Quantum Mechanics for Mathematicians

where the last of these equations is the equation for the time dependence of a
Heisenberg picture observableO(t) in quantum mechanics. Dirac’s suggestion
was that given any classical Hamiltonian system, one could “quantize” it by
finding a rule that associates to a functionf on phase space a self-adjoint
operatorOf(in particularOh=H), acting on a state spaceHsuch that

O{f,g}=−

i

~

[Of,Og]

This is completely equivalent to asking for a unitary representation (π′,H)
of the infinite dimensional Lie algebra of functions on phase space (with the
Poisson bracket as Lie bracket). To see this, note that units for momentum
pand positionq can be chosen such that~= 1. Then, as usual getting a
skew-adjoint Lie algebra representation operator by multiplying a self-adjoint
operator by−i, setting
π′(f) =−iOf

the Lie algebra homomorphism property

π′({f,g}) = [π′(f),π′(g)]

corresponds to
−iO{f,g}= [−iOf,−iOg] =−[Of,Og]

so one has Dirac’s suggested relation.
Recall that the Heisenberg Lie algebra is isomorphic to the three dimen-
sional sub-algebra of functions on phase space given by linear combinations of
the constant function, the functionqand the functionp. The Schr ̈odinger rep-
resentation Γ′Sprovides a unitary representation not of the Lie algebra of all
functions on phase space, but of these polynomials of degree at most one, as
follows
O 1 = 1 , Oq=Q, Op=P

so

Γ′S(1) =−i 1 , Γ′S(q) =−iQ=−iq, Γ′S(p) =−iP=−

d

dq

Moving on to quadratic polynomials, these can also be quantized, as follows

Op^2

2

=

P^2

2

, Oq^2

2

=

Q^2

2

For the functionpqone can no longer just replacepbyPandqbyQsince the
operatorsPandQdon’t commute, so the ordering matters. In addition, neither
PQnorQPis self-adjoint. What does work, satisfying all the conditions to give
a Lie algebra homomorphism, is the self-adjoint combination

Opq=

1

2

(PQ+QP)

This shows that the Schr ̈odinger representation Γ′Sthat was defined as a
representation of the Heisenberg Lie algebrah 3 extends to a unitary Lie algebra