23.2 Free Fermi Gas in the Thermodynamical Limit
161
the finite volume dynamics converges uniformly to the dynamics defined by
α t (a(g)) = a(e
i t h
g).
(23.4)
For this result a crucial role is played by the fact that a(g), a(g)
∗ are bounded
operators and that the free Fermi algebra is generated by them through products,
linear combinations and norm closures.
We can now discuss the thermodynamical limit of the Gibbs (quasi- free) states
given by (23.2), with a label V understood.
It is not difficult to see that the correlation functions converge as V → ∞. In
particular the limit of the two-point function is given by (for simplicity, we put the
fermion mass m = 1/2)
ω(a
∗
( f ) a(g)) = (2π)
−s
d
s p ˜
g( p) ˜
f ( p) z e
−β p
2 (1 + ze
−β p
2 )
−1
,
(23.5)
∀ f , g ∈ L
2
(R
s
). It is also easy to see that in the infinite volume limit one has a
quasi-free state.
The chemical potential μ, which enters into z, is determined by the condition that
the average density
ρ(β, z) = (4π
2
β)
−s/2
d
s x z e
−x
2 (1 + ze
−x
2 )
−1
takes the given value ρ.
140 In the limit of zero temperature, (β → ∞), one has
ρ(∞, μ) = (2π)
−s
p 2 ≤μ
d
s p
and one recovers the analog of (17.8), with μ = k
2
F , i.e. the one-particle states with
p
2
≤ μ are occupied (Fermi sphere).
The GNS representation defined by the state (23.5) has a Fock-type interpretation in terms of occupation numbers of particles and “holes”. For this purpose, one
introduces new annihilation and creation operators ( f, g ∈ L
2
(R
s
))
a ω ( f ) = a(
√
1 − T f ) ⊗ 1 + θ ⊗ a
∗
(K
√
T f ),
(23.6)
a
∗
ω (g) = a
∗
(
√
1 − T g) ⊗ 1 + θ ⊗ a(K
√
T g),
(23.7)
where T is the positive self-adjoint bounded operator, ||T || ≤ 1, defined by
ω(a
∗
( f ) a(g)) = (T
1/2
g, T
1/2 f ),
140 For the explicit inversion, see A. Leonard, Phys. Rev. 175, 221 (1968).
161
the finite volume dynamics converges uniformly to the dynamics defined by
α t (a(g)) = a(e
i t h
g).
(23.4)
For this result a crucial role is played by the fact that a(g), a(g)
∗ are bounded
operators and that the free Fermi algebra is generated by them through products,
linear combinations and norm closures.
We can now discuss the thermodynamical limit of the Gibbs (quasi- free) states
given by (23.2), with a label V understood.
It is not difficult to see that the correlation functions converge as V → ∞. In
particular the limit of the two-point function is given by (for simplicity, we put the
fermion mass m = 1/2)
ω(a
∗
( f ) a(g)) = (2π)
−s
d
s p ˜
g( p) ˜
f ( p) z e
−β p
2 (1 + ze
−β p
2 )
−1
,
(23.5)
∀ f , g ∈ L
2
(R
s
). It is also easy to see that in the infinite volume limit one has a
quasi-free state.
The chemical potential μ, which enters into z, is determined by the condition that
the average density
ρ(β, z) = (4π
2
β)
−s/2
d
s x z e
−x
2 (1 + ze
−x
2 )
−1
takes the given value ρ.
140 In the limit of zero temperature, (β → ∞), one has
ρ(∞, μ) = (2π)
−s
p 2 ≤μ
d
s p
and one recovers the analog of (17.8), with μ = k
2
F , i.e. the one-particle states with
p
2
≤ μ are occupied (Fermi sphere).
The GNS representation defined by the state (23.5) has a Fock-type interpretation in terms of occupation numbers of particles and “holes”. For this purpose, one
introduces new annihilation and creation operators ( f, g ∈ L
2
(R
s
))
a ω ( f ) = a(
√
1 − T f ) ⊗ 1 + θ ⊗ a
∗
(K
√
T f ),
(23.6)
a
∗
ω (g) = a
∗
(
√
1 − T g) ⊗ 1 + θ ⊗ a(K
√
T g),
(23.7)
where T is the positive self-adjoint bounded operator, ||T || ≤ 1, defined by
ω(a
∗
( f ) a(g)) = (T
1/2
g, T
1/2 f ),
140 For the explicit inversion, see A. Leonard, Phys. Rev. 175, 221 (1968).