54
4 Examples of Classical Mechanical Projections
classical projections of QM. Let us write λ
2
(λ ∈ (0, 1]) instead of in all formulas
of the subsections 4.1.1–4.1.7. Let X j (λ) be the Schrödinger realizations of the CCRgenerators in L
2
(R
n
) =: H and let us apply to them the transformation U λ from
(4.1.7) for each value of λ. Let us denote X
λ
j := U λ X j (λ)U
−1
λ . We obtain:
Q
λ
j ϕ(q) = λq j ϕ(q), P
λ
j ϕ(q) = −i λ
∂
∂q j
ϕ(q),
(4.1.24)
where
Q
λ
j := X
λ
j , P
λ
j := X
λ
j+n ( j = 1, 2, . . . n).
Let us fix ϕ = ϕ 0 ∈ H according to (4.1.15), which will be held unchanged for all
the values of λ. Let W
λ be the unitary representation from 4.1.3:
W
λ
x := exp
i
λ 2
X
λ
· S · x
.
(4.1.25)
Let ϕ
λ
x := W
λ
x ϕ, i.e. for x := (q; p) ∈ R
2n we have
ϕ
λ
x (q
) = exp
−
i
2λ 2
q · p
exp
i
λ
q
· p
ϕ(q
−
q
λ
); q, p, q
∈ R
n
.
(4.1.26)
Let
P
(λ)
x
be the projector onto ϕ
λ
x , P
(λ)
0 = P ϕ ≡ P ϕ for all λ.
The correlations of all orders are for any λ independent of x:
T r
P
(λ)
x (X
λ
j − x j )(X
λ
k − x k ) . . . (X
λ
r − x r )
= T r( P ϕ X
λ
j X
λ
k . . . X
λ
r ). (4.1.27)
The right hand side of (4.1.27) is proportional to λ
s , where s is the number of
X
λ in the right hand side of (4.1.27). From this we see that the
algebra E(g)
λ of quantal observables consisting of polynomials in X
λ
is mapped onto a set of functions on O
λ
ϕ := W
λ
(G)ϕ:
f
λ
: E → f
λ
E (x) := T r( P
(λ)
x E), E ∈ E(g)
λ
; F
λ
X j
(x) = x j ,
(4.1.28)
and this mapping f
λ becomes in the limit λ → 0 a homomorphism of associative
algebras.
For the generator f
λ
A of the ‘projected’ evolution in time, corresponding to the
quantal generator (4.1.18), i.e., for each λ, to the operator
4 Examples of Classical Mechanical Projections
classical projections of QM. Let us write λ
2
(λ ∈ (0, 1]) instead of in all formulas
of the subsections 4.1.1–4.1.7. Let X j (λ) be the Schrödinger realizations of the CCRgenerators in L
2
(R
n
) =: H and let us apply to them the transformation U λ from
(4.1.7) for each value of λ. Let us denote X
λ
j := U λ X j (λ)U
−1
λ . We obtain:
Q
λ
j ϕ(q) = λq j ϕ(q), P
λ
j ϕ(q) = −i λ
∂
∂q j
ϕ(q),
(4.1.24)
where
Q
λ
j := X
λ
j , P
λ
j := X
λ
j+n ( j = 1, 2, . . . n).
Let us fix ϕ = ϕ 0 ∈ H according to (4.1.15), which will be held unchanged for all
the values of λ. Let W
λ be the unitary representation from 4.1.3:
W
λ
x := exp
i
λ 2
X
λ
· S · x
.
(4.1.25)
Let ϕ
λ
x := W
λ
x ϕ, i.e. for x := (q; p) ∈ R
2n we have
ϕ
λ
x (q
) = exp
−
i
2λ 2
q · p
exp
i
λ
q
· p
ϕ(q
−
q
λ
); q, p, q
∈ R
n
.
(4.1.26)
Let
P
(λ)
x
be the projector onto ϕ
λ
x , P
(λ)
0 = P ϕ ≡ P ϕ for all λ.
The correlations of all orders are for any λ independent of x:
T r
P
(λ)
x (X
λ
j − x j )(X
λ
k − x k ) . . . (X
λ
r − x r )
= T r( P ϕ X
λ
j X
λ
k . . . X
λ
r ). (4.1.27)
The right hand side of (4.1.27) is proportional to λ
s , where s is the number of
X
λ in the right hand side of (4.1.27). From this we see that the
algebra E(g)
λ of quantal observables consisting of polynomials in X
λ
is mapped onto a set of functions on O
λ
ϕ := W
λ
(G)ϕ:
f
λ
: E → f
λ
E (x) := T r( P
(λ)
x E), E ∈ E(g)
λ
; F
λ
X j
(x) = x j ,
(4.1.28)
and this mapping f
λ becomes in the limit λ → 0 a homomorphism of associative
algebras.
For the generator f
λ
A of the ‘projected’ evolution in time, corresponding to the
quantal generator (4.1.18), i.e., for each λ, to the operator