The Larson Research Center

In the eighth post copied from the BAUT forum thread (see the previous 10 posts), I briefly explained how the Bott Periodicty theorem is easily seen in the Reciprocal System of Mathematics (RSM), used in the scalar science of the LRC. However, I would like to demonstrate it more completely by focusing on the theorem’s role in current legacy system (LST) research, to contrast the use of the math of the two systems.

We can start with the Wikipedia article on Bott Periodicty, recalling that what the theorem says in short is that there is no new phenomena beyond three dimensions. This implies then that the concept of 10 dimensions, which is the foundation of string theory, has to cope in some manner with the theorem. The Wikipedia article states:

In mathematics, the Bott periodicity theorem is a result from homotopy theory discovered by Raoul Bott during the latter part of the 1950s, which proved to be of foundational significance for much further research, in particular in K-theory of stable complex vector bundles, as well as the stable homotopy groups of spheres. Bott periodicity can be formulated in numerous ways, with the periodicity in question always appearing as a period 2 phenomenon, with respect to dimension, for the theory associated to the unitary group. See for example topological K-theory.

Now, homotopy theory has to do with functions in topology, and K-theory has to do with topology, as a branch of algebraic K-theory, which is used to study topological spaces. I seem to remember reading Michael Atiyah, the originator of the Index theorem, stating that it is used to find the number of solutions to a system of differential equations in a given algebraic space, which can be useful in finding the actual solutions themselves, something like knowing that there are 180 degrees in the angles of a triangle is helpful in finding the value of a given angle that otherwise would be much more difficult to do.

With the index theorem, the shape, or the topology, of the space under study provides the clues needed to get at the solutions to differential equations that the vectorial system of physical theory requires. What Atiyah (and Singer) did, was to use K-theory to demonstrate that the index could be described topologically. This was huge and led to the popular connection between topology and theoretical physics. Indeed, from what I understand, it was the mathematical methods derived from the index theorem that enabled Witten to make his significant contribution to string theory.

Anyway, the Wikipedia article on algebraic topology states:

The goal is to take topological spaces and further categorize or classify them. An older name for the subject was combinatorial topology, implying an emphasis on how a space X was constructed from simpler ones. The basic method now applied in algebraic topology is to investigate spaces via algebraic invariants, by mapping them, for example, to groups which have a great deal of manageable structure in a way that respects the relation of homeomorphism of spaces. This allows one to recast statements about topological spaces into statements about groups, which are often easier to prove.

Of course, it is the study of groups, in group theory, that connects all this to modern physics, especially to string theory. I’ve written several times about the connection of the octonions to modern physics and their connection through the periodicity theorem that John Baez finds so mysterious (Period Eight, or Bott Periodicity, Strikes Again!), but back in May of 2000, Dr. Rick Ye, gave a talk at what is now the KITP in Santa Barbara entitled “The Bott periodicity theorem,” where he explains that there are various ways to state the theorem in terms of the topological language of groups, and later he goes through proofs of these, but, after he explains three versions of this period eight theorem, he talks about why it is important, and gives four reasons:

1. It is the fundamental structure theorem for K-theory, which enables computations to be made in homotopy structure.
2. Homotopy structure of the unitary groups is the cornerstone for everything based on these groups (standard model).
3. It leads very quickly to Thom isomorphism and then to the Atiyah-Singer Index theorem.
4. It’s needed for D-branes!

The relation between the period eight of Bott periodicity, K-theory, mathematical groups, Index theorem, and D-branes in string theory, seems to be simply that it maps the D-branes of different dimension, as seen through the eyes of vectorial physics, using vectorial spaces, and the system of differential equations clothed in the language of the shapes of spaces, or topology.

It’s all very complicated and involved. However, what we have discovered in scalar physics is that vectorial physics appears to be the wrong system for describing the internal degrees of freedom of physical entities, and the vectorial mathematics of differential equations is the wrong language for investigation of these properties, and the vectorial geometry of vectorial motion is the wrong geometry. Thus, the science of using complex concepts of topology, connected to complex concepts of groups, connected to complex systems of differential equations, are simply inappropriate for calculating the properties of these entities. Physicists may be able to get there from here, eventually, or not, but why beat our heads against the wall? It appears that scalar science offers a much simpler route to understanding these properties.

Take Bott periodicty for instance. When viewed from the standpoint of scalar mathematics, this phenomenon becomes trivially simple to comprehend. Recall that the first reciprocal number (RN) is

ds/dt = (1/2 + 1/1 + 2/1) = 4/4 nm,

where nm is natural units of motion. As explained previously, this equation exploits the operational interpretation of numbers of the RSM to express the motion combination of the two unit speed-displacements of the RST, which we have called the SUDR and TUDR, into one composite unit, designated as S|T. The RN is a three-dimensional number that can express the period eight phenomenon in terms of a power expansion, by first showing that the quadratic expansion of the RN is isomorphic to the binomial expansion, given the OI of RSM:

This shows how each dimension, n, can be expressed as a power of the first RN, or 2ⁿ ~ 4ⁿ = RNⁿ. Finally,

shows why the period eight is fundamental to higher dimensional numbers. And we have always thought that the 8 bit byte was a necessary, but rather arbitrary, basis for the binary language of computers!

Anyway, the bottom line here is that because we don’t need the systems of differential equations upon which vectorial physics is based, we obviously don’t need all that complex homotopy stuff either. Evidently, all that we need is some good computer engineers! LOL

I was watching an interview of Bill Gates on cable last night. I wonder how intrigued he would be with this? It really doesn’t matter, but he does have the economic clout to gather some really competent mountain climbers at the foot of this mountain, a mountain that the amatuer Larson discovered, but that the professionals like Lee Smolin should prepare to climb (to use his own metaphor).

Alas, we probably can’t even get Smolin’s attention, let alone Gate’s attention. Pity.