We shall next prove a number of important properties of the Liouville operator and the classical propagator. First, we show that the Liouville operator is Hermitian:
In order to prove this, we introduce a set of orthonormal functions, 
on the phase space. These satisfy
We shall also assume that 
as 
. That is, the phase
space is bounded.
The matrix elements of L in this basis are:
Thus, the Hermitian condition on L can be expressed as
Interchanging k and l in the matrix element expression gives
Now integrate the last line by parts. When this is done, the boundary term vanishes due to our assumption about the boundedness of phase space, and we obtain
Therefore, L is Hermitian.
Given that L is Hermitian, it follows that the propagator 
is a unitary operator. This means
To show this, note that
Finally, we shall show that unitarity of the propagator implies time reversal symmetry in the equations of motion.
Time reversal symmetry means that if the direction of the flow of time is reversed, the equations of motion
take the same form. Note that under a time reversal transformation, 
and 
.
Thus, while 
, 
. Substituting these into Hamilton's equations gives
so that Hamilton's equations take the same form under a time-reversal transformation.
The implication of time-reversal symmetry is that if the system is propagated forward in time up to a time t and then
the clock is allowed to run backwards for a time -t, the system will evolve according to the same equations of motion
but the direction of the velocities will be reversed, so that the system will simply return to its initial condition.
To see that this is implied by the unitarity of the propagator, note that 
. Now applying U(t)
to 
to yield 
followed by application of U(-t) gives
Thus, the operator U(-t)U(t)=I, which implies that 
, since U(t) is unitary. Since 
is
equivalent to backward propagation in time, unitarity implies reversibility since it is always true that 
for a unitary operator.
Since U(t) is unitary, it is possible to show that its determinant is 1. In order to show this, consider working in a basis in which
U(t) is diagonal with diagonal elements 
. The determinant of U(t) is
Therefore, the determinant of is
Finally, since 
, and the diagonal elements of 
are 
. However,
since , this implies that 
so that 
.
determinant of both sides gives
However, since , it follows that the determinant is 1. But since 
, we see that
the matrix U(t) is just the Jacobian matrix 
, and we already
know that the determinant of this matrix is 1 for Hamiltonian systems. Therefore, unitarity of the propagator is consistent
with Liouville's theorem, which states that the volume in phase space is preserved under Hamilton's equations.
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Mark Tuckerman