The evolution of the phase space vector that we have been examining via the classical propagator:

might appear to be of only formal significance because we cannot evaluate the action of for a general Liouville operator.

The difficulty in evaluating the action of can be
seen by noting that *iL* is of the form

where

Thus, for any phase space function, *f*(*x*,*p*), it follows that

Therefore, we see that or

The term in parentheses is known as the commutator between and and is denoted

Thus, , so that the two operators do not *commute* with each other.

Unfortunately, this makes classical mechanics difficult. If we could
split the classical propagator according to ,
everything would be simple because we can evaluate the action of the two operators
on the right on a phase space vector . However, because and
do not commute, the classical propagator *cannot* be factorized in this
simple way. This can be easily verified by Taylor expanding both sides:

However,

Fortunately, there is a theorem that we can use to factorize the classical propagator.
This is the *Trotter theorem*, which states that

Although the proof of the theorem is somewhat beyond the scope of these
lectures, it can be found in several places, in particular, see
the appendix to Chapter 1 of *Techniques and Applications of Path Integration*
by L. S. Schulman.

How does the Trotter theorem help us? Consider the approximation to
obtained by choosing *M* large but finite:

which can be rearranged to yield

The first of these expressions looks like approximate propagation of the
system up to a time *t* by *M* applications of the operator in brackets.
The second, therefore, looks like an approximation to the propagator for
a small time interval *t*/*M*. If we interpret this as a single
time step, , then we have a single-time-step approximation
to the propagator of the form

The left side is the exact propagator, , for a time step, . The right side represents an approximate propagator .

There are several things to note about . First, . Thus, is unitary and, therefore, preserves the time-reversibility of the dynamics:

Second, by Taylor expansion, it can be shown that is accurate to order :

where the last line indicates equivalence up to second order in . Thus,

Thu Oct 17 21:36:23 EDT 2002