B.8 Chapter 17
Problem 1:
This is part of a proof of the Groenewold-van Hove theorem.
- Show that one can writeq^2 p^2 in two ways as a Poisson bracket
q^2 p^2 =
1
3
{q^2 p,p^2 q}=
1
9
{q^3 ,p^3 }
- Assume that we can quantize any polynomial inq,pby a Lie algebra
homomorphismπ′that takes polynomials inq,pwith the Poisson bracket
to polynomials inQ,Pwith the commutator, in a way that extends the
standard Schr ̈odinger representation
π′(q) =−iQ, π′(p) =−iP, π′(1) =−i 1
Further assume that the following relations are satisfied for low degree
polynomials (it actually is possible to prove that these are necessary):
π′(qp) =−i
1
2
(QP+PQ), π′(q^2 ) =−iQ^2 , π′(p^2 ) =−iP^2
π′(q^3 ) =−iQ^3 , π′(p^3 ) =−iP^3
Then show that
π′(q^2 p) =−i
1
2
(Q^2 P+PQ^2 )
(Hint: use{q^3 ,p^2 }= 6q^2 p)
- Also show that
π′(qp^2 ) =−i
1
2
(QP^2 +P^2 Q)
- Finally, show that
π′
(
1
9
{q^3 ,p^3 }
)
=−i
(
−
2
3
1 − 2 iQP+Q^2 P^2
)
and
π′
(
1
3
{q^2 p,p^2 q}
)
=−i
(
−
1
3
1 − 2 iQP+Q^2 P^2
)
which demonstrates a contradiction.