The Larson Research Center

It took me several months to complete the FQXI essay, but most of the time was spent trying to compress the main idea into something that would fit into the contest constraints. Now I realize that I should have spent more time on thinking about the idea itself and how it applies to the topic.

If I had done that, I would have no doubt realized the main point of the RST, that space and time are quantized, could have been brought to the forefront in a much more dramatic and convincing manner than the wishy-washy way I ended up presenting it. I posted a comment that hints of something to that effect on my FQXI discussion thread here.

In that post, I wrote:

The point is, that there is no use trying to define a point in space that has any extent, or an instant of time that has some duration. This contradiction at the foundation of our science and mathematics cannot help manifest itself in terrible ways later on. Our concept of the electron is the best example, but there are many others. 

A really advanced alien society would no doubt laugh at our pathetic theories that we take so seriously that we build silly machines like the LHC, going to astronomical expense to look for figments of our imaginations.

Why look for the Higgs, when we can’t even understand the electron? If there is a discrete unit of space, then, by definition, it means that it cannot be subdivided. Yet, we can represent any magnitude with figure 1 of my essay. ANY geometric length magnitude whatsoever, including the so-called Planck length, can be represented by the radius of the unit circle. This means that the radius of the square root of 2 circle can be represented as well. With these two radii and the eight cubes between them, we have both digital and analog 1D, 2D and 3D geometric quantities such as circumference, area and volume, represented. So, how can we say space and time are doomed at some length, as today’s leading theoreticians contend?

I went on to try to explain that, choosing a unit, which we must do, subordinates any subdivisions of that unit to the unit ratio. Since the unit space/time ratio is the unit speed, then the speed (time) of any subdivision thereof is necessarily greater (less). For example, if we use Larson’s space and time unit to form the unit ratio (light speed, c, based on the Rydberg frequency of hydrogen), then there would be about 2.8205 x 1031 subdivisions of the Planck length within the space unit of that unit speed. Hence, the time it takes the last of those Planck lengths to collapse, or the first to expand, in the course of the unit oscillation, is miniscule indeed.

But the point is that even if we took the Planck length as the unit length, along with the corresponding time unit to give us the unit ratio of speed c, the construction of figure 1 of my essay is still valid, and it could be subdivided into even smaller units. Then the ratio of those units could be calculated and taken as standard, and the process repeated ad infinitum.

This is the crux of the problem, when subdividing the continuum: There is no conceivable mathematical limit to the size of divisions. The question is, though, is there a physical limit? Larson’s RST, our new system of physical theory, assumes that there is, and Larson calculated it based on the Rydberg frequency of hydrogen and the speed of light.

The key consequence of this assumption, which Larson described, is that space/time motion limits the distance between two entities to the unit of space, after which time/space motion reduces the distance (to zero if need be).

Now, the question is, how can we express this inverse relation mathematically? The problem we run into is that s/t = 1/1 is equal to t/s = 1/1, mathematically. With Larson’s “direction” reversals, this unit ratio gives us s/t = 1/2 and t/s = 1/2 (assuming space reversals in the former and time reversals in the latter).

In our development at the LRC, we try to treat these two values as negative and positive units of motion, by taking the arithmetic ratio (difference between denominator and numerator) as the operational differential. But how do we do that with figure 1 of the essay, since inverting the radius and diameter is impossible (we can’t have a radius of 2 and a diameter of 1). Therefore, what we have done up until now is switch the labels. We say that space (diameter) is now time (radius) and time (radius) is now space, but what justification do we have to do this? How does space in the material sector become time in the cosmic sector and vice-versa?

The problem is the 1/1 ratio. If we take the ratio of the advanced function of the expansion/contraction (e/c), which is the square root of 2r2, (the outer circle radius of figure 1) and the retarded function of the e/c, which is the inverse of the advanced function (the radius of a third circle, smaller than the inner circle of figure 1 that is not shown), we get two positive units in the two “directions” of the numbers. The number 2 for the greater than unit numbers and the number 1/2 for the less than unit numbers . To see this for yourself, just take the inverse of the square root of two over the square root of 2, and then the inverse of this, the square root of 2 over its inverse. You get the following number line:

…, 2.5, 2.0, 1.5, 1.0, 0.5 | 2, 4, 6, 8, 10, …

This number line has advantages. First of all, it preserves the greater than, less than, relation between the inverses, inherent in the geometric ratio (numerator/denominator quotient relation), while at the same time it incorporates the unit symmetry of the arithmetic ratio (numerator/denominator difference relation).

Second, the numbers are both functions of the unit radius (the inner circle of figure 1): The square root of 2 is the unit advanced function, we might say, while its inverse is the unit retarded function. But why call them advanced and retarded? In reality, mathematically, one is the inverse of the other, just as space is the inverse of time. True, these numbers are not the elements of a group, but since their respective units are inverses, we need only to invert their multiples to form the group under division, given the quotient interpretation:

…, 1/5(1/2), 1/4(1/2), 1/3(1/2), 1/2(1/2), 1/1(1/2) | 1/1(2/1), 2/1(2/1), 3/1(2/1), 4/1(2/1), 5/1(2/1), …

The beauty of this unorthodox group is that the identity element itself is a product of the two units, not an explicit element of the group. It’s a product of the two inverse units; that is, 1/2 / 2/1 = 1/4 and 2/1 / 1/2 = 4/1, but the product of these two, 1/4 *4/1 = 4/4 = 1/1, acts as the identity element of the group. Moreover, taking the difference interpretation of the numbers, they form a group under addition, the integer group, where the identity element is 0.

Now, a very interesting observation is that, with the difference interpretation, we can build the standard model of particle physics, as shown here, and, with the quotient interpretation, we can build the periodic table of elements.

Larson’s mathematical pattern for the periodic table is 4n2, to which he gives a physical meaning in order to obtain the four periods, 22 = 4; 42 = 16; 62 = 36; 82 = 64 of the table. These are the double periods of the Wheel of Motion, as opposed to the 2n2 half-periods of quantum mechanics, obtained using the four quantum numbers, n, l, m and s. 

The trouble is, in each case, the periodic nature of the table is built on the variability of constituent particles of atomic structure (QM), or on the variability of constituent motions of the atom (RSt). Both work out fairly well in accounting for the order of the elements, but neither theory conforms to the new empirical data shown by Le Cornec, which uses a ratio of square roots of atomic ionization potentials to order the elements by their energy levels. Le Cornec shows that in the QM theory of the atom, the “s” and “p” energy level groups are reversed, while in the development of his RSt, Larson altogether abandoned the effort to explain the spectroscopic data of the elements, using scalar rotations.

With the new numbers, however, the quantitative relations of the periods fall out from the physical concept of the 3D oscillation. The sequence of 2, 4, 6, 8 of the new number series under the quotient interpretation, shown above, contains 4, 16, 36 and 64 positive “slots” for the respective, positive, inverses. Since these results come from the advanced/retarded functions of the 3D oscillation, the relation of square roots and their inverses, as described above, it’s easy to conclude that a connection with Le Cornec’s work is plausible.

At least that’s what I’m hoping, but this unorthodox view of a mathematical group may end up derailing the whole thing. I guess we’ll see if it’s valid or not, in the end.