76
5 Macroscopic Limits
X ξ N :=
1
N
X
N
ξ , ξ ∈ g, N = 1, 2, . . . .
(5.1.4)
In terms of [155] X
N
ξ (resp. X ξ N ) are ‘extensive (resp. intensive) observables’
but, contrary to [155], they can be unbounded in our case. The limits for large N
of X ξ N ’s could exist in some convenient sense, but they are not generators of any
unitary representation of the group G. Due to the commutation relations
[X ξ N , X η N ] =
i
N
X [ξ,η]N ,
(5.1.5)
the limits of X ξ N (ξ ∈ g) will be mutually commuting operators. To obtain correct
classical commutation relations (i.e. the Poisson brackets, see 1.3.5) for functions
f ξ N on the orbits O
N
x (x ∈ H N ),
f ξ N : x → f ξ N (x) := T r(P x X ξ N ),
(5.1.6)
in the limit N → ∞, the two-form
N from (5.1.2) should be ‘renormalized’. We
define
N :=
1
N
N
.
(5.1.7)
The form N (if restricted onto the symplectic manifold M
N
x obtained from O
N
x
as in Sect. 3.2) associates with the Hamiltonian function f ξ N the vector field σ ξ
(restricted to M
N
x ) given by the flow (5.1.3). It is
N • (σ ξ , σ η ) = i T r(P • [X
N
ξ , X η N ]) = −T r(P • X [ξ,η]N ).
(5.1.8)
We intend to develop a corresponding formalism for infinite systems, i.e. a suitable
one for the work in the limit N = ‘actual infinity’.
5.1.3 Let U (G) be a continuous unitary representation of a connected Lie group G
on a separable Hilbert space H. We shall use notation of Chap. 4 for concepts related
to U (G). Let be an index set (of arbitrary cardinality) and H j ( j ∈ ) be copies
of H. Let us fix unitary maps
u j : H → H j , j ∈ ,
(5.1.9)
of H onto H j ’s. Let
H :=
j∈
H j
(5.1.10)
be the tensor product defined according to von Neumann [227] and known as CTPS
(:= complete tensor product space—see also notes in the text in 5.1.4 below and
[35, 106, 274]). For ϕ j ∈ H j let
5 Macroscopic Limits
X ξ N :=
1
N
X
N
ξ , ξ ∈ g, N = 1, 2, . . . .
(5.1.4)
In terms of [155] X
N
ξ (resp. X ξ N ) are ‘extensive (resp. intensive) observables’
but, contrary to [155], they can be unbounded in our case. The limits for large N
of X ξ N ’s could exist in some convenient sense, but they are not generators of any
unitary representation of the group G. Due to the commutation relations
[X ξ N , X η N ] =
i
N
X [ξ,η]N ,
(5.1.5)
the limits of X ξ N (ξ ∈ g) will be mutually commuting operators. To obtain correct
classical commutation relations (i.e. the Poisson brackets, see 1.3.5) for functions
f ξ N on the orbits O
N
x (x ∈ H N ),
f ξ N : x → f ξ N (x) := T r(P x X ξ N ),
(5.1.6)
in the limit N → ∞, the two-form
N from (5.1.2) should be ‘renormalized’. We
define
N :=
1
N
N
.
(5.1.7)
The form N (if restricted onto the symplectic manifold M
N
x obtained from O
N
x
as in Sect. 3.2) associates with the Hamiltonian function f ξ N the vector field σ ξ
(restricted to M
N
x ) given by the flow (5.1.3). It is
N • (σ ξ , σ η ) = i T r(P • [X
N
ξ , X η N ]) = −T r(P • X [ξ,η]N ).
(5.1.8)
We intend to develop a corresponding formalism for infinite systems, i.e. a suitable
one for the work in the limit N = ‘actual infinity’.
5.1.3 Let U (G) be a continuous unitary representation of a connected Lie group G
on a separable Hilbert space H. We shall use notation of Chap. 4 for concepts related
to U (G). Let be an index set (of arbitrary cardinality) and H j ( j ∈ ) be copies
of H. Let us fix unitary maps
u j : H → H j , j ∈ ,
(5.1.9)
of H onto H j ’s. Let
H :=
j∈
H j
(5.1.10)
be the tensor product defined according to von Neumann [227] and known as CTPS
(:= complete tensor product space—see also notes in the text in 5.1.4 below and
[35, 106, 274]). For ϕ j ∈ H j let