The Larson Research Center

The importance of groups in mathematical physics is hard to overstate. A group is defined as a set of elements with a binary operation that conforms to the laws of a group, which include certain algebraic properties and the notion of an identity element that relates to the reciprocal elements of the group in a specified way.

Here at the LRC, we have discovered that the reciprocal numbers of our new system of mathematics are group representations. These consist of two operational interpretations (OI) of reciprocal numbers: The first OI number is the difference relation of the reciprocal number that is isomorphic to the integers, a|b, and the second is the quotient relation that is isomorphic to the non-zero rational numbers, a/b.

We’ve seen that the first generates a number system, with 0 as the identity element, and the second generates a different number system, with 1 as the identity element. The former is a representation of a group under addition, while the latter is a representation of a group under multiplication.  We have seen how the SUDRs and TUDRs are combined as elements of the additive group, forming SUDR|TUDR (S|T) combos, and how these S|T units combine as elements of the multiplicative group, forming S|T triplets, which have the properties of spin, charge, and mass, observed in the standard model of particle physics.

Of course, the particles of the standard model are based on complex numbers and rotation in the complex plane, in what is known as gauge theory, while the S|T units are based on the reciprocal numbers that are generated by the “direction” reversals of the RST. Obviously, the straightforward concepts of the RST stand in stark contrast to the esoteric concepts of gauge theory, but the two approaches have to be related somehow, if both are valid to some extent.

The connection is enlightening.  While the LRC investigations of the RST have revealed that the multidimensional magnitudes of motion shown in the chart of motions are fundamental modes of motion that don’t include rotational motion, it is also clear that rotational motion has enabled LST physicists to progress without recognizing the principles inherent in the chart of motion; that is, they have been able to construct their theories on one-dimensional magnitudes, by resorting to the two-dimensional rotation of the complex plane.

In the past, we’ve discussed how this works from a physical perspective, since frequency, 1/dt, is actually a velocity, ds/dt, considered as a rotation. However, recognizing that the unification of binary sums and products, via the OI reciprocal number representations of the two, reciprocal, groups, under addition and multiplication, dramatically shows how this is possible from a mathematical perspective.  

Recall that the identity element of the additive group is 0, resulting in an infinite expansive system,

-n, …, -1, 0, 1, …, n

while the identity element of the multiplicative group is 1, resulting in the reciprocal, infinite, contractive system,

-0. …, 1, …, 0 

where the order of a given finite subgroup depends on the reciprocal number n|n = 1|1 = 0, or n/n = 1/1 = 1. In the additive group, then,

-1 + 1 = 0, and 1 -1 = 0,

while in the multiplicative group,

.5 x 2 = 1 and 2 x .5 = 1,

when n = 2. Thus, in the binary operations of the two groups of order 2, the sum is equal to the product; that is,

2+2 = 4, and

2x2 = 4 

The question is, why is this significant? And the answer is that it reveals the mathematical connection between rotation and vibration, or between the standard model’s gauge theory, and the LRC’s S|T theory.  To understand this it is only necessary to recognize that the reciprocity of the two representations of OI reciprocal numbers, which makes them dual to one another, is inherent in the rotations of the complex plane as well.

This is seen, in the first instance, by rotating from zero to π in one of two “directions,” positive, or negative:

-π, 0, π

or rotating from π/2 to 2π:

π/2, π, 2π

What the complex number did was give the LST physicists the ability to define an infinite expanse of numbers in the first case, and an infinite contraction of numbers in the second case. On this basis, they can construct representations of the same two groups that we have constructed using OI reciprocal numbers. In the first case,

-π + π = 0, and π - π = 0. and in the second case,

.5π x 2 = π, and 2π x .5 = π 

Clearly, then, this implies that the ad hoc invention of complex numbers is no longer needed.  We should be able to do quantum physics without complex numbers and the formalism that is based on them.  Time will tell, but certainly these insights are encouraging indications.

Addendum:

To be more explicit, the above analysis for the multiplicative group of π rotations requires that the identity element be 2π/2π = 1/1 = 1. In other words, when we use the reciprocal number identity element, n/n = 1/1 =1, n is the order of the finite subgroup (the number of elements in the subgroup, not counting the identity element). In this case, we are choosing n = 2. The elements of this subgroup are therefore

π/2π, 2π/2π, 2π/π = .5, 1, 2,

which, as shown previously, is taken to be

-0, 1, 0, 

under the quotient OI. In gauge theory, this would be equivalent to a phase shift, where the identity element is “unit” phase, and the inverse elements are 180 degrees out of phase in opposite “directions.”