The choice of a definite S-function, which fixes the velocity field, determines the initial momenta but leaves the initial positions free. Choose a probability distribution for these initial positions. This is less general than classical statistical mechanics which starts with a probability distribution f(q,p) on phase space which corresponds to a set of S-functions. Our more specialized procedure for a single S-function leads to a class of classical phase space distributions of the functional form
f(q,p,t) = g(q,t)Dirac delta function of p - gradient of the S-function. (i.e., p. 49)This corresponds to a classically pure state in which there is a unique momentum at each space point. The really interesting idea here is that this carries over to the quantum realm which talks of pure versus mixed states. Of particular importance in quantum measurement and quantum gravity theories is how pure states convert to mixed states. Mixed states have several S-functions with different gradient fields.
Liouville’s theorem is that for a cloud of representative points in phase space whose boundary is defined by the individual phase space orbits, the volume and the total number of ensemble elements in the volume are constants. Hence df/dt = 0 or f is a constant along a phase space path. This should be compared with the case of the space-projected density P’ for which dP’/dt =/ 0 ... P’ is not conserved along a mean spacetime orbit. p.50
Although the pure subensembles are not individually conserved by the Liouville equation (p.52) as they are in quantum mechanics, and the disitnction ... between pure and mixed states is not [classically] canonically invariant (p.52), nevertheless, from a formal point of view the single-valued classical pure case provides a key to understanding the connection with quantum mechanics (p.52).The above remarks are for a general distribution which is decomposed into a classical superposition of pure distributions. In contrast, if we have a single pure distribution to start with, then, the Liouville equation does preserve it. There is no linear superposition principle at the local classical level. Superposition is highly nonlinear there. Quantum nonlocality is necessary to enforce linear quantum superposition.
.. the existence of a delta distribution has nothing at all to do with the issue of whether the particle actually has a well-defined position and momentum at each instant - this is always so. It merely tells us that our knowledge of the system is precise and remains so. p.53
There is no diffraction or spreading of the classical wave packet. (p.53)One can make a Schrodinger type equation in classical statistical mechanics but it has an additional nonlinear term of the form (hbar^2/2m) Laplacian of mode psi divided by mod psi all multiplied by psi. (i.e., eq 2.6.7, p.56) This suggests that gravitationally induced nonlinearities create the classical limit as conjectured by Penrose and others.
There is a wave-particle duality in classical mechanics, but the wave is purely passive ... it is not a causal agent of change and it does not contribute to the physical state of a system, which is completely defined by its position and momentum at each instant. p.57