Appendix B: Breaking of the Galilei Group and Plasmon Energy Spectrum
239
The term σ L ∗ ρ is a consequence of the (necessary) infrared regularization and
describes a largely delocalized variable which converges to 4πe
2
ρ ∞ , in the limit
L → ∞, (in expectations on the states of Σ),
18 so that T L (x) → i4π e
2
(ρ(x) − ρ ∞ ).
The other terms in Eq. (B.2) converge to
−2i[ ∂ i U ∗ (ρ − ρ B ) ]∂ i ψ − iU ∗ (ρ − ρ B )Δψ,
and according to the above discussion, they may be considered as elements of an
essentially localized algebra A l , since they involve the operator ρ(y) − ρ B → 0, for
|y| → ∞, convoluted with functions decreasing at least as |x|
−1 .
The occurrence of variables at infinity in the time evolution of local variables is
not peculiar to gauge dependent variables like Δψ. In fact, for the current
j i (x) = (i/2m)
d
3 x [(∇ i ψ
∗
ψ − ψ
∗
∇ i ψ](x),
one has
m
d
2
α
t
L ( j i (x))
dt 2
| t=0 =: ∂ k j k E i,L : (x)+ : ρ ∂ i U L ∗ ∂ k j k : (x) + ∂ k T ki (x), (B.3)
where : : denotes Wick ordering, E i,L ≡ −∂ i U L ∗ (ρ − ρ B ) and T ik is a local gauge
invariant function of ψ
∗
, ψ, E i,L also in the limit L → ∞. The variable j i (x) and
the gauge invariant functions of ψ
∗
, ψ at t = 0, belong to A 0 and therefore to A l .
The delicate term is the second one; a careful removal of the infrared cutoff gives
19
lim
L→∞
(∂ i U L ∗ ∂ k j k )(x) = 4π e
2
( j
∞
i − j
long
i
(x)),
where j
∞
i is the variable at infinity given by the ergodic mean of j i and
−4πe
2 j
long
i
(x) ≡ lim
L→∞
lim
M→∞
d
3 y ∂ i ∂ k U (x − y) j k (y) ×
× f L (|(x k − y k |) f M (|(x − y)
⊥
k |), y
⊥
k ≡ (|y|
2
− y
2
k )
1
2 .
The variable j
long
k
describes the longitudinal current and, by the above arguments, it
may be taken to belong to A l .
18 In fact, putting ω AB (C) ≡ ω(AC B), one has (since σ L is positive)
|ω A A (σ L ∗ ρ − 4πe
2 ρ ∞ )| ≤
d
3 y σ L (x − y) sup L<|y|
the IX International Conference on Mathematical Physics, Swansea (Wales) July 17–27, 1988, B.
Simon et al. eds., Adam Hilger, 1989, p. 490.
239
The term σ L ∗ ρ is a consequence of the (necessary) infrared regularization and
describes a largely delocalized variable which converges to 4πe
2
ρ ∞ , in the limit
L → ∞, (in expectations on the states of Σ),
18 so that T L (x) → i4π e
2
(ρ(x) − ρ ∞ ).
The other terms in Eq. (B.2) converge to
−2i[ ∂ i U ∗ (ρ − ρ B ) ]∂ i ψ − iU ∗ (ρ − ρ B )Δψ,
and according to the above discussion, they may be considered as elements of an
essentially localized algebra A l , since they involve the operator ρ(y) − ρ B → 0, for
|y| → ∞, convoluted with functions decreasing at least as |x|
−1 .
The occurrence of variables at infinity in the time evolution of local variables is
not peculiar to gauge dependent variables like Δψ. In fact, for the current
j i (x) = (i/2m)
d
3 x [(∇ i ψ
∗
ψ − ψ
∗
∇ i ψ](x),
one has
m
d
2
α
t
L ( j i (x))
dt 2
| t=0 =: ∂ k j k E i,L : (x)+ : ρ ∂ i U L ∗ ∂ k j k : (x) + ∂ k T ki (x), (B.3)
where : : denotes Wick ordering, E i,L ≡ −∂ i U L ∗ (ρ − ρ B ) and T ik is a local gauge
invariant function of ψ
∗
, ψ, E i,L also in the limit L → ∞. The variable j i (x) and
the gauge invariant functions of ψ
∗
, ψ at t = 0, belong to A 0 and therefore to A l .
The delicate term is the second one; a careful removal of the infrared cutoff gives
19
lim
L→∞
(∂ i U L ∗ ∂ k j k )(x) = 4π e
2
( j
∞
i − j
long
i
(x)),
where j
∞
i is the variable at infinity given by the ergodic mean of j i and
−4πe
2 j
long
i
(x) ≡ lim
L→∞
lim
M→∞
d
3 y ∂ i ∂ k U (x − y) j k (y) ×
× f L (|(x k − y k |) f M (|(x − y)
⊥
k |), y
⊥
k ≡ (|y|
2
− y
2
k )
1
2 .
The variable j
long
k
describes the longitudinal current and, by the above arguments, it
may be taken to belong to A l .
18 In fact, putting ω AB (C) ≡ ω(AC B), one has (since σ L is positive)
|ω A A (σ L ∗ ρ − 4πe
2 ρ ∞ )| ≤
d
3 y σ L (x − y) sup L<|y|
the IX International Conference on Mathematical Physics, Swansea (Wales) July 17–27, 1988, B.
Simon et al. eds., Adam Hilger, 1989, p. 490.