4.1 The Heisenberg Group (CCR)
57
with the same initial values z = q − i p. Elementary calculations give:
z
λ
(t, z) =
1 −
1
a(λ)
f
λ
A (z)
z +
1
a(λ)
f
λ
A (z) exp
−
it
λ 2 a(λ)
z,
(4.1.43)
z
λ
(t, z) cl = exp
−
it
λ 2 f
λ
A (z)
z.
(4.1.44)
We see that (4.1.43) and (4.1.44) describe motions on mutually tangent circles
in C with different radii and different dependence of frequencies on initial conditions as well as on the parameter λ. For λ → 0 the quantum evolution vanishes
independently on the ‘renormalization’ a(λ). For slowly varying a(λ), the classical
evolution vanishes too, but the way of this vanishing looks qualitatively differently.
If, e.g. a(λ) = λ
2 exp(b/2λ
2
), b > 0, then zz = b is a critical value for λ → 0.
4.2 Extension of CCR by a Quadratic Generator
4.2.1 All the orbits O ϕ occurring in Sect. 4.1 were mutually homeomorphic (and
homeomorphic to R
2n ). In this section, we shall give examples of irreducible representations U (G) of some Lie groups G in a Hilbert space H containing various
mutually nonhomeomorphic orbits O j := U(G)ϕ j in P(H), ( j = 1, 2, . . . ). Let G
be a connected Lie group containing the 2n + 1—dimensional Heisenberg group
G n as an invariant (i.e. normal) subgroup (G will be specified later). Let U be such
a unitary continuous representation of G, the restriction of which to G n coincides
with the irreducible representation described in Sect. 4.1 with = 1 = λ. With the
notation of the previous section, for m = 1, 2, . . . K , A m ∈ U (g), set
A m :=
1
2
h
jk
m X j X k , (summation over j, k = 1, 2, . . . 2n),
(4.2.1)
with any h m a real symmetric 2n × 2n—matrix; the formally defined operator A m is
symmetric on the Gårding domain of U (G n ). From (4.1.1a) we have commutation
relations (cf. also 4.1.3):
[X j , X k ] = i S jk I, [X j , A m ] = i S jk h
kl
m X l =: i (S · h m · X ) j ,
(4.2.2)
[A m , A k ] =
i
2
X · (h m · S · h k − h k · S · h m ) · X, m, k = 1, 2, . . . K . (4.2.3)
Assume that for any m, k there are reals c
j
mk such, that
h m · S · h k − h k · S · h m =
K
j=1
c
j
mk h j , m, k = 1, 2, . . . K .
(4.2.4)
57
with the same initial values z = q − i p. Elementary calculations give:
z
λ
(t, z) =
1 −
1
a(λ)
f
λ
A (z)
z +
1
a(λ)
f
λ
A (z) exp
−
it
λ 2 a(λ)
z,
(4.1.43)
z
λ
(t, z) cl = exp
−
it
λ 2 f
λ
A (z)
z.
(4.1.44)
We see that (4.1.43) and (4.1.44) describe motions on mutually tangent circles
in C with different radii and different dependence of frequencies on initial conditions as well as on the parameter λ. For λ → 0 the quantum evolution vanishes
independently on the ‘renormalization’ a(λ). For slowly varying a(λ), the classical
evolution vanishes too, but the way of this vanishing looks qualitatively differently.
If, e.g. a(λ) = λ
2 exp(b/2λ
2
), b > 0, then zz = b is a critical value for λ → 0.
4.2 Extension of CCR by a Quadratic Generator
4.2.1 All the orbits O ϕ occurring in Sect. 4.1 were mutually homeomorphic (and
homeomorphic to R
2n ). In this section, we shall give examples of irreducible representations U (G) of some Lie groups G in a Hilbert space H containing various
mutually nonhomeomorphic orbits O j := U(G)ϕ j in P(H), ( j = 1, 2, . . . ). Let G
be a connected Lie group containing the 2n + 1—dimensional Heisenberg group
G n as an invariant (i.e. normal) subgroup (G will be specified later). Let U be such
a unitary continuous representation of G, the restriction of which to G n coincides
with the irreducible representation described in Sect. 4.1 with = 1 = λ. With the
notation of the previous section, for m = 1, 2, . . . K , A m ∈ U (g), set
A m :=
1
2
h
jk
m X j X k , (summation over j, k = 1, 2, . . . 2n),
(4.2.1)
with any h m a real symmetric 2n × 2n—matrix; the formally defined operator A m is
symmetric on the Gårding domain of U (G n ). From (4.1.1a) we have commutation
relations (cf. also 4.1.3):
[X j , X k ] = i S jk I, [X j , A m ] = i S jk h
kl
m X l =: i (S · h m · X ) j ,
(4.2.2)
[A m , A k ] =
i
2
X · (h m · S · h k − h k · S · h m ) · X, m, k = 1, 2, . . . K . (4.2.3)
Assume that for any m, k there are reals c
j
mk such, that
h m · S · h k − h k · S · h m =
K
j=1
c
j
mk h j , m, k = 1, 2, . . . K .
(4.2.4)