162
23 Fermi and Bose Gases at Non-zero Temperature
θ is an operator which anticommutes with a, a
∗ and K is an antilinear involution
(K f, K g) = (g, f ). Then, by introducing the state Ω ω ≡ Ω F ⊗ Ω F on the a ω , a
∗
ω ,
where Ω F is the Fock vacuum on the a, a
∗ , and θΩ F = Ω F (this requirement fully
determines θ), we have
ω(a
∗
( f ) a(g)) = (Ω F ⊗ Ω F )(a
∗
ω ( f ) a ω (g)).
(23.8)
At zero temperature, T is the multiplication by the characteristic function of the
Fermi sphere (in momentum space) and a ω ( f ) has the physical interpretation of
destroying a particle outside the Fermi sphere, with wave function
√
1 − T f and of
creating a “hole” inside the Fermi sphere with wave function K
√
T f .
The above equation (23.8) displays the general properties of a KMS state at inverse
temperature β, on the Von Neumann algebra π ω (A)
generated by the a ω , a
∗
ω .
The representation is reducible; in fact the Von Neumann algebra π ω
(A) generated by the operators
a
ω ( f ) = 1 ⊗ a(
√
1 − T f ) + a
∗
(K
√
T f ) ⊗ θ,
(23.9)
α
∗
ω
(g) = 1 ⊗ a
∗
(
√
1 − T g) + a(K
√
T g) ⊗ θ,
(23.10)
commutes with the Von Neumann algebra π ω (A)
generated by a ω , a
∗
ω .
The representation is primary. In fact, since π ω (A)
⊆ π ω
(A)
, we have
π ω (A)
∩ π ω (A)
⊆ π ω (A)
∩ π ω
(A)
= (π ω (A)
∪ π ω
(A))
and since π ω (A)
∪ π ω
(A) is a doubled fermionic canonical algebra, which is irreducibly represented by ω, its commutant consists of multiples of the identity.
Furthermore, the two equations
π ω (A)
∪ π ω
(A) = B(H),
π ω (A)
∪ π ω (A)
= (π ω (A) ∩ π ω (A)
)
= B(H)
imply
π ω (A)
= π ω
(A).
(23.11)
Since π ω
(A) is isomorphic to π ω (A)
, the Von Neumann algebra π ω (A)
is isomorphic to its commutant.
It is an instructive exercise to explicitly derive the properties 1–3 listed in
Sect. 22.3, for this specific example.