14 The Poisson Bracket and Symplectic Geometry

We have seen that the quantum theory of a free particle corresponds to the construction of a representation of the Heisenberg Lie algebra in terms of operators $Q$ and P, together with a choice of Hamiltonian $\begin{array}{r}H=\frac{1}{2m}{P}^2\end{array}$ . 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 the $Q$ and $P$ operators somehow corresponded to the Poisson bracket relations between the position and momentum coordinates on phase space in the Hamiltonian formalism for classical mechanics. In this chapter we’ll give an outline of the topic of Hamiltonian mechanics and the Poisson bracket, including an introduction to the symplectic geometry that characterizes phase space.

The Heisenberg Lie algebra ${\mathfrak{h}}_{2d+1}$ is 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.

中文目录

Chapter contents

• 14.1 Classical mechanics and the Poisson bracket
• 14.2 The Poisson bracket and the Heisenberg Lie algebra
• 14.3 Symplectic geometry
• 14.4 For further reading

Previous chapter: 13 The Heisenberg group and the Schrödinger Representation

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