The Acceleration Operator and the Classical Potential
In conventional quantum theory, one gets the acceleration operator by taking the commutator of x (as a position operator) with the Hamiltonian and then the resulting p (or mv) with the Hamiltonian again. In the position representation, one gets something like dV(x)/dx, and in the momentum representation, one gets something like pV(ihbar d/dp). In the momentum representation, all the differential action is in the potential term, rather than the kinetic term.
Now we compare this dual action of the classical potential to the action of the quantum potential. The latter cannot seemingly be expressed in the momentum representation. Unlike the classical potential, it cannot be regarded as an operator in the Hilbert space. And yet it acts on the acceleration part of the particle and has the same form as the acceleration operator for the classical potential, namely dQ(x)/dx.
So, I guess I would like to suggest that one should treat the classical potential as an operator on the Hilbert space and only the quantum potential as an operator on the sub-Hilbert space, on the noumenal part of the particle, which is an acceleration part of the particle.
The idea seems to be that there are three acceleration parts of the particle:
1. the noumenal part, which is physical and is acted upon or guided in a determinate way by the quantum potential in the Bohm decomposition of the representation of the Hilbert space,
2. the part that is acted upon or guided in an indeterminate way by the classical potential via the acceleration operator in the position representation of the Hilbert space - this exists in the second or emotional world,
3. the part that has to do with the acceleration operator and the classical potential in the momentum representation - this acceleration exists in the seventh or meta-physical world, and the classical potential, as the form of the acceleration operator, exists in the sixth or causal world.
The momentum part of the particle in the Bohm picture exists in the seventh or meta-physical world (it actually functions as part of the acceleration operator), so we have the rather interesting result that acceleration caused by the classical potential in the momentum representation seems to be contiguous with momentum caused by the phase guidance condition in the Bohm picture.
This, I believe, is a non-trivial idea that is the real line of thinking undertaken by Arthur Young. For contiguous with this momentum and this acceleration is a jerk of an unknown source. It just so happens that this is the great unknown that, when known, will answer Henry Stapp's question about the cause of choices. But the first step is to understand how and why the conventional theory of the classical potential interpenetrates with the Bohm description of a noumenal reality to give us a complete picture of what is happening to a particle that is both in and out of its Hilbert space.