1.2 Quantum Mechanics
11
unitary representation U . Since any bounded operator is weakly approximated by
linear combinations
λ j U (g j ) we can hope to obtain some insight into possible
interpretations of other operators. The complete answer to this problem of interpretation needs, probably, an analysis of possible interactions of the system under
consideration with all other systems, or, at least with systems which could be used in
the role of measuring instruments. The choice of G together with (eventually) some
other assumptions on the physical properties of the system (e.g. the value of spin)
might also determine the dimension of H.
The proper choice of the representation of G depends on comparison of consequences of the chosen ‘interpretation U ’ with empirical data; this step contains,
e.g. the choice of the correct value of the Planck constant, if G is the Heisenberg
group (i.e. a central extension of the classical phase space R
2n considered as the
commutative group of translations).
1.2.9 It will be further assumed that the time evolution of the system is either
a one-parameter subgroup of G, or it is separately defined as a one-parameter
w*-continuous subgroup τ of the group of *-automorphisms of L(H) t → τ t ∈
∗ -aut (L(H)), τ t+u = τ t ◦ τ u (t, u ∈ R), τ o := identity. Note that for each automorphism α ∈
∗ -aut (L(H)) there is some unitary U α ∈ L(H) such that for all
A ∈ L(H) : α(A) ≡ U α A U
∗
α , i.e. the automorphisms of L(H) are inner automorphisms, cf. e.g. [274, Corollary 2.9.32].
1.3 Classical Hamiltonian Mechanics
1.3.1 In this section, we shall outline the formal scheme of classical Hamiltonian
mechanics (CM) parallel to the exposition of QM in the preceding section. We
shall restrict our considerations to the case of systems with finite number of degrees
of freedom. We shall use the language of differential geometry (for pedagogically
well written course of differential geometry we refer to [111]). A technically more
complicated quantum theory of systems with infinite number of degrees of freedom
will be described later. For classical theory of infinite systems, i.e. classical field
theory, see the corresponding monographs, or also e.g. [1, II.5.5], [7, Append.2], [37].
1.3.2 To any physical system there corresponds in CM a symplectic manifold
(M; ) (cf. [1, 7, 178]). M is here an (even dimensional) infinitely differentiable
Hausdorff second countable connected manifold modeled by R
2n and is a nondegenerate closed two-form on M, the symplectic form, cf. also [37, A.3]. Observables
in CM are represented by real-valued functions f on M; for technical convenience,
we shall assume usually f to be infinitely differentiable, f ∈ C
∞
(M, R). These
observables constitute a real associative algebra F(M) with respect to the ordinary
multiplication of functions: f.g(x) := f (x)g(x) ( f, g ∈ F(M), x ∈ M). This algebra has the natural complexification F C (M).
11
unitary representation U . Since any bounded operator is weakly approximated by
linear combinations
λ j U (g j ) we can hope to obtain some insight into possible
interpretations of other operators. The complete answer to this problem of interpretation needs, probably, an analysis of possible interactions of the system under
consideration with all other systems, or, at least with systems which could be used in
the role of measuring instruments. The choice of G together with (eventually) some
other assumptions on the physical properties of the system (e.g. the value of spin)
might also determine the dimension of H.
The proper choice of the representation of G depends on comparison of consequences of the chosen ‘interpretation U ’ with empirical data; this step contains,
e.g. the choice of the correct value of the Planck constant, if G is the Heisenberg
group (i.e. a central extension of the classical phase space R
2n considered as the
commutative group of translations).
1.2.9 It will be further assumed that the time evolution of the system is either
a one-parameter subgroup of G, or it is separately defined as a one-parameter
w*-continuous subgroup τ of the group of *-automorphisms of L(H) t → τ t ∈
∗ -aut (L(H)), τ t+u = τ t ◦ τ u (t, u ∈ R), τ o := identity. Note that for each automorphism α ∈
∗ -aut (L(H)) there is some unitary U α ∈ L(H) such that for all
A ∈ L(H) : α(A) ≡ U α A U
∗
α , i.e. the automorphisms of L(H) are inner automorphisms, cf. e.g. [274, Corollary 2.9.32].
1.3 Classical Hamiltonian Mechanics
1.3.1 In this section, we shall outline the formal scheme of classical Hamiltonian
mechanics (CM) parallel to the exposition of QM in the preceding section. We
shall restrict our considerations to the case of systems with finite number of degrees
of freedom. We shall use the language of differential geometry (for pedagogically
well written course of differential geometry we refer to [111]). A technically more
complicated quantum theory of systems with infinite number of degrees of freedom
will be described later. For classical theory of infinite systems, i.e. classical field
theory, see the corresponding monographs, or also e.g. [1, II.5.5], [7, Append.2], [37].
1.3.2 To any physical system there corresponds in CM a symplectic manifold
(M; ) (cf. [1, 7, 178]). M is here an (even dimensional) infinitely differentiable
Hausdorff second countable connected manifold modeled by R
2n and is a nondegenerate closed two-form on M, the symplectic form, cf. also [37, A.3]. Observables
in CM are represented by real-valued functions f on M; for technical convenience,
we shall assume usually f to be infinitely differentiable, f ∈ C
∞
(M, R). These
observables constitute a real associative algebra F(M) with respect to the ordinary
multiplication of functions: f.g(x) := f (x)g(x) ( f, g ∈ F(M), x ∈ M). This algebra has the natural complexification F C (M).