50
4 Examples of Classical Mechanical Projections
Setting
Q j := X j , P j := X j+n for j = 1, 2, . . . n
(4.1.4)
we obtain from (4.1.1a) the usual form of the canonical commutation relations
(CCR). There is only one physically admissible choice of the constant : it is the
Planck constant divided by 2π (its numerical value depends on a choice of physical
units for determination of which it is necessary to consider also dynamics). Operators
Q j (resp. P j ) are interpreted to correspond to observables called ‘coordinates of the
configuration’ (resp. ‘coordinates of the linear momentum’), in a cartesian basis.
Note, that this representation of G can be considered as a projective representation
of R
2n , as it was described in 1.2.7.
4.1.2 The Schrödinger form of the above mentioned representation of G consists of
the realization of the Hilbert space H of the representation as L
2
(R
n
, d
n q) (d
n q is the
Lebesgue measure) and the action of X j ’s can be defined on such ϕ ∈ L
2
(R
n
, d
n q),
which belong to Schwartz test functions:
(X j ϕ)(q 1 , q 2 , . . . q n ) := q j ϕ(q 1 , q 2 , . . . q n )
(4.1.5a)
and
(X j+n ϕ)(q 1 , q 2 , . . . q n ) := −i
∂
∂q j
ϕ(q 1 , q 2 , . . . q n )
(4.1.5b)
for j = 1, 2, . . . n. An equivalent realization of CCR is obtained by an arbitrary
unitary transformation U of H onto itself, e.g. by the scaling U = U λ (λ ∈ R + \ {0}):
(U λ ϕ)(q) := λ
n/2
ϕ(λq).
(4.1.6)
It is U
−1
λ = U 1/λ and we have:
X
j := U λ X j U
−1
λ = λX j , X
j+n := U λ X j+n U
−1
λ =
1
λ
X j+n , j = 1, 2, . . . n.
(4.1.7)
These transformations are useful for taking limits → 0, compare [154] and also
our 4.1.8.
4.1.3 Let X · S · x := X j S jk x k with summation over j, k = 1, 2, . . . 2n, where x k ∈
R for all k. Let W x (x ∈ R
2n
) be unitary operators of the above mentioned projective
representation (cf. 1.2.7) :
W x := exp
i
X · S · x
.
(4.1.8)
4 Examples of Classical Mechanical Projections
Setting
Q j := X j , P j := X j+n for j = 1, 2, . . . n
(4.1.4)
we obtain from (4.1.1a) the usual form of the canonical commutation relations
(CCR). There is only one physically admissible choice of the constant : it is the
Planck constant divided by 2π (its numerical value depends on a choice of physical
units for determination of which it is necessary to consider also dynamics). Operators
Q j (resp. P j ) are interpreted to correspond to observables called ‘coordinates of the
configuration’ (resp. ‘coordinates of the linear momentum’), in a cartesian basis.
Note, that this representation of G can be considered as a projective representation
of R
2n , as it was described in 1.2.7.
4.1.2 The Schrödinger form of the above mentioned representation of G consists of
the realization of the Hilbert space H of the representation as L
2
(R
n
, d
n q) (d
n q is the
Lebesgue measure) and the action of X j ’s can be defined on such ϕ ∈ L
2
(R
n
, d
n q),
which belong to Schwartz test functions:
(X j ϕ)(q 1 , q 2 , . . . q n ) := q j ϕ(q 1 , q 2 , . . . q n )
(4.1.5a)
and
(X j+n ϕ)(q 1 , q 2 , . . . q n ) := −i
∂
∂q j
ϕ(q 1 , q 2 , . . . q n )
(4.1.5b)
for j = 1, 2, . . . n. An equivalent realization of CCR is obtained by an arbitrary
unitary transformation U of H onto itself, e.g. by the scaling U = U λ (λ ∈ R + \ {0}):
(U λ ϕ)(q) := λ
n/2
ϕ(λq).
(4.1.6)
It is U
−1
λ = U 1/λ and we have:
X
j := U λ X j U
−1
λ = λX j , X
j+n := U λ X j+n U
−1
λ =
1
λ
X j+n , j = 1, 2, . . . n.
(4.1.7)
These transformations are useful for taking limits → 0, compare [154] and also
our 4.1.8.
4.1.3 Let X · S · x := X j S jk x k with summation over j, k = 1, 2, . . . 2n, where x k ∈
R for all k. Let W x (x ∈ R
2n
) be unitary operators of the above mentioned projective
representation (cf. 1.2.7) :
W x := exp
i
X · S · x
.
(4.1.8)