130
19 Examples
h
0 (α t (A)) ≡ lim
V →∞
lim
V →∞
V,h
0 (α
V
t (A))
= lim
V →∞
V,h
0 (α
V
t (A)) = lim
V →∞
V,h
0 (A) =
h
0 (A).
Moreover, by keeping n fixed and letting h → 0, the correlation functions of
h
0
converge and define a state
n
0 on A, which is not invariant under spin rotations.
This gives rise to a symmetry breaking order parameter (magnetization)
n
0 (n · σ j ) = lim
h→0
lim
V →∞
(Ψ 0
V,h
, n · σ j Ψ 0
V,h
) = 1.
(19.2)
Each
n
0 defines a (physically relevant) representation π n of A with cyclic vector Ψ 0
n .
Different directions n give rise to inequivalent representations of A, each labelled
by a different symmetry breaking order parameter. In fact, by asymptotic abelianess,
the ergodic limits
lim
V →∞
V
−1
i∈V
n · σ i ≡ (n · σ) ∞
exist in any π n and belong to the centre (by the proof of Proposition 16.3); since they
take different values in representations labelled by different n, such representations
cannot be unitarily related. Such representations are physically equivalent in the
sense that one goes from one to the other by a different choice of the coordinate
axes (which leaves the Hamiltonian invariant); the physically relevant point is the
existence of a symmetry breaking order parameter in each π n .
By taking rotationally invariant averages of the states
n
0 , similar to (17.6) for the
free Bose gas, one may obtain a state
inv
0 whose correlation functions are rotationally
invariant and do not provide a symmetry breaking order parameter. However,
inv
0
is not a pure state on A and symmetry breaking order parameters emerge if one
decomposes
inv
0 into the pure states of which is a mixture.
19.2 Bose–Einstein Condensation
As discussed in Sect. 17.2, the gauge transformations define a one-parameter group
of algebraic symmetries of the field algebra A, which describes a system of free
bosons.
In each representation π θ defined by θ , the gauge symmetry is spontaneously
broken with order parameter < ψ > θ . The occurrence of symmetry breaking also for
a free system is due to the fact that for non-zero density the total number operator
does not exist, the generalized version of Von Neumann’s theorem does not apply
and inequivalent representations of the Weyl field algebra are allowed.
On the other hand, all the correlation functions of the gauge invariant state are
by construction invariant under gauge transformations and one may wonder about
their breaking. The point is that is a pure state on the observable algebra but
19 Examples
h
0 (α t (A)) ≡ lim
V →∞
lim
V →∞
V,h
0 (α
V
t (A))
= lim
V →∞
V,h
0 (α
V
t (A)) = lim
V →∞
V,h
0 (A) =
h
0 (A).
Moreover, by keeping n fixed and letting h → 0, the correlation functions of
h
0
converge and define a state
n
0 on A, which is not invariant under spin rotations.
This gives rise to a symmetry breaking order parameter (magnetization)
n
0 (n · σ j ) = lim
h→0
lim
V →∞
(Ψ 0
V,h
, n · σ j Ψ 0
V,h
) = 1.
(19.2)
Each
n
0 defines a (physically relevant) representation π n of A with cyclic vector Ψ 0
n .
Different directions n give rise to inequivalent representations of A, each labelled
by a different symmetry breaking order parameter. In fact, by asymptotic abelianess,
the ergodic limits
lim
V →∞
V
−1
i∈V
n · σ i ≡ (n · σ) ∞
exist in any π n and belong to the centre (by the proof of Proposition 16.3); since they
take different values in representations labelled by different n, such representations
cannot be unitarily related. Such representations are physically equivalent in the
sense that one goes from one to the other by a different choice of the coordinate
axes (which leaves the Hamiltonian invariant); the physically relevant point is the
existence of a symmetry breaking order parameter in each π n .
By taking rotationally invariant averages of the states
n
0 , similar to (17.6) for the
free Bose gas, one may obtain a state
inv
0 whose correlation functions are rotationally
invariant and do not provide a symmetry breaking order parameter. However,
inv
0
is not a pure state on A and symmetry breaking order parameters emerge if one
decomposes
inv
0 into the pure states of which is a mixture.
19.2 Bose–Einstein Condensation
As discussed in Sect. 17.2, the gauge transformations define a one-parameter group
of algebraic symmetries of the field algebra A, which describes a system of free
bosons.
In each representation π θ defined by θ , the gauge symmetry is spontaneously
broken with order parameter < ψ > θ . The occurrence of symmetry breaking also for
a free system is due to the fact that for non-zero density the total number operator
does not exist, the generalized version of Von Neumann’s theorem does not apply
and inequivalent representations of the Weyl field algebra are allowed.
On the other hand, all the correlation functions of the gauge invariant state are
by construction invariant under gauge transformations and one may wonder about
their breaking. The point is that is a pure state on the observable algebra but