14.2 The Poisson bracket and the Heisenberg Lie algebra

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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 function $f$ acts 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 Jacobi 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 functions $q$ and $p,$ for instance:

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

$Ω (\cdot, \cdot)$ is the restriction of the Poisson bracket to $M^{*}$ , the linear functions on $M$ . Taking as basis vectors of $M^{*}$ the coordinate functions $q$ and $p,$ Ω is given on basis vectors by
$Ω (q, q) = Ω (p, p) = 0, Ω (q, p) = - Ω (p, q) = 1$

A general element of $M^{*}$ will be a linear combination $c_{q} q + c_{p} p$ for some constants $c_{q}, c_{p}$ . For general pairs of elements in $M^{*}$ , Ω will be given by

Ω (c_{q} q + c_{p} p, c_{q}^{'} q + c_{p}^{'} p) = c_{q} c_{p}^{'} - c_{p} c_{q}^{'} (14.1)

We will often write elements of $M^{*}$ as the column vector of their coeficients $c_{q}, c_{p}$ , identifying

c_{q} q + c_{p} p \leftrightarrow (c _{p} c _{q})

Then one has

Ω ((c _{p} c _{q}), (c _{p}^{'} c _{q}^{'})) = c_{q} c_{p}^{'} - c_{p} c_{q}^{'}

Taking together linear functions on $M$ and the constant function, one gets a three dimensional space with basis elements $q, p, 1$ , and this space is closed under Poisson bracket. This space is thus a Lie algebra, and is isomorphic to the Heisenberg Lie algebra $h_{3}$ (see section 13.1), with the isomorphism given on basis elements by

X \leftrightarrow q, Y \leftrightarrow p, Z \leftrightarrow 1

This isomorphism preserves the Lie bracket relations since

[X, Y] = Z \leftrightarrow {q, p} = 1

It is convenient to choose its own notation for the dual phase space, so we will often write $M^{*} = M$ . The three dimensional space we have identified with the Heisenberg Lie algebra is then

M \oplus R

We will denote elements of this space in two diferent ways

• As functions $c_{q} q + c_{p} p + c,$ with Lie bracket the Poisson bracket

{c_{q} q + c_{p} p + c, c_{q}^{'} q + c_{p}^{'} p + c^{'}} = c_{q} c_{p}^{'} - c_{p} c_{q}^{'}

• As pairs of an element of M and a real number

((c _{p} c _{q}), c)

In this second notation, the Lie bracket is

[((c _{p} c _{q}), c), ((c _{p}^{'} c _{q}^{'}), c^{'})] = ((0 0), Ω ((c _{p} c _{q}), (c _{p}^{'} c _{q}^{'})))

which is identical to the Lie bracket for $h_{3}$ of equation 13.1. Notice that the Lie bracket structure is determined purely by Ω.

In higher dimensions, coordinate functions $q_{1}, \dots, q_{d}, p_{1}, \dots, p_{d}$ on $M$ provide a basis for the dual space $M$ . Taking as an additional basis element the constant function 1, we have a $2 d + 1$ dimensional space with basis

q_{1}, \dots, q_{d}, p_{1}, \dots, p_{d}, 1

The Poisson bracket relations

{q_{j}, q_{k}} = {p_{j}, p_{k}} = 0, {q_{j}, p_{k}} = δ_{j k}

turn this space into a Lie algebra, isomorphic to the Heisenberg Lie algebra $h_{2 d + 1}$ . On general functions, the Poisson bracket will be given by the obvious generalization of the $d = 1$ case

{f_{1}, f_{2}} = j = 1 \sum d (\frac{\partial f _{1}}{\partial q _{j}} \frac{\partial f _{2}}{\partial p _{j}} - \frac{\partial f _{1}}{\partial p _{j}} \frac{\partial f _{2}}{\partial q _{j}}) (14.2)

Elements of $h_{2 d + 1}$ are functions on $M = R^{2 d}$ of the form

c_{q_{1}} q_{1} + \dots + c_{q_{d}} q_{d} + c_{p_{1}} p_{1} + \dots + c_{p_{d}} p_{d} + c = c_{q} \cdot q + c_{p} \cdot p + c

(using the notation $c_{q} = (c_{q_{1}}, \dots, c_{q_{d}}), c_{p} = (c_{p_{1}}, \dots, c_{p_{d}}))$ . We will often denote these by

((c _{p} c _{q}), c)

This Lie bracket on $h_{2 d + 1}$ is given by

[((c _{p} c _{q}), c), ((c _{p}^{'} c _{q}^{'}), c^{'})] = ((0 0), Ω ((c _{p} c _{q}), (c _{p}^{'} c _{q}^{'}))) (14.3)

which depends just on the antisymmetric bilinear form

Ω ((c _{p} c _{q}), (c _{p}^{'} c _{q}^{'})) = c_{q} \cdot c_{p}^{'} - c_{p} \cdot c_{q}^{'} (14.4)

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正文：英文 · OCR 机器稿 · 待校对

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原书 PDF · 印刷页 163、164、165、166、167、168、169、170

来源版本：2025-10-20

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14.2 The Poisson bracket and the Heisenberg Lie algebra

14.2 The Poisson bracket and the Heisenberg Lie algebra

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