Quantum Mechanics for Mathematicians

which says that the momentum is the mass times the velocity, and is conserved.
For a particle subject to a potentialV(q) one has

h=

p^2

2 m

+V(q)

and the trajectories are the solutions to

q ̇=

p

m

, p ̇=−

∂V

∂q

This adds Newton’s second law

F=−

∂V

∂q

=ma=mq ̈

to the relation between momentum and velocity.
One can easily check that the Poisson bracket has the properties

• Antisymmetry
  {f 1 ,f 2 }=−{f 2 ,f 1 }


• Jacobi identity

{{f 1 ,f 2 },f 3 }+{{f 3 ,f 1 },f 2 }+{{f 2 ,f 3 },f 1 }= 0

These two properties, together with the bilinearity, show that the Poisson
bracket fits the definition of a Lie bracket, making the space of functions on
phase space into an infinite dimensional Lie algebra. This Lie algebra is respon-
sible for much of the structure of the subject of Hamiltonian mechanics, and it
was historically the first sort of Lie algebra to be studied.
From the fundamental dynamical equation
df
dt
={f,h}

we see that

{f,h}= 0 =⇒

df

dt

= 0

and in this case the functionfis called a “conserved quantity”, since it does
not change under time evolution. Note that if we have two functionsf 1 andf 2
on phase space such that

{f 1 ,h}= 0, {f 2 ,h}= 0

then using the Jacobi identity we have

{{f 1 ,f 2 },h}=−{{h,f 1 },f 2 }−{{f 2 ,h},f 1 }= 0

This shows that iff 1 andf 2 are conserved quantities, so is{f 1 ,f 2 }. As a
result, functionsfsuch that{f,h}= 0 make up a Lie subalgebra. It is this Lie
subalgebra that corresponds to “symmetries” of the physics, commuting with
the time translation determined by the dynamical law given byh.