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< ψ
+
ψ >
< ψ
+
ψ > G
(−∂
t
+
ˆ
H
1
)
−1
ψ
(−∂
t
+
ˆ
H
1
)G(t, t
0
, x, x
0
) = δ(t − t
0
)δ(x − x
0
), G(t, t
0
, x, x
0
) ≡< ψ
+
(x, t)ψ(x
0
, t
0
) > .
ε
α
Φ
α
ˆ
H
1
Φ
α
(x) = ε
α
Φ
α
(x)
G(t, t
0
, x, x
0
) =
X
α
Φ
∗
α
(x)Φ
α
(x
0
)e
ε
α
(t−t
0
)
[Θ(t
0
− t) ∓ n(ε
α
)],
n(ε
α
) = 1/(e
βε
α
± 1)
α
t t
0
C β µ
t
G(t = t
0
, x, x
0
) ≡ G(t = t
0
+ 0, x, x)
C
ρ(x, t) = ∓ < ψ
+
(x, t)ψ(x, t
0
) > |
t
0
=t−0
= ∓G(t, t
0
= t − 0, x, x) =
X
α
|Φ
α
(x)|
2
n(ε
α
).
|Φ
α
(x)|
2
α
x
C
)
u
ρ = −
1
V
∂Ω
∂µ
T,V
u = −
1
V
∂ ln Σ
∂β
µ,V
,
ψ
+
ˆ
H
1
ψ
Z
dt
Z
dx
Z
dx
0
ψ
+
(x, t)ψ(x, t)V (x − x
0
)ψ
+
(x
0
, t)ψ(x
0
, t),
V t t
0
)
