Thus Feynman required extra energy to appear from somewhere. A more detailed discussion of this feature appears in Feynman and Hibbs [ 35 ]. The Bohm approach indicates that some ‘extra’ energy appears in the form of the quantum potential energy at the expense of the kinetic energy. Could it be that the source of the energy is the same?
see Feynman Paths and Weak Values (Robert Flack and Basil J. Hiley)
we see that the quantum potential is playing a similar role as the mass/energy fluctuation in Feynman’s approach. In fact, de Broglie’s original suggestion was that the quantum potential could be associated with a change of the rest mass [36]
Notice that the quantum potential appears essentially as a derivative of the osmotic velocity,
which in turn is obtained from the imaginary part of S′(x, x′). Any fluctuating term added to the real part of Se(x, x′) should also be added to the imaginary part. This would also introduce some change in the energy relation shown in Equation (20). This interplay between the real components of the complex Se(x, x′) is clearly presented as an average over fluctuations arising from some background.
Here we can recall Bohr insisting that quantum phenomena must include a description of the whole experimental arrangement. More details will be found in Smolin [37] and in Hiley [38].
de Broglie, General Covariance and a Geometric Background to Quantum Mechanics
Therefore, writing the rest energy in the form of Equation (13) not only covers the Feynman case but is more general, arising whenever the mass-flow involves acceleration from whatever cause. We therefore
have the emergence of the extra energy term ¯h2Q, which DeWitt [5] showed must be added to the classical Hamiltonian to obtain a general covariant quantum Hamiltonian. Hence, the quantum potential energy is a necessary part of the covariant formalism and not ad hoc as originally claimed by Heisenberg [14].
.............
In this paper, we have seen how the quantum potential arises naturally from an algebraic approach which replaces the wave function with an element of the left ideal from the algebra itself. This emphasizes the role of the individual dynamical process rather than having to rely only on the statistical features of an ensemble of individual processes...
This confirms the view developed in this paper, namely that the dynamics of the
individual quantum process is contained in the non-commutative algebra itself. Hilbert
space is not necessary.
...................
No explanation of the mass fluctuation was given. What was actually missing was quantum potential energy which naturally cancels out the divergence without the need for an unexplained mass fluctuation. It is interesting to note that de Broglie [61] claimed that the quantum potential could be regarded as a change in rest mass.
The role of geometric and dynamical phases in the Dirac–Bohm picture
"it blows up but was Richard Feynman worried? Not at all! What he said is you know, "how do i?, where is this what about how can I get rid of this energy?" and then he said "Let me change the rest mass of the particle to the rest mass one plus delta where delta is a small chain for a short time say epsilon that's my epsilon." So the major thing is why and he doesn't tell us - he just says "do it" and nobody seems to say why, but we now have a story about mass re-normalization blah blah blah
But hold it - this is a squared term and what we find look at the work of Maurice and myself, what we find is that the quantum potential never appears until you go to order h squared. So what he's missing here is the quantum potential and if you put the quantum potential in you get the right result and now the interesting thing is de Broglie, remember chris was saying, it's the de broglie-Bohm theory that that he was talking about"
Basil J. Hiley - How does the Classical World Emerge from the Implicate Order?
It is not the momentum of a single ‘particle’ passing the
point Q, but the mean momentum flow at the point in question.
The question remains as to the nature of the underlying reality. Is it
particulate in nature or is it a more subtle notion of a quantised process
involving a novel organisation of energy and momentum? The Bohm ap-
proach was taken as support for a particle-like picture, even though the
appearance of the quantum potential suggested that there was an element
of non-locality present, an element of the wholeness Bohr talked about.
The identification of the canonical relation PB = ∇S, an unjustified
assumption in Bohm’s original paper [23], was always a worrying feature of
the approach. Now we see that it has its origins in the averaging over a
deeper fundamental non-commutative stochastic process, being related to
the infinitesimal transition amplitude shown in equation (13).....
However the spray of possible momenta
emanating from a region cannot be completely random since, as Feynman has shown, the transition amplitudes satisfy the Schr¨odinger equation under
certain assumptions.
Dirac, Bohm and the Algebraic Approach
If they do not commute, the eigenvalues cannot be simultaneously speci-
fied. However in the Heisenberg algebra these operators depend on time and,
in the non-relativistic theory, all operators that occur at different times com-
mute2. Thus if we write them in a time order, then a meaning can be given
to the expectation value.
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