Quantum Mechanics for Mathematicians

14.2 The Poisson bracket and the Heisenberg Lie algebra

A third fundamental property of the Poisson bracket that can easily be checked
is the

• Leibniz rule

{f 1 f 2 ,f}={f 1 ,f}f 2 +f 1 {f 2 ,f}, {f,f 1 f 2 }={f,f 1 }f 2 +f 1 {f,f 2 }

This property says that taking Poisson bracket with a functionfacts on a
product of functions in a way that satisfies the Leibniz rule for what happens
when you take the derivative of a product. Unlike antisymmetry and the Ja-
cobi identity, which reflect the Lie algebra structure on functions, the Leibniz
property describes the relation of the Lie algebra structure to multiplication of
functions. At least for polynomial functions, it allows one to inductively reduce
the calculation of Poisson brackets to the special case of Poisson brackets of the
coordinate functionsqandp, for instance:

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

The Poisson bracket is thus determined by its values on linear functions
(thus by the relations{q,q}={p,p}= 0,{q,p}= 1). We will define:

Definition.Ω(·,·)is the restriction of the Poisson bracket toM∗, the linear
functions onM. Taking as basis vectors ofM∗the coordinate functionsqand
p,Ωis given on basis vectors by

Ω(q,q) = Ω(p,p) = 0, Ω(q,p) =−Ω(p,q) = 1

A general element ofM∗will be a linear combinationcqq+cppfor some
constantscq,cp. For general pairs of elements inM∗, Ω will be given by

Ω(cqq+cpp,c′qq+c′pp) =cqc′p−cpc′q (14.1)

We will often write elements ofM∗as the column vector of their coefficients
cq,cp, identifying

cqq+cpp↔

(

cq

cp

)

Then one has

Ω

((

cq

cp

)

,

(

c′q

c′p

))

=cqc′p−cpc′q

Taking together linear functions onMand the constant function, one gets
a three dimensional space with basis elementsq,p,1, and this space is closed
under Poisson bracket. This space is thus a Lie algebra, and is isomorphic to
the Heisenberg Lie algebrah 3 (see section 13.1), with the isomorphism given on
basis elements by
X↔q, Y↔p, Z↔ 1