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## 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:

 (4.10)

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 qi and pi itself to re-express the Hamilton equations in the following form:
 = (4.23) = (4.24)

In turn, substituting H in place of F yields:

 (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 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

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.

Next: Hamilton-Jacobi Equation Up: Kinematics and Dynamics of Previous: Lagrange and Hamilton Equations
Zdzislaw Meglicki
2001-02-26