1.4 Quantum Theory of Large Systems
19
Here C
for a subset C ⊂ L(H) denotes the commutant of C in L(H): C
:=
{B ∈ L(H) : [B, A] ≡ B A − AB = 0, ∀A ∈ C}, and C
:= (C
)
. The representations π 1 , π 2 satisfying the conditions of the Theorem are called mutually disjoint
representations. If the GNS representations determined by the two states ω 1 , ω 2 :
{π ω 1 ; H ω 1 }, {π ω 2 ; H ω 2 }, are mutually disjoint, then we call these two states also
mutually disjoint : ω 1 ⊥ ω 2 .
1.4.5 An abstractly defined symmetry of the system in QTLS is any
∗ -automorphism
of the algebra A of bounded observables. Let τ be a representation of the group R
as a group of symmetries, i.e. a homomorphism t (∈ R) → τ t ∈
∗ - Aut A, which is
‘conveniently continuous’, e.g. functions t → ω(τ t x) are continuous for all x ∈ A
and all ω ∈ S ph (A). It is often assumed, that the group τ corresponding to a oneparameter group of empirically defined transformations is σ(A, A
∗
)-continuous (i.e.
S ph replaced by S(A) in the last mentioned case), but this assumption might be too
stringent. Let S be a ‘sufficiently large’ subset of states containing S ph and denote by
σ(A, S) the topology on A determined by functions x(∈ A) → ω(x) for all ω ∈ S.
We shall assume, that τ is σ(A, S)-continuous in the sense:
(i) f unctions t → ω(τ t x) ar e continuous f or all ω ∈ S, x ∈ A,
(ii) f unctions x → τ t x are σ(A, S) − σ(A, S) − continuous
f or all t ∈ R,
(iii) ω ∈ S ⇒ ω ◦ τ t ∈ S f or all t ∈ R.
The last condition (iii) allows us to define a σ(S,A)-continuous group of transformations of S by
τ
∗
t ω := ω ◦ τ t (for all t ∈ R), ω ∈ S.
(1.4.5)
Any selfadjoint element a ∈ A generates a σ(A,S) (i.e. σ(A,A
∗ ))-continuous
group of inner
∗ -automorphisms of A, τ
a , by
τ
a
t x := exp(ita)x exp(−ita), for all x ∈ A.
(1.4.6)
A one-parameter group of inner automorphisms of A cannot represent some of
physically important global transformations of quasilocal algebras, e.g. Euclidean
or Poincaré transformations, cf. e.g. [106, Ch.4,Thm.3]. For a general (sufficiently
continuous) one-parameter group τ of automorphisms of A we can define a generator
δ τ - a densely defined derivation on A, [53, 274].
12 The connection of such generators with physically measurable quantities is in general in QTLS less transparent then
it is in QM or in CM. If the state ω ∈ S is τ −invariant, i.e. τ
∗
t ω ≡ ω, then there is
unique weakly continuous unitary group U
ω acting on H ω (cf. 1.4.4) such that [271]:
π ω (τ t x) = U
ω
−t π ω (x)U
ω
t , U
ω
t ω = ω for all t ∈ R.
(1.4.7)
12 A densely defined linear mapping δ : D(δ) ⊂ A → A is a derivation on A if it satisfies the
Leibniz rule : δ(xy) = δ(x)y + xδ(y) ∀ x, y ∈ D(δ) ⊂ A.
19
Here C
for a subset C ⊂ L(H) denotes the commutant of C in L(H): C
:=
{B ∈ L(H) : [B, A] ≡ B A − AB = 0, ∀A ∈ C}, and C
:= (C
)
. The representations π 1 , π 2 satisfying the conditions of the Theorem are called mutually disjoint
representations. If the GNS representations determined by the two states ω 1 , ω 2 :
{π ω 1 ; H ω 1 }, {π ω 2 ; H ω 2 }, are mutually disjoint, then we call these two states also
mutually disjoint : ω 1 ⊥ ω 2 .
1.4.5 An abstractly defined symmetry of the system in QTLS is any
∗ -automorphism
of the algebra A of bounded observables. Let τ be a representation of the group R
as a group of symmetries, i.e. a homomorphism t (∈ R) → τ t ∈
∗ - Aut A, which is
‘conveniently continuous’, e.g. functions t → ω(τ t x) are continuous for all x ∈ A
and all ω ∈ S ph (A). It is often assumed, that the group τ corresponding to a oneparameter group of empirically defined transformations is σ(A, A
∗
)-continuous (i.e.
S ph replaced by S(A) in the last mentioned case), but this assumption might be too
stringent. Let S be a ‘sufficiently large’ subset of states containing S ph and denote by
σ(A, S) the topology on A determined by functions x(∈ A) → ω(x) for all ω ∈ S.
We shall assume, that τ is σ(A, S)-continuous in the sense:
(i) f unctions t → ω(τ t x) ar e continuous f or all ω ∈ S, x ∈ A,
(ii) f unctions x → τ t x are σ(A, S) − σ(A, S) − continuous
f or all t ∈ R,
(iii) ω ∈ S ⇒ ω ◦ τ t ∈ S f or all t ∈ R.
The last condition (iii) allows us to define a σ(S,A)-continuous group of transformations of S by
τ
∗
t ω := ω ◦ τ t (for all t ∈ R), ω ∈ S.
(1.4.5)
Any selfadjoint element a ∈ A generates a σ(A,S) (i.e. σ(A,A
∗ ))-continuous
group of inner
∗ -automorphisms of A, τ
a , by
τ
a
t x := exp(ita)x exp(−ita), for all x ∈ A.
(1.4.6)
A one-parameter group of inner automorphisms of A cannot represent some of
physically important global transformations of quasilocal algebras, e.g. Euclidean
or Poincaré transformations, cf. e.g. [106, Ch.4,Thm.3]. For a general (sufficiently
continuous) one-parameter group τ of automorphisms of A we can define a generator
δ τ - a densely defined derivation on A, [53, 274].
12 The connection of such generators with physically measurable quantities is in general in QTLS less transparent then
it is in QM or in CM. If the state ω ∈ S is τ −invariant, i.e. τ
∗
t ω ≡ ω, then there is
unique weakly continuous unitary group U
ω acting on H ω (cf. 1.4.4) such that [271]:
π ω (τ t x) = U
ω
−t π ω (x)U
ω
t , U
ω
t ω = ω for all t ∈ R.
(1.4.7)
12 A densely defined linear mapping δ : D(δ) ⊂ A → A is a derivation on A if it satisfies the
Leibniz rule : δ(xy) = δ(x)y + xδ(y) ∀ x, y ∈ D(δ) ⊂ A.