Quantum Mechanics for Mathematicians

The Jacobi identity implies

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

so
X{f 1 ,f 2 }=Xf 2 Xf 1 −Xf 1 Xf 2 =−[Xf 1 ,Xf 2 ] (15.5)
This shows that the mapf→Xfof equation 15.2 that we defined between
these Lie algebras is not quite a Lie algebra homomorphism because of the -
sign in equation 15.5 (it is called a Lie algebra “antihomomorphism”). The map
that is a Lie algebra homomorphism is

f→−Xf (15.6)

To keep track of the minus sign here, one needs to keep straight the difference
between

• The functions on phase spaceMare a Lie algebra, with a functionfacting
  on the function space by the adjoint action

ad(f)(·) ={f,·}

and

• The functionsfprovide vector fieldsXfacting on functions onM, where

Xf(·) ={·,f}=−{f,·}

As a simple example, the functionpsatisfies

{p,F(q,p)}=−

∂F

∂q

so

{p,·}=−

∂(·)

∂q

=−Xp

Note that acting on functions withpin this way is the Lie algebra version
of the representation of the translation group on functions induced from the
translation action on the position (see equations 10.1 and 10.2).
It is important to note that the Lie algebra homomorphism 15.6 from func-
tions to vector fields is not an isomorphism, for two reasons:

• It is not injective (one-to-one), since functionsfandf+Cfor any constant
  Ccorrespond to the sameXf.


• It is not surjective since not all vector fields are Hamiltonian vector fields
  (i.e., of the formXffor somef). One property that a vector fieldXmust
  satisfy in order to possibly be a Hamiltonian vector field is

X{g 1 ,g 2 }={Xg 1 ,g 2 }+{g 1 ,Xg 2 } (15.7)

forg 1 andg 2 onM. This is the Jacobi identity forf,g 1 ,g 2 , whenX=Xf.