Poisson brackets

Poisson brackets are mathematical expressions initially introduced into analytical mechanics. They make it possible to formulate the equations of motion of a mechanical system that are particularly suitable for finding laws of conservation and above all, they are a central part of quantum mechanics with the canonical quantification conditions of Born, Jordan and Dirac.

They are at the origin of the Heisenberg equations and the role of commutators in quantum mechanics.

Finally, Poisson brackets are closely related to group theory and Lie algebras and therefore occur everywhere in theoretical physics and differential geometry.

Siméon Denis Poisson (21 June 1781 - 25 April 1840)

To introduce them, consider any two functions:

$F(q_i,p_i, t)$

$G(q_i,p_i, t)$

conjugate canonical coordinates of a mechanical system in Hamiltonian form. Remind that $q_i$ and $p_i$ can be interpreted as the positions and momentums of N material points.

Poisson then introduced the following construction:

$[F , G]=\sum_{i=1}^{3N}\frac{\partial F}{\partial q_{i}} \frac{\partial G}{\partial p_{i}} - \frac{\partial F}{\partial p_{i}} \frac{\partial G}{\partial q_{i}}$

which is called the Poisson bracket.

Differential equations can then be written in the form:

$\frac{dF}{dt}=[F,H]+\frac{\partial F}{\partial t}$

where $H(q_i,p_i)$ is the total energy of a Hamiltonian mechanical system (a time dependence of this Hamiltonian function can also be introduced).

In particular, if $F(q_i,p_i, t)$ is equal to $q_{i}$ or $p_{i}$ , the Hamilton equations :

$\frac{dp_i}{dt}=[p_i,H]=-\frac{\partial H}{\partial q_i}$ $\frac{dq_i}{dt}=[q_i,H]=\frac{\partial H}{\partial p_i}$

$\dot{p_{i}}=-\frac{\partial H}{\partial q_{i}}$ $\dot{q_{i}}=\frac{\partial H}{\partial p_{i}}$

are found. We shall quickly see that these equations indeed coincide with the equations of motion of a particle in a static scalar potential V.

Let H be the Hamiltonian function of this particle

$H= \frac{p_x^2}{2m} +\frac{p_y^2}{2m} +\frac{p_z^2}{2m}+V(x,y,z)$

then by substitution in the above equations we obtain:

$\frac{dp_x}{dt}=[p_x,H]=-\frac{\partial H}{\partial x}$

$\frac{dp_y}{dt}=[p_y,H]=-\frac{\partial H}{\partial y}$

$\frac{dp_z}{dt}=[p_z,H]=-\frac{\partial H}{\partial z}$

If $F(q_i,p_i)$ therefore no longer depends implicitly on time and that

$[F, H]=0$

then, according to the Poisson equations, we have

$\frac{dF}{dt}=0$

The quantity $F(q_i,p_i)$ is therefore constant over time and this is therefore how conservation laws in the Hamiltonian formalism are contained.

In particular, the conservation of energy by setting $F(q_i,p_i)$ equal to $H(q_i,p_i)$ is obtained.

The equations and Poisson brackets are the basis of the fundamental equations of matrix mechanics, the Heisenberg equations.

In this respect, they play an equally important role in quantum field theory.

Poisson brackets are canonical invariants because they do not change form during a canonical transformation. They have the following remarkable property

$[q_{i},p_{j}]=\delta_{ij}$ ( $\delta_{ij}=1$ if $i=j$ and 0 otherwise)

Transposed into matrices, or operators, and using Planck's constant, this relationship is the key to the quantification of mechanical systems as shown by Born and Dirac.

And in the form of operators, this relation will become

$[\hat x_j, \hat p_k]=i\hbar\delta_{jk}$ $=\hat x_j \hat p_k- \hat p_k \hat x_j$