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:

This operation has the following neat properties:

= | (4.11) | ||

= | 0 | (4.12) | |

= | (4.13) | ||

= | (4.14) | ||

= | (4.15) | ||

= | (4.16) | ||

= | 0 | (4.17) | |

= | 0 | (4.18) | |

= | (4.19) | ||

0 | = | (4.20) | |

= | (4.21) |

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

= | |||

= | |||

= | (4.22) |

This equation can then be applied to

= | (4.23) | ||

= | (4.24) |

In turn, substituting

(4.25) |

Expressions such as
ought to tug at the heart of everyone acquainted with Quantum
Mechanics, where one of the expressions of the Heisenberg
Uncertainty Principle is

where

is a

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:

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.