12
1 Introduction
Remark: In the larger algebra B(M) of all bounded Borel functions on M we
can associate to any f ∈ B(M) the projector-valued measure E f defined on Borel
subsets of R (for a real-valued f ):
E f : B → χ f −1 (B) for any Borel B ⊂ R, where χ N is the characteristic function
of the Borel subset N ⊂ M.
It is clear that E f (B) := χ f −l (B) are projectors in B(M) and the association
B → E f (B) is σ-additive, with E f (R) = χ M = the unit element of B(M). The
real-valued Borel functions f can be also considered as selfadjoint operators on a
Hilbert space H := L
2
(R, μ) acting as the multiplication operators, and E f ’s are
their canonical spectral measures.
1.3.3 States in CM are probability Borel measures μ on M, which form a convex
set S cl with extremal points consisting of all measures concentrated at one-point sets
in M, i.e. of all Dirac measures on M. Hence, pure states are identified with points
x ∈ M. Any measure μ ∈ S cl has a unique decomposition into (an integral of) Dirac
measures, i.e. it is a simplex, contrary to the state space of QM. This has serious
consequences for different possibilities of statistical interpretations of states in CM
and QM, cf. 1.2.3, also footnote 8.
1.3.4 According to CM, the disturbance of the state connected with the measurement
of arbitrary observables can be made negligibly small. Because of uniqueness of the
decomposition of an arbitrary state to its extremal components we can interpret any
μ ∈ S cl as a representative of a statistical ensemble of a large number of copies of the
considered system, each being in an (its own) pure state. Repeated measurements on
the state μ have to be understood now as a repeated random choice (with probability
corresponding to the probability measure μ on M) from the ensemble of a system
appearing in a pure state x ∈ M and measuring precise values f (x) of observables
f ∈ F(M) afterwards. For such a measurement procedure the probability of finding
the value of an observable f in a Borel set B ⊂ R is μ(E f (B)) (compare Remark
in 1.3.2), where μ( f ) for f ∈ B(M) means the integral of f with the measure μ
on M. The value μ( f ) for f ∈ F(M) is then the expectation value of f in the state
μ ∈ S cl . The mapping μ : f → μ( f ) is a positive normalized linear functional on
F(M) (and also on B(M)), which is continuous with respect to the usual sup-norm
on B(M). Better continuity properties have, e.g. functionals μ which are absolutely
continuous (as measures) with respect to the natural measure
n on the symplectic
manifold (M; ).
1.3.5 A symmetry of a system in CM is defined as a symplectomorphism F of
(M; ), i.e. F is such a diffeomorphism of M onto itself which leaves the symplectic
form unchanged: F
∗
= , where F
∗ is the pull-back on M, see e.g. [1], or also
[37, A]. For f ∈ F(M) let F
∗ f := f ◦ F; such an action of F onto the algebra
F(M) is an automorphism. It conserves, moreover, another structure on F(M)—the
Poisson algebra structure defined below.
1 Introduction
Remark: In the larger algebra B(M) of all bounded Borel functions on M we
can associate to any f ∈ B(M) the projector-valued measure E f defined on Borel
subsets of R (for a real-valued f ):
E f : B → χ f −1 (B) for any Borel B ⊂ R, where χ N is the characteristic function
of the Borel subset N ⊂ M.
It is clear that E f (B) := χ f −l (B) are projectors in B(M) and the association
B → E f (B) is σ-additive, with E f (R) = χ M = the unit element of B(M). The
real-valued Borel functions f can be also considered as selfadjoint operators on a
Hilbert space H := L
2
(R, μ) acting as the multiplication operators, and E f ’s are
their canonical spectral measures.
1.3.3 States in CM are probability Borel measures μ on M, which form a convex
set S cl with extremal points consisting of all measures concentrated at one-point sets
in M, i.e. of all Dirac measures on M. Hence, pure states are identified with points
x ∈ M. Any measure μ ∈ S cl has a unique decomposition into (an integral of) Dirac
measures, i.e. it is a simplex, contrary to the state space of QM. This has serious
consequences for different possibilities of statistical interpretations of states in CM
and QM, cf. 1.2.3, also footnote 8.
1.3.4 According to CM, the disturbance of the state connected with the measurement
of arbitrary observables can be made negligibly small. Because of uniqueness of the
decomposition of an arbitrary state to its extremal components we can interpret any
μ ∈ S cl as a representative of a statistical ensemble of a large number of copies of the
considered system, each being in an (its own) pure state. Repeated measurements on
the state μ have to be understood now as a repeated random choice (with probability
corresponding to the probability measure μ on M) from the ensemble of a system
appearing in a pure state x ∈ M and measuring precise values f (x) of observables
f ∈ F(M) afterwards. For such a measurement procedure the probability of finding
the value of an observable f in a Borel set B ⊂ R is μ(E f (B)) (compare Remark
in 1.3.2), where μ( f ) for f ∈ B(M) means the integral of f with the measure μ
on M. The value μ( f ) for f ∈ F(M) is then the expectation value of f in the state
μ ∈ S cl . The mapping μ : f → μ( f ) is a positive normalized linear functional on
F(M) (and also on B(M)), which is continuous with respect to the usual sup-norm
on B(M). Better continuity properties have, e.g. functionals μ which are absolutely
continuous (as measures) with respect to the natural measure
n on the symplectic
manifold (M; ).
1.3.5 A symmetry of a system in CM is defined as a symplectomorphism F of
(M; ), i.e. F is such a diffeomorphism of M onto itself which leaves the symplectic
form unchanged: F
∗
= , where F
∗ is the pull-back on M, see e.g. [1], or also
[37, A]. For f ∈ F(M) let F
∗ f := f ◦ F; such an action of F onto the algebra
F(M) is an automorphism. It conserves, moreover, another structure on F(M)—the
Poisson algebra structure defined below.