Part 2 of commentary on Hestenes vote.
With this much understood, we can now look at Hestenes' introduction of de Broglie's equations, and try to clarify some of the dimensional/conceptual confusion that the RST approach brings to light.
When we consider Planck's constant, we see that it has the dimensions of action, t^2/s, which, when multiplied by frequency, nu, with dimensions 1/t, yields the dimensions of energy, t/s.
As Hestenes introduces the fundamental equations in his essay, he annotates them with remarks, indicating the familiar interpretations, such as "Energy is frequency" (Planck's equation, E = hv), "Mass is energy" (Einstein's equation, E = mc^2), and then the less familiar, "Mass is frequency" (de Broglie's equation, ωB = mec^2/hbar).
However, in the first and last equation, unlike in the second, the dimensions are not the dimensions of energy, mass and velocity alone, where velocity merely seems to act as the mysterious conversion constant between mass and energy, but they also include the dimensions of Planck's constant, h, and since the dimensions of h are the dimensions of action, which are the dimensions of area times mass per unit of time, or (s^2 * t^3/s^3)/t = t^2/s, which really are the dimensions of inertia, t^3/s, times frequency, 1/t, we can rewrite the de Broglie frequency equation as the energy equation:
E = Iω^2,
where 'I' is inertia, and ω^2 is the 1/t term in Planck's constant h, times nu, times 2π.
Nevertheless, as noted by Larson, frequency, as cycles per second, 1/t, is really a velocity, s/t, in space and time terms, where the "direction" of the motion reverses every 180 degrees, and as such the term 1/t is actually an auxiliary device, used as a mathematical expedient to express the units of periodicity in rotational motion.
When this fact is recognized, and the proper space/time dimensions of velocity are substituted for the dimensions of frequency, in the radiation equation of energy, the dimensions of Planck's constant become the dimensions of momentum, not the dimensions of action: Thus, the energy equation of radiation becomes t^2/s^2 * "s/t" = t/s, in space/time terms, where "s/t," the velocity term, is the "velocity" of oscillation, in which the "direction" of the motion is periodically changing; It is not the uniform translational velocity, as indicated by the quotation marks.
In the same way, the energy of the de Broglie equation can be viewed as t^3/s^3 * "s^2/t^2" = t/s, conforming to the dimensions of the Einstein equation, when the periodic "velocity" term is substituted for the frequency term.
Consequently, we see that the use of the mathematics of frequency hides the fact that these three equations are not all that different. The radiation equation, E = hv, can be rewritten as E = Iω^2, which in space/time terms is equivalent to E = mc^2, when the "velocity," ω, has the value c (in units of 2π). Likewise, the de Broglie equation, ωB = mec^2/hbar, can be rewritten as E = Iω^2, where the same thing holds, when the "velocity," ωB, has the value c; That is to say, there appears to be only one, underlying, energy equation, not three.
Ironically enough, however, equating translational c-speed, with oscillating c-speed, is mathematically problematic, when "direction" reversals of c-speed are used to define the quanta of speed-displacement, as we do in our RST-based theory. This is due to the fact that, upon reflection, we can see that this "folding" of c-speed through oscillation means that ω, the frequency of the oscillation, in terms of units of 2π, can never really equal c-speed, since by definition, oscillation can only be achieved in one or two ways: Either the space aspect of the motion has to oscillate (the oscillating spatial pseudoscalar), or the time aspect of the motion has to oscillate (the oscillating temporal pseudoscalar), while the reciprocal aspect (the temporal or spatial scalar) continues to increase uniformly.
In the former case, the frequency of the oscillation is 1/2 c-speed, and, in the latter case, the frequency is 2/1 c-speed. Hence, if we double 1/2, we get 2 * 1/2 = 2/2 = 1, and, if we divide 2/1 in half, we get 1/2 * 2/1 = 2/2 = 1, but, in terms of cycles, this makes no sense, because, under this familiar mathematical operation, the number of cycles is doubled, while the number of time (space) units required to complete these two cycles is not. The number of reciprocal units, time or space, in each case does not change, under the doubling (halving) operation. The same number of scalar units is required to complete two cycles as is required to complete one cycle.
Clearly, something is wrong with this. If the number of cycles is doubled, then the number of time (space) units to complete those cycles must be doubled as well, when we are adding these ratios together as units of oscillation. So, the only thing we can do, to be mathematically consistent, in doubling (halving) the 1/2 and 2/1 space/time ratios as units, is to multiply both the numerator and the denominator by 2; that is, in both cases, the operator must be 2/2, not 2/1 and 1/2: In this way, we get 2/2 * 1/2 = 2/4, and 2/2 * 2/1 = 4/2, which, in effect, conserves the frequency, since 2/4 = 1/2, and 4/2 = 2/1.
But, then, if the frequency is conserved, what changes, when the number of these pseudoscalar oscillations is doubled (halved)? Well, obviously, given the energy equations just discussed, the only thing that can change is the energy/mass term, but in terms of the pseudoscalar oscillation's geometric properties, the SIZE of the pseudoscalar changes.
Under the doubling operation, the radius of the sphere, isomorphic to the scalar (either 0D time or 0D space) is doubled and, consequently, so is the diameter of the sphere, which is always twice the size of the radius and generates the 1D, 2D and 3D pseudoscalars, increasing them exponentially, according to their dimensions.
In Hestenes' model, when the energy/mass of a particle changes through interaction, the radius and zitter frequency vary, in order to maintain the velocity of light. Now we can see that, in the RST model, it is the 1:2 ratio of the pseudoscalar's radius and diameter that is maintained, as the properties of the system change, which is tantamount to maintaining the speed of light, only in this case, it's the "velocity" of light in the periodic variation that is maintained.
In a former paper, Hestenes writes:
'The zitterbewegung, if it turns out to be physically real, is belated confirmation of de Broglie's original hypothesis that the electron has an internal clock with period precisely equal to twice the zitter period, precisely the relation between the period of a rotor and that of a vector it rotates. As we have seen, the physical signature of zitter is a rotating electric dipole with ultra high frequency. If this exists, its implications for quantum mechanics will be far-reaching."
Indeed.
