7.3 Quantum Domino
177
considered physical system appear. In another setting, we can speak instead of a
selfadjoint operator A about its projection-valued measure (≡ projector-valued
measure) (PM) → E A (() for ≡ the set (with a given σ-algebra structure)
of possible values of the observable (specifying the operator uniquely); here E A (()
are mutually commuting orthogonal projectors satisfying σ-additivity with respect
to set unions of various disjoint arguments ⊂ , with E A (() = I H .
More general concept of “observable” in QM is again σ-additive positive operator valued measure (POVM) → A((), with A(() ∈ L(H), 0 ≤ A(() ≤ A(() =
I H , , i ∩ j = ∅ (∀i, j) ⇒ A(∪ k k ) =
k A(( k ), which also specifies a selfadjoint operator A, but is not specified by it uniquely. The different A((), , ⊂ , need
not be now mutually commutative. According to a general ‘philosophy’ of QM, to
each observable corresponds a measuring apparatus (better: a class of equivalent
apparatuses) characterized abstractly by the observable, by which it can be measured. Conversely, if we perform a measurement on some quantum-mechanical system, some observable is measured. The results of the measurement of A on the state
is found in the set with the probability pr A (, ,) = T r(A(()). If →
pr (, ,) (( ) is a probability measure for any and this mapping depends on
affinely: pr (λλ 1 + (1 − λ) 2 , ,) ≡ λ pr ( 1 , ,) + (1 − λ) pr ( 2 , ,), then there
is a unique observable A of the measured system such that pr (, ,) ≡ T r(A(()).
If the distribution of the results of a measurement is expressed in this way by some
POVM A = E A , the measurement is often called a nonideal measurement. For
more complete formulations cf. [84, 149].
We are dealing in this work with infinite quantal systems described by C
∗ -algebras
having many mutually inequivalent representations. Hence, we cannot restrict the
concept of observables to operators acting e.g. on a Hilbert space H ω of a specific
cyclic representation. If we want stay in a framework of the above presented scheme,
we can, and we presently shall, use the universal representation of C
∗ -algebra A in
H u , resp. of its weak closure, which is a W
∗ -algebra isomorphic to the double dual
A
∗∗ of A. For some comments on this reformulation see e.g. [84, Sect. 2.5].
7.3.7 We can now ask, which observable (in the sense of 7.3.6) was measured by
the ‘measuring apparatus’ modeled by our QD, as it was sketched in 7.3.5. The
‘microsystem’ being measured consists in the spin sitting in the point j = 0 of
the infinite spin-chain and the rest of the chain is the ‘measuring apparatus’. Let
us consider as the apparatus the half-infinite chain with spins sitting in the points
numbered by j = 1, 2, . . . ∞ only, because the spins sitting in the points with j < 0
do not take part in these measurements.
4 An integral part of the characterization of
the apparatus is, however, also its initial state ‘with all spins pointing down’, as well
as its dynamics including the interaction with the measured spin. The results of these
measurements are read by looking at the final states of the apparatus.
5 There are
just two possibilities in this process: The state ω ↓ with all spins pointing down, i.e.
4 In accordance with that, the notation in this section will be changed slightly with respect to the
notation in the Sect. 7.3.5.
5 We are speaking here about the states on the algebra generated by a j , a ∗
j with j > 0 only.
177
considered physical system appear. In another setting, we can speak instead of a
selfadjoint operator A about its projection-valued measure (≡ projector-valued
measure) (PM) → E A (() for ≡ the set (with a given σ-algebra structure)
of possible values of the observable (specifying the operator uniquely); here E A (()
are mutually commuting orthogonal projectors satisfying σ-additivity with respect
to set unions of various disjoint arguments ⊂ , with E A (() = I H .
More general concept of “observable” in QM is again σ-additive positive operator valued measure (POVM) → A((), with A(() ∈ L(H), 0 ≤ A(() ≤ A(() =
I H , , i ∩ j = ∅ (∀i, j) ⇒ A(∪ k k ) =
k A(( k ), which also specifies a selfadjoint operator A, but is not specified by it uniquely. The different A((), , ⊂ , need
not be now mutually commutative. According to a general ‘philosophy’ of QM, to
each observable corresponds a measuring apparatus (better: a class of equivalent
apparatuses) characterized abstractly by the observable, by which it can be measured. Conversely, if we perform a measurement on some quantum-mechanical system, some observable is measured. The results of the measurement of A on the state
is found in the set with the probability pr A (, ,) = T r(A(()). If →
pr (, ,) (( ) is a probability measure for any and this mapping depends on
affinely: pr (λλ 1 + (1 − λ) 2 , ,) ≡ λ pr ( 1 , ,) + (1 − λ) pr ( 2 , ,), then there
is a unique observable A of the measured system such that pr (, ,) ≡ T r(A(()).
If the distribution of the results of a measurement is expressed in this way by some
POVM A = E A , the measurement is often called a nonideal measurement. For
more complete formulations cf. [84, 149].
We are dealing in this work with infinite quantal systems described by C
∗ -algebras
having many mutually inequivalent representations. Hence, we cannot restrict the
concept of observables to operators acting e.g. on a Hilbert space H ω of a specific
cyclic representation. If we want stay in a framework of the above presented scheme,
we can, and we presently shall, use the universal representation of C
∗ -algebra A in
H u , resp. of its weak closure, which is a W
∗ -algebra isomorphic to the double dual
A
∗∗ of A. For some comments on this reformulation see e.g. [84, Sect. 2.5].
7.3.7 We can now ask, which observable (in the sense of 7.3.6) was measured by
the ‘measuring apparatus’ modeled by our QD, as it was sketched in 7.3.5. The
‘microsystem’ being measured consists in the spin sitting in the point j = 0 of
the infinite spin-chain and the rest of the chain is the ‘measuring apparatus’. Let
us consider as the apparatus the half-infinite chain with spins sitting in the points
numbered by j = 1, 2, . . . ∞ only, because the spins sitting in the points with j < 0
do not take part in these measurements.
4 An integral part of the characterization of
the apparatus is, however, also its initial state ‘with all spins pointing down’, as well
as its dynamics including the interaction with the measured spin. The results of these
measurements are read by looking at the final states of the apparatus.
5 There are
just two possibilities in this process: The state ω ↓ with all spins pointing down, i.e.
4 In accordance with that, the notation in this section will be changed slightly with respect to the
notation in the Sect. 7.3.5.
5 We are speaking here about the states on the algebra generated by a j , a ∗
j with j > 0 only.