5.11  Exponential Identities

The composition of Lie transforms can be written as products of exponentials of Lie derivative operators. In general, Lie derivative operators do not commute. If A and B are non-commuting operators, then the exponents do not combine in the usual way:

So it will be helpful to recall some results about exponentials of non-commuting operators.

We introduce the commutator

The commutator is bilinear and satisfies the Jacobi identity

which is true for all A, B, and C.

We introduce a notation A for the commutator with respect to the operator A:

In terms of the Jacobi identity is

An important identity is

We can check this term by term.

We see that

using e-C eC = I, the identity operator. Using the same trick, we find

More generally, if f can be represented as a power series then

For instance, applying this to the exponential function yields

Using equation (5.467), we can rewrite this as

Exercise 5.30.  Commutators of Lie derivatives

a.  Let W and W' be two phase-space state functions. Use the Poisson-bracket Jacobi identity (3.92) to show

b.  Consider the phase-space state functions that give the components of the angular momentum in terms of rectangular canonical coordinates


c.  Relate the Jacobi identity for operators to the Poisson-bracket Jacobi identity.

Exercise 5.31.  Baker-Campbell-Hausdorff
Derive the rule for combining exponentials of non-commuting operators: