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
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:
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,