Chapter 17

Quantization

Given any Hamiltonian classical mechanical system with phase spaceR^2 d, phys-
ics textbooks have a standard recipe for producing a quantum system, by a
method known as “canonical quantization”. We will see that for linear func-
tions on phase space, this is just the construction we have already seen of a
unitary representation Γ′Sof the Heisenberg Lie algebra, the Schr ̈odinger repre-
sentation. The Stone-von Neumann theorem assures us that this is the unique
such construction, up to unitary equivalence. We will also see that this recipe
can only ever be partially successful: the Schr ̈odinger representation gives us
a representation of a sub-algebra of the Lie algebra of all functions on phase
space (the polynomials of degree two and below), but a no-go theorem shows
that this cannot be extended to a representation of the full infinite dimensional
Lie algebra. Recipes for quantizing higher-order polynomials will always suffer
from a lack of uniqueness, a phenomenon known to physicists as the existence
of “operator ordering ambiguities.”
In later chapters we will see that this quantization prescription does give
unique quantum systems corresponding to some Hamiltonian systems (in par-
ticular the harmonic oscillator and the hydrogen atom), and does so in a manner
that allows a description of the quantum system purely in terms of representa-
tion theory.

17.1 Canonical quantization

Very early on in the history of quantum mechanics, when Dirac first saw the
Heisenberg commutation relations, he noticed an analogy with the Poisson
bracket. One has

{q,p}= 1 and −

i

~

[Q,P] = 1

as well as
df
dt

={f,h} and

d

dt

O(t) =−

i

~

[O,H]