14.2 Asymptotic Abelianess
99
In fact, ∀ Ψ , Φ ∈ H π and ∀ε > 0, there exists a B 1 ∈ π(A) such that ||(B −
B 1 )Φ|| ≤ ε, ||(B − B 1 )Ψ || ≤ ε, so that
|(Φ, [π(A x ), B]Ψ )| ≤ |(Φ, [π(A x ), B 1 ]Ψ )| + ε||A||(||Φ|| + ||Ψ ||)
and therefore in the limit |x| → ∞ the right hand side can be made as small as one
likes.
79
As we shall see below, the above property of asymptotic abelianess in the weak
form (14.6) will play a crucial role in the analysis of the physically relevant representations of the observable algebra.
79 By a similar argument one can also prove asymptotic abelianess when A and B are strong limits
of elements of some A(V ), on a common dense domain D stable under the implementers of the
space translations.
99
In fact, ∀ Ψ , Φ ∈ H π and ∀ε > 0, there exists a B 1 ∈ π(A) such that ||(B −
B 1 )Φ|| ≤ ε, ||(B − B 1 )Ψ || ≤ ε, so that
|(Φ, [π(A x ), B]Ψ )| ≤ |(Φ, [π(A x ), B 1 ]Ψ )| + ε||A||(||Φ|| + ||Ψ ||)
and therefore in the limit |x| → ∞ the right hand side can be made as small as one
likes.
79
As we shall see below, the above property of asymptotic abelianess in the weak
form (14.6) will play a crucial role in the analysis of the physically relevant representations of the observable algebra.
79 By a similar argument one can also prove asymptotic abelianess when A and B are strong limits
of elements of some A(V ), on a common dense domain D stable under the implementers of the
space translations.