160
23 Fermi and Bose Gases at Non-zero Temperature
where h is the restriction of H to the one-particle subspace. By using the above
equation one can easily compute the two-point function (z ≡ e
βμ , Z = Tr e
−β H (μ) )
Z Ω(a
∗
( f ) a(g)) = Tr (e
−β H (μ) a
∗
( f )a(g)) =
= z Tr (a
∗
(e
−β h f ) e
−β H (μ) a(g)) = z Tr (e
−β H (μ) a(g) a
∗
(e
−β h f )) =
= z Tr (e
−β H (μ)
{[a(g), a
∗
(e
−β h f )] ∓ ± a
∗
(e
−β h f )a(g)}) =
= z Z(g, e
−β h f ) ± z Z Ω(a
∗
(e
−β h f ) a(g)).
In conclusion, one has
Ω(a
∗
((1 ∓ z e
−β h
) f ) a(g)) = z (g, e
−β h f ),
i.e. letting f → (1 ∓ ze
−β h
)
−1 f ,
Ω(a
∗
( f ) a(g)) = (g, ze
−β h
(1 ∓ ze
−β h
)
−1 f ).
(23.2)
By a similar trick, one can easily compute the 2n-point functions and prove that
they can be expressed in terms of products of two- point functions. A state with this
property is called a quasi-free state. Furthermore, the correlation function of a product
containing a different number of a
∗ and a vanishes, because exp(−β H (μ)) commutes
with the number operator and by computing the trace on a basis of eigenvectors of
N , each matrix element vanishes.
Equation (23.2) yields in particular the expectations
< n k >= Ω(a
∗
(k) a(q)) = δ k,q
e
−β (ω(k)−μ)
1 ∓ e −β (ω(k)−μ) ,
(23.3)
which are the basis of the elementary treatment of the free Bose and Fermi gas, but
(23.2) provides much more detailed information since it determines all the correlation
functions.
23.2 Free Fermi Gas in the Thermodynamical Limit
We start by discussing the thermodynamical limit of the finite volume dynamics
α
V
t (a(g)) = a(e
i t h V g), g ∈ L
2
(V ),
where the label V has been spelled out to distinguish quantities in the volume V .
Now, since a( f )
2
= 0, a( f )
4
= =(a( f )
∗ a( f ))
2
= =a( f )
∗
{a( f ), a( f )
∗
}a( f )
= = f
2
a( f )
2 , i.e. a( f ) = = f . Then, since
||α
V
t (a(g)) − a(e
ith
g)|| = ||(e
i t h V − e
i t h
) g|| −→
V →∞
0,