96
5 Macroscopic Limits
χ λ is pure, it is a character on M. Consequently for any projector q ∈ M it is
χ λ (q) = [χ λ (q)]
2
, i.e. χ λ (q) ∈ {0, 1}, q
∗
= q
2
= q ∈ M.
(5.1.111)
We obtain from the Schwarz inequality, for any x ∈ A,
ω λ (x) = ω λ (qx) = ω λ (xq), for all {q = q
∗
= q
2
: q ∈ M, χ λ (q) = 1}.
(5.1.112)
Any element z ∈ M can be expressed as a norm limit of finite linear combinations
of projectors q ∈ M. Since the product z → xz is norm-continuous and the state ω λ
is also continuous in the norm of A, we obtain from (5.1.112):
ω λ (xz − zx) ≡ ω λ ([x, z]) = 0, ∀x ∈ A, and ∀z ∈ M.
(5.1.113)
Due to CCR we have
[exp(it P), exp(iτ Q)] = (e
itτ
− 1) exp(iτ Q) exp(it P),
(5.1.114)
and after the substitution to (5.1.113):
0 = (e
itτ
− 1) ω λ (exp(iτ Q) exp(it P)).
(5.1.115)
The relation (5.1.115) is valid for all real t and τ . From an application of (5.1.113)
to
ω λ (exp(−iτ Q)[exp(it P), exp(iτ Q)]) = ω λ (exp(−iτ Q) exp(it P) exp(iτ Q))
− ω λ (exp(it P))
we obtain the invariance of ω λ with respect to the group σ
∗ of affine isometries of
S(A),
σ
∗
τ ω(x) := ω(exp(−iτ Q)x exp(iτ Q)), τ ∈ R, x ∈ A.
(5.1.116)
This leads, together with the formula (4.1.9), to
ω λ (exp(it P)) = e
itτ
ω λ (exp(it P)) for all t, τ ∈ R,
(5.1.117)
what implies the discontinuity of (5.1.110).
The obtained formulas show also uniqueness of the extension ω λ of χ λ to the
CCR-subalgebra of A defined as the norm closed algebra generated by exp(iτ Q)
and exp(it P) (t, τ ∈ R).
5.1.29 We shall now change the definition 5.1.12 of the macroscopic algebra of
the system (A
, σ G ) in such a way that a larger subset of states from S(A
) will
be mapped onto probability measures on g
∗ than it was before, according to the