58
4 Examples of Classical Mechanical Projections
then the linear hull of the operators X j ( j = 1, 2, . . . 2n), A m (m = 1, 2, . . . K )
and I := I H := id H forms the Lie algebra U (g). We have (cf. also [37, Proposition
3.3.12]):
4.2.2 Proposition. Let U (g) be the above defined representation of a Lie algebra
g in H and let G be the corresponding simply connected Lie group with the Lie
algebra g. Then the representation U (G n ) of the Heisenberg group G n has a unique
extension to the representation U (G) of G in H such, that the closures of the operators
X j ( j = 1, 2, . . . 2n), A m (m = 1, 2, . . . K ) and I H are selfadjoint generators of
U (G) corresponding to basis vectors in g according to (2.3.20). In particular the
operators A m (m = 1, 2, . . . K ) are essentially selfadjoint on the Gårding domain
of U (G n ).
Proof. The Gårding domain of U (G n ) is a common dense invariant domain of all
the operators in U (g). According to a Nelson’s theorem (see [13, Theorem 11.5.2.])
it suffices to prove essential selfadjointness of the operator ,
:=
2n
j=1
X
2
j +
K
m=1
A
2
m ,
(4.2.5)
on the invariant domain. First we shall choose m := ( j; k) with j, k = 1, 2, . . . 2n
and
A m := A ( j;k) :=
1
2
(X j X k + X k X j ).
(4.2.6)
In this case the operator in (4.2.5) can be expressed in the form
=
3
2
n I H +
n
j=1
(P
2
j + Q
2
j )(I H +
n
k=1
(P
2
k + Q
2
k )),
(4.2.7)
where we used CCR and the notation (4.1.4). From the known properties of the
Hamiltonians P
2
j + Q
2
j of independent linear oscillators, we conclude (with a help,
e.g., of [262, Theorem VIII.33]) that is essentially selfadjoint.
Denote the Lie algebra generated by X j ’s and A ( j;k) ( j, k = 1, 2, . . . 2n) by
g max and the corresponding simply connected group by G max . Any A m of the
form (4.2.1) is a linear combination of A ( j;k) ’s. Consequently, any Lie algebra U (g)
from 4.2.1 is a subalgebra of U (g max ) and the corresponding group G is a subgroup
of G max . From this just proved integrability of U (g max ) to a unitary representation
U (G max ), it follows integrability of U (g) for any g introduced in 4.2.1. This implies
the selfadjointness of (4.2.5) with arbitrary A m of the form (4.2.1) and this, in turn,
implies uniqueness of U (G).
4 Examples of Classical Mechanical Projections
then the linear hull of the operators X j ( j = 1, 2, . . . 2n), A m (m = 1, 2, . . . K )
and I := I H := id H forms the Lie algebra U (g). We have (cf. also [37, Proposition
3.3.12]):
4.2.2 Proposition. Let U (g) be the above defined representation of a Lie algebra
g in H and let G be the corresponding simply connected Lie group with the Lie
algebra g. Then the representation U (G n ) of the Heisenberg group G n has a unique
extension to the representation U (G) of G in H such, that the closures of the operators
X j ( j = 1, 2, . . . 2n), A m (m = 1, 2, . . . K ) and I H are selfadjoint generators of
U (G) corresponding to basis vectors in g according to (2.3.20). In particular the
operators A m (m = 1, 2, . . . K ) are essentially selfadjoint on the Gårding domain
of U (G n ).
Proof. The Gårding domain of U (G n ) is a common dense invariant domain of all
the operators in U (g). According to a Nelson’s theorem (see [13, Theorem 11.5.2.])
it suffices to prove essential selfadjointness of the operator ,
:=
2n
j=1
X
2
j +
K
m=1
A
2
m ,
(4.2.5)
on the invariant domain. First we shall choose m := ( j; k) with j, k = 1, 2, . . . 2n
and
A m := A ( j;k) :=
1
2
(X j X k + X k X j ).
(4.2.6)
In this case the operator in (4.2.5) can be expressed in the form
=
3
2
n I H +
n
j=1
(P
2
j + Q
2
j )(I H +
n
k=1
(P
2
k + Q
2
k )),
(4.2.7)
where we used CCR and the notation (4.1.4). From the known properties of the
Hamiltonians P
2
j + Q
2
j of independent linear oscillators, we conclude (with a help,
e.g., of [262, Theorem VIII.33]) that is essentially selfadjoint.
Denote the Lie algebra generated by X j ’s and A ( j;k) ( j, k = 1, 2, . . . 2n) by
g max and the corresponding simply connected group by G max . Any A m of the
form (4.2.1) is a linear combination of A ( j;k) ’s. Consequently, any Lie algebra U (g)
from 4.2.1 is a subalgebra of U (g max ) and the corresponding group G is a subgroup
of G max . From this just proved integrability of U (g max ) to a unitary representation
U (G max ), it follows integrability of U (g) for any g introduced in 4.2.1. This implies
the selfadjointness of (4.2.5) with arbitrary A m of the form (4.2.1) and this, in turn,
implies uniqueness of U (G).