116
17 Examples
so that for different θ the states Ω θ assign different values to an element of the centre
and therefore the corresponding representations are inequivalent.
The algebra A contains a (pointwise) gauge invariant subalgebra A obs , which has
the meaning of the algebra of observables. All the states Ω θ and therefore Ω define
equivalent representations of A obs .
The ground state correlation functions for the free Fermi gas can be computed
by putting the system in a box of volume V with periodic boundary conditions. The
ground state Ω V is completely characterized by the two-point function
Ω V (a
∗
k a k ) = δ k,k θ(k
2
F − k
2
), k
3
F ≡ 3π
2 n,
(17.7)
where θ denotes the Heaviside step function and n the density. Thus, in the thermodynamical limit
< ψ( f )
∗
ψ(g) > Ω V = V
−1
j
˜
f (k j ) ˜
g(k j ) θ(k
2
F − k
2
j )
→ (2π)
−3
d
3 k ˜
f (k) ˜
g(k) θ(k
2
F − k
2
).
(17.8)
17.3 ∗ Appendix: The Infinite Volume Dynamics
for Short-Range Spin Interactions
We consider a spin system on a lattice Z
d with many-body “potentials”
Φ
k
(x 1 , . . . x k ) = v(x 1 , . . . x k )σ(x 1 ) . . . σ(x k ),
where x i denote the lattice points and for simplicity the spin components are not
spelled out. Briefly, if X = {x 1 , . . . x k } denotes a set of lattice points, we denote by
Φ(X ) the corresponding interaction energy. For example, in the case of a spin system
interacting only via a two-body potential, one has Φ(X ) = 0, unless X = {x 1 , x 2 }
and Φ(X ) = J (x 1 , x 2 )σ(x 1 ) σ(x 2 ).
The potentials are assumed to describe translationally invariant interactions, i.e.
α a (Φ(X )) = Φ(X + a).
The interaction is said to be of finite range if, given a lattice point x, the number of
sets X , which contain x and for which Φ(X ) = 0, is finite; the union of such sets is
denoted by Δ and called the range of Φ; N (Δ) denotes the number of points of Δ.
The finite volume Hamiltonian is therefore of the following form
H V =
X ⊂V
Φ(X ).