Chapter 14

The Poisson Bracket and

Symplectic Geometry

We have seen that the quantum theory of a free particle corresponds to the con-
struction of a representation of the Heisenberg Lie algebra in terms of operators
QandP, together with a choice of HamiltonianH= 21 mP^2. One would like to
use this to produce quantum systems with a similar relation to more non-trivial
classical mechanical systems than the free particle. During the earliest days of
quantum mechanics it was recognized by Dirac that the commutation relations
of theQandP operators somehow corresponded to the Poisson bracket rela-
tions between the position and momentum coordinates on phase space in the
Hamiltonian formalism for classical mechanics. In this chapter we’ll give an out-
line of the topic of Hamiltonian mechanics and the Poisson bracket, including
an introduction to the symplectic geometry that characterizes phase space.
The Heisenberg Lie algebrah 2 d+1is usually thought of as quintessentially
quantum in nature, but it is already present in classical mechanics, as the Lie
algebra of degree zero and one polynomials on phase space, with Lie bracket
the Poisson bracket. In chapter 16 we will see that degree two polynomials on
phase space also provide an important finite dimensional Lie algebra.
The full Lie algebra of all functions on phase space (with Lie bracket the
Poisson bracket) is infinite dimensional, so not the sort of finite dimensional Lie
algebra given by matrices that we have studied so far. Historically though, it
is this kind of infinite dimensional Lie algebra that motivated the discovery of
the theory of Lie groups and Lie algebras by Sophus Lie during the 1870s. It
also provides the fundamental mathematical structure of the Hamiltonian form
of classical mechanics.

14.1 Classical mechanics and the Poisson bracket

In classical mechanics in the Hamiltonian formalism, the space M = R^2 d
that one gets by putting together positions and the corresponding momenta