Quantum Mechanics for Mathematicians

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.