Response to Hiley:
Seeing into the real nature of a particle
[Peter Mutnick, prev.]
In the position representation, Bohm showed that velocity and acceleration are determined, in principle, by the guidance conditions. If we try to measure the velocity, however, we will be up against the uncertainty principle. Question: how exactly did Bohm explain that?
See our discussion in "The Undivided Universe" p. 114-6
Secondly, what is the role of acceleration and position in the momentum representation? In other words, what are the deterministic Bohm-like "guidance" laws (if any) pertaining to the momentum representation?
You will find a detailed answer in quant-ph/0005026
Thirdly, what is therefore the over-all role of acceleration in the Bohm interpretation of quantum theory?
Isn't it clear from point 3, p. 29 of UU?
Can it be regarded as an observable operator? If so, what is the precise form of the operator?
Why do we need it to be an operator in the causal interpretation described in section 3.1 of UU?
If not, how is it then to be regarded?
Simply as the rate of change of the momentum along the trajectory.
Of course, the conventional answer for getting such operators is arrived at by taking the commutator of x any number of times with the Hamiltonian, but this is not a fundamental answer.
It is the fundamental answer in the conventional theory.
The dV/dx form of the acceleration operator is only part of the real answer. It represents the guidance condition exercised by the classical potential over the acceleration part of the particle in the sense of an indeterminate quantum particle, but it does not describe the full guidance condition over the determinate quantum particle, if such a thing exists when it comes to the acceleration in the general representation-free sense.
I am sorry but I cannot make any sense of this comment.
I will clarify it below.
Note: I suspect that these are the deepest of non-trivial questions pertaining to quantum ontology. Even as classical physics stopped at the second derivative (acceleration), ignoring except for engineering applications the "jerk", so I suspect that quantum physics superficially stops at the first derivative (ignoring even acceleration), but in a deep and ontological way allows us to go further and include even the "jerk"). This has never, to my knowledge, been done however. I believe it is quintessential work that remains to be done. Any comments or ideas?
Again I do not know what is troubling you. Sorry.
What is troubling me is that I see very clearly that the classical potential does not act on the "particle" (or even on the same "part" of the "particle") in the same way as does the quantum potential.
So, when you treat Q and V on the same footing, that does not seem correct or accurate to me. V is a different kind of beast from Q. The fact that it is easily transformable into the momentum representation in the conventional way, while Q is not (at least according to Bohm), should be a clue. I intuit that V should be regarded as acting in the conventional way, within the context of the Hilbert space, with its dual representations, with rather diverse meanings, while Q must be regarded as acting on a noumenal sub-Hilbert space.
Moreover, if we just look at the forms, we see something interesting:
In the position representation, the action of the classical potential in the conventional theory is just grad V(x), while the action of the quantum potential in Bohm theory is also just grad Q(x). However, we can regard the former as an operator in the Hilbert space, while we cannot regard the latter as such. This speaks to me. Moreover, we can look at the action of the classical potential in the momentum representation, where the acceleration operator is something like pV(ihbar d/dp). Again, the quantum potential has no corresponding representation.
However, and here is the wild thing, I believe that the p in pV(ihbar d/dp) is more or less the same momentum as the one that is determined by the other guidance condition, grad S = p, in Bohm theory. Of course, one is an operator and the other is a c-number, but I believe they are ontologically and in reality one and the same, whereas for instance the part of the particle that undergoes acceleration due to the quantum potential is NOT the same as the part of the particle that undergoes acceleration due to the classical potential in the position or momentum representations.
My vision tells me that the "particle", like ourselves, is not a metaphysically simple entity - it exists on many levels of reality. To me, this is a central message of quantum theory - there is a physical noumenal level that is quantal in nature *and* a meta-physical classical level wherein dwells the actual observer, *and* quite a bit in between. These are the hypostases of von Neumann: I (the actually observed system), II (the measuring instrument), and III (the actual observer).
Specifically, the noumenal part of the particle at the base of the physical world is affected by the quantum potential, while it is a different part of the acceleration mode of the particle that is affected by the classical potential in the position representation in the second or emotional world. It is yet a different part of the acceleration mode of the particle that is affected by the classical potential in the momentum representation in the sixth or causal world.
Moreover, the action of the second guidance condition, grad S = p, on the momentum mode of the particle is on the *same* level as the action of the classical potential in the momentum representation on that particular acceleration mode of the particle. This seems to suggest a close identity between the p in pV(ihbar d/dp) and the p in p = grad S, despite the fact that the former is an operator in the Hilbert space and the latter is a c-number. What are NOT the *same*, however, are the acceleration modes of the particle affected by the quantum potential and the classical potential in the position and momentum representations, respectively.
Moreover, one may view all these elements of reality in a nomological rather than an ontological system of worlds (system of organization). Then one sees very clearly the levels upon which the acceleration and velocity modes of the particle are determined by the guidance conditions, leaving open a final actualization of the position mode of the particle due to observation. Thus, according to Bell, all genuine measurements are position measurements. The essential idea of Bohm, from this perspective, is that there are hidden modes of the particle, especially of the position mode of the particle. The position mode is physical, but the hidden position mode is between the etheric acceleration mode and the mental momentum mode, synthesizing them into the notion a real but occult particle, unified in its acceleration, momentum, and position modes. Configuration Space, BTW, in this nomological system of worlds, is in the astral world, while Motion and Momentum are in the mental world, and configuration Phase Space would be in the intermediate causal world.
The secret place of the hidden particle, in the context of the ontological system of worlds, is the oroboric space between the quantum physical world and the meta-physical classical world (worlds 1 and 7, respectively). The State Vector is in the mental world and its representations are in the physical world, and we have stated that the noumenal mode of the particle is an acceleration mode, so the real metaphysical character of the momentum and hidden position modes have everything to do with defining conjugate sets of basis states, i.e., the uncertainty principle and complementarity.
Moreover, seeing into the real character of the particles may enable us to combine formalisms in a creative way that goes deeper than either formalism by itself. If the real particles exist both in and out of their Hilbert spaces, simultaneously, then our theory should describe that dual existence and not attempt to confine the particles to one or the other. They would not like that. :-)(-:
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Peter Joseph Mutnick 1949 - 2000