrared cutoff.
To this purpose, as a family Σ of infrared regular states we consider those ω
such that a) ∀A, B ∈ A 0 , the correlation functions |x|
−2
ω(A (ρ(x, t) − ρ B ) B) ≡
|x|
−2
ω AB (ρ(x, t) − ρ B )are absolutely integrable in x.
16
By exploiting this condition, one can control the removal of the infrared cutoff at
each order of the Taylor expansion in t.
The family Σ is also assumed to be invariant under space translations.
ii) Time evolution and variables at infinity
To display the occurrence of variables at infinity in the time evolution of local variables, we consider the equation of motion for the local variable Δψ at t = 0:
dΔψ
dt
| t=0 =
i
2m
Δ
2
ψ − 2i(∂ i U L ∗ ˜
ρ) ∂ i ψ − i(U L ∗ ˜
ρ) Δψ − i(ΔU L ∗ ˜
ρ) ψ,
(B.2)
where ˜
ρ ≡ ρ − ρ B , ρ ≡ ψ
∗
ψ and ∗ denotes the (spacial) convolution product.
Now, as a consequence of condition a), with A, B polynomials of ψ
∗
, ψ, putting
ρ V ≡ V
−1
V d
3 x ρ(x), one has that lim V →∞ ρ V ≡ ρ ∞ exists with respect to the
family Σ and that ρ ∞ = ρ B .
17 Furthermore, one has
ΔU L (x) = −4π e
2
δ(x) + σ L (x),
where, by the (compact) support of f L , supp σ L ⊂ [ L < |x| < L(1 + ε) ], and
d
3 x σ L (x) = 4π e
2 , since
d
3 x Δ(U f L ) = 0.
Then, the last term on the right-hand side of Eq. (B.2) becomes i [ 4πe
2
ρ(x) −
σ L ∗ ρ ]ψ(x) ≡ T L (x)ψ(x).
16 The existence of the correlation functions of the electric field
E(x, t) = −
d 3 y ∇U L (x − y) (ρ(y, t) − ρ B ) in the limit L → ∞, i.e. the weak convergence
of α t
L (E), is guaranteed by the (infrared regularity) condition that the correlation functions
|x| −1 ω AB (ρ(x) − ρ B ), with A, B ∈ A 0 , are absolutely integrable in x. This means that at large
distances, the electron density approaches the background density ρ B faster than |x| −2 .
To get the group law for the dynamics in the limit L → ∞, one should actually have the ultrastrong convergence of α t
L (e.g. for the two-point function of the electric field one should have the
integrability in x, y of the functions |x| −2 |y| −2 ω AB ((ρ(x, t) − ρ B ) (ρ(y, t) − ρ B ))). This is the
reason for the above condition a).
17 In fact, one has
|ω(ρ V − ρ B )| ≤ sup x∈V |x| 2 V −1
V d 3 x |x| −2 |ω(ρ(x) − ρ B )| ≤ sup x∈V |x| 2 V −1 C → V →∞ 0.