angle from 0 to 2π), the phase angle only goes from 0 toπ, demonstrating the
same problem that occurs in the case of the spinor representation.
When we study the Schr ̈odinger representation using its action on the quan-
tum harmonic oscillator state spaceHin chapter 22 we will see that the operator
1
2(Q^2 +P^2 )
is the Hamiltonian operator for the quantum harmonic oscillator, and all of
its eigenvectors (not justψ 0 (q)) have half-integer eigenvalues. In chapter 24
we will go on to discuss in more detail the construction of the metaplectic
representation, using methods developed to study the harmonic oscillator.
17.2 The Groenewold-van Hove no-go theorem
If one wants to quantize polynomial functions on phase space of degree greater
than two, it quickly becomes clear that the problem of “operator ordering am-
biguities” is a significant one. Different prescriptions involving different ways
of ordering theP andQoperators lead to differentOffor the same function
f, with physically different observables (although the differences involve the
commutator ofPandQ, so higher-order terms in~).
When physicists first tried to find a consistent prescription for producing an
operatorOfcorresponding to a polynomial function on phase space of degree
greater than two, they found that there was no possible way to do this consistent
with the relation
O{f,g}=−i
~[Of,Og]for polynomials of degree greater than two. Whatever method one devises for
quantizing higher degree polynomials, it can only satisfy that relation to lowest
order in~, and there will be higher order corrections, which depend upon one’s
choice of quantization scheme. Equivalently, it is only for the six dimensional Lie
algebra of polynomials of degree up to two that the Schr ̈odinger representation
gives one a Lie algebra representation, and this cannot be consistently extended
to a representation of a larger subalgebra of the functions on phase space. This
problem is made precise by the following no-go theorem
Theorem(Groenewold-van Hove).There is no mapf→Offrom polynomials
onR^2 to self-adjoint operators onL^2 (R)satisfying
O{f,g}=−i
~[Of,Og]and
Op=P, Oq=Q
for any Lie subalgebra of the functions onR^2 for which the subalgebra of poly-
nomials of degree less than or equal to two is a proper subalgebra.