Quantum Mechanics for Mathematicians

Proof.For a detailed proof, see section 5.4 of [8], section 4.4 of [26], or chapter
16 of [37]. In outline, the proof begins by showing that taking Poisson brack-
ets of polynomials of degree three leads to higher order polynomials, and that
furthermore for degree three and above there will be no finite dimensional sub-
algebras of polynomials of bounded degree. The assumptions of the theorem
force certain specific operator ordering choices in degree three. These are then
used to get a contradiction in degree four, using the fact that the same degree
four polynomial has two different expressions as a Poisson bracket:

q^2 p^2 =

1

3

{q^2 p,p^2 q}=

1

9

{q^3 ,p^3 }

17.3 Canonical quantization inddimensions

The above can easily be generalized to the case ofddimensions, with the
Schr ̈odinger representation Γ′Snow giving a unitary representation of the Heisen-
berg Lie algebrah 2 d+1determined by

Γ′S(qj) =−iQj, Γ′S(pj) =−iPj

which satisfy the Heisenberg relations

[Qj,Pk] =iδjk

Generalizing to quadratic polynomials in the phase space coordinate func-
tions, we have

Γ′S(qjqk) =−iQjQk, Γ′S(pjpk) =−iPjPk, Γ′S(qjpk) =−

i
2
(QjPk+PkQj)
(17.1)
These operators can be exponentiated to get a representation on the sameH
ofMp(2d,R), a double cover of the symplectic groupSp(2d,R). This phe-
nomenon will be examined carefully in later chapters, starting with chapter 20
and the calculation in section 20.3.2, followed by a discussion in chapters 24
and 25 using a different (but unitarily equivalent) representation that appears
in the quantization of the harmonic oscillator. The Groenewold-van Hove the-
orem implies that we cannot find a unitary representation of a larger group of
canonical transformations extending this one of the Heisenberg and metaplectic
groups.

17.4 Quantization and symmetries

The Schr ̈odinger representation is thus a Lie algebra representation providing
observables corresponding to elements of the Lie algebrash 2 d+1(linear combi-
nations ofQjandPk) andsp(2d,R) (linear combinations of degree-two com-
binations ofQjandPk). The observables that commute with the Hamiltonian