Poisson Brackets

For functions that are defined on the phase space we can define the following operation. Let F = F(q, p, t) and G = G(q, p, t). Then a Poisson bracket of these two functions is defined by:

$$\left\{ F, G \right\} = \sum_{i=1}^n \left( \frac{\pa... ...al G}{\partial q_i} \frac{\partial F}{\partial p_i} \right)$$ (4.10)

This operation has the following neat properties:

$$\left\{ F, G \right\} - \left\{ G, F \right\}$$ = (4.11)

$$\left\{ F, F \right\}$$ = 0 (4.12)

$$\left\{ F_1 + F_2, G \right\} \left\{ F_1, G \right\} + \left\{ F_2, G \right\}$$ = (4.13)

$$\left\{ F_1 F_2, G \right\} F_1 \left\{ F_2, G \right\} + F_2 \left\{ F_1, G \right\}$$ = (4.14)

$$\left\{ F, q_i \right\} - \frac{\partial F}{\partial p_i}$$ = (4.15)

$$\left\{ F, q_i \right\} \frac{\partial F}{\partial q_i}$$ = (4.16)

$$\left\{ q_i, q_j \right\}$$ = 0 (4.17)

$$\left\{ p_i, p_j \right\}$$ = 0 (4.18)

$$\left\{ q_i, p_j \right\} \delta_{ij}$$ = (4.19)

$$\left\{ F_1, \left\{ F_2, F_3 \right\} \right\} + \left\{ F_2, \left\{ F_3, F_1 \right\} \right\} + \left\{ F_3, \left\{ F_1, F_2 \right\} \right\}$$ 0 = (4.20)

$$\frac{\partial}{\partial t} \left\{ F, G \right\} \left\{ \frac{\partial F}{\partial t}, G \right\} + \left\{ F, \frac{\partial G}{\partial t} \right\}$$ = (4.21)

Poisson brackets can be used to express time derivatives of phase space functions:

$$\frac{\textrm{d} F}{\textrm{d} t} \sum_{i=1}^n\left(\frac{\partial F}{\partial q_i} \frac{\textrm{d... ...p_i} \frac{\textrm{d} p_i}{\textrm{d} t}\right) + \frac{\partial F}{\partial t}$$ =

$$\sum_{i=1}^n\left( \frac{\partial F}{\partial q_i} \frac{\partial... ...al p_i} \frac{\partial H}{\partial q_i} \right) + \frac{\partial F}{\partial t}$$ =

$$\left\{F, H\right\} + \frac{\partial F}{\partial t}$$ = (4.22)

This equation can then be applied to q_i and p_i itself to re-express the Hamilton equations in the following form:

$$\frac{\textrm{d} q_i}{\textrm{d} t} \left\{ q_i, H \right\}$$ = (4.23)

$$\frac{\textrm{d} p_i}{\textrm{d} t} \left\{ p_i, H \right\}$$ = (4.24)

In turn, substituting H in place of F yields:

$$\frac{\textrm{d} H}{\textrm{d} t} = \left\{H, H\right\} + \frac{\partial H}{\partial t} = \frac{\partial H}{\partial t}$$ (4.25)

Expressions such as $\left\{q_i, p_j\right\} = \delta_{ij}$ ought to tug at the heart of everyone acquainted with Quantum Mechanics, where one of the expressions of the Heisenberg Uncertainty Principle is

$$\left[\hat{q}_i, \hat{p}_j\right] = i\hbar\delta_{ij},$$

where

$$\left[\hat{q}_i, \hat{p}_j\right] = \hat{q}_i \hat{p}_j - \hat{p}_j \hat{q}_i$$

is a commutator of operators that represent position and momentum. Similarly time evolution of any Quantum Mechanical operator that does not depend on time explicitly is given by

$$\left[\hat{\Psi}, \hat{H}\right] = i\hbar \frac{\textrm{d} \hat{\Psi}}{\textrm{d} t}$$

This is not entirely an accident. Poisson brackets lead directly to the so called canonical quantization. Canonical quantization is a procedure which converts a classical field theory or a classical mechanical theory into the corresponding Quantum theory. One of its rules is:

$$\left\{\Psi, \Phi\right\} \rightarrow \frac{1}{i\hbar} \left[\hat{\Psi}, \hat{\Phi}\right]$$

But the truth about canonical quantization carried out like that is that it has to be interfered with frequently in order to deliver a meaningful Quantum theory, and the reason for that is that Quantum theories cannot be derived formally from classical theories. The opposite is the case, i.e., Quantum theories are a lot richer than classical theories, and it is the latter that are derivable from the former in thermodynamic limit. But canonical quantization was useful in its day in providing a bridge between XIXth century classical physics and XXth century quantum physics.