10
1 Introduction
some way (we shall not specify it here) to a measurable quantity connected with the
macroscopic motions. We shall assume (and this is really fulfilled for many finite
systems) that the group G obtained in this way is large enough to determine all the
‘basic observables’; all the other observables are supposed to be functions of these
basic ones (see the following subsections).
1.2.7 We shall assume that a w*-continuous representation σ of a connected Lie
group G in the group of *-automorphisms of L(H) is given and that the group {σ g ∈
∗ - Aut L(H) : g ∈ G} acts on L(H) irreducibly: there is no nontrivial von Neumann
subalgebra of L(H) which is left invariant by the all σ g (g ∈ G). One-parameter
subgroups of G are in bijective correspondence with elements ξ of the Lie algebra
g of G to which, in turn, correspond selfadjoint generators X ξ of unitary groups
determined by σ ex p(tξ) , cf. [37, A.4.8].
Since the unitary operators U (g) determined by automorphisms σ g (g ∈ G) via
the relation
U (g)
∗ BU (g) = σ g (B), ∀B ∈ L(H)
(1.2.5)
are only defined up to a phase factor, in general case, the representation σ leads only
to a projective representation g → U (g) of G in the unitary group of H, i.e.
U (g 1 g 2 ) = m(g 1 , g 2 )U (g 1 )U (g 2 ),
(1.2.6)
where m : G × G → S
1 (:= the complex numbers of unit modulus) is a multiplier
of the projective representation, cf. [37, 3.3.6]. Such a representation can be always
extended to a unitary representation of a group G m , which is the central extension of
G [174, 15.2, Thm. 1] by the multiplicative group S
1 corresponding to the multiplier
m, [37, 1.5-c]. The group multiplication in G m (which can be identified, as a set,
with G × S
1
) is
(g 1 ; λ 1 )(g 2 ; λ 2 ) = (g l g 2 ; m(g 1 , g 2 )λ 1 λ 2 ), λ j ∈ S
1
.
(1.2.7)
In the unitary extension of the projective representation U (G) the elements of
the center of G m are represented by the numbers from S
1 (‘phase factors’) acting by
multiplication of the vectors x ∈ H. All the extensions G m of G (corresponding to
various multipliers m) are classified by the second cohomology group H
2
(G, S
1
) of
the group G with values in S
1 , for details see [174, 321]. We shall assume that the
unitary representation U (G m ) corresponding to the representation σ of G according
to (1.2.5) can be (and really is) chosen strongly continuous. In the following we
shall usually write G instead of G m .
A natural consequence of irreducibility of σ is the irreducibility of corresponding
unitary representation U . Hence, the weak-operator closure of the linear hull of the
subset {U (g) : g ∈ G} of L(H) in the von Neumann algebra L(H) is L(H) itself.
1.2.8 The interpretation of G as a group of (empirically defined) physical symmetries of the system leads to a natural interpretation of generators X ξ (ξ ∈ G) of the
1 Introduction
some way (we shall not specify it here) to a measurable quantity connected with the
macroscopic motions. We shall assume (and this is really fulfilled for many finite
systems) that the group G obtained in this way is large enough to determine all the
‘basic observables’; all the other observables are supposed to be functions of these
basic ones (see the following subsections).
1.2.7 We shall assume that a w*-continuous representation σ of a connected Lie
group G in the group of *-automorphisms of L(H) is given and that the group {σ g ∈
∗ - Aut L(H) : g ∈ G} acts on L(H) irreducibly: there is no nontrivial von Neumann
subalgebra of L(H) which is left invariant by the all σ g (g ∈ G). One-parameter
subgroups of G are in bijective correspondence with elements ξ of the Lie algebra
g of G to which, in turn, correspond selfadjoint generators X ξ of unitary groups
determined by σ ex p(tξ) , cf. [37, A.4.8].
Since the unitary operators U (g) determined by automorphisms σ g (g ∈ G) via
the relation
U (g)
∗ BU (g) = σ g (B), ∀B ∈ L(H)
(1.2.5)
are only defined up to a phase factor, in general case, the representation σ leads only
to a projective representation g → U (g) of G in the unitary group of H, i.e.
U (g 1 g 2 ) = m(g 1 , g 2 )U (g 1 )U (g 2 ),
(1.2.6)
where m : G × G → S
1 (:= the complex numbers of unit modulus) is a multiplier
of the projective representation, cf. [37, 3.3.6]. Such a representation can be always
extended to a unitary representation of a group G m , which is the central extension of
G [174, 15.2, Thm. 1] by the multiplicative group S
1 corresponding to the multiplier
m, [37, 1.5-c]. The group multiplication in G m (which can be identified, as a set,
with G × S
1
) is
(g 1 ; λ 1 )(g 2 ; λ 2 ) = (g l g 2 ; m(g 1 , g 2 )λ 1 λ 2 ), λ j ∈ S
1
.
(1.2.7)
In the unitary extension of the projective representation U (G) the elements of
the center of G m are represented by the numbers from S
1 (‘phase factors’) acting by
multiplication of the vectors x ∈ H. All the extensions G m of G (corresponding to
various multipliers m) are classified by the second cohomology group H
2
(G, S
1
) of
the group G with values in S
1 , for details see [174, 321]. We shall assume that the
unitary representation U (G m ) corresponding to the representation σ of G according
to (1.2.5) can be (and really is) chosen strongly continuous. In the following we
shall usually write G instead of G m .
A natural consequence of irreducibility of σ is the irreducibility of corresponding
unitary representation U . Hence, the weak-operator closure of the linear hull of the
subset {U (g) : g ∈ G} of L(H) in the von Neumann algebra L(H) is L(H) itself.
1.2.8 The interpretation of G as a group of (empirically defined) physical symmetries of the system leads to a natural interpretation of generators X ξ (ξ ∈ G) of the