The Larson Research Center

In the development of quantum field theory (QFT), one of the most important advances is the development of so-called gauge theories. Gauge theories are described in Wikipedia as “… a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally.” Normally, applying a scale factor to a system applies globally, to every component of that system. Otherwise, if the change of scale applies locally, where it affects only some of a system’s components, a distortion results. For example, a square remains a square when the scale of the 2D spatial coordinate system, in which it is defined, is doubled, or halved, for some reason, but if only the scale of the x axis in the system is affected, the square will be distorted and its symmetry lost.

However, a local change of scale that is a relative change can leave the symmetry unchanged, For instance, when the scale of the reciprocal B and E fields of an electromagnetic wave equation change, the change in one, is compensated by a change in the other. Such a symmetrical change in a system is viewed in physics as an immunity to change, in the words of Joe Rosen; that is, it’s a change that is not really a change, or we can say that it is the possibility of making a change that leaves some aspect of the physical state of a system unchanged, in the words of Alexandre Guay (see here).

According to Guay, in QFT, the application of gauge, or local, symmetry principles, permits certain transformations in the fields that have no empirical consequences, but the trouble is that, while these changes are the foundation of successful theory, there is no way to tell if they have any physical meaning, because they are based on unobservable quantities. In other words, the question, “What is changed under a gauge transformation?”, in these QFT  theories, cannot be definitively answered.

This state of affairs is unacceptable, because it implies that the QFT formalism, used to model the micro-structure of the physical universe, may have superfluous mathematical structure. Guay quotes Ismael and van Fraassen on this who write:

Formalisms with little superfluous structure are nice, of course, because they reflect cleanly the structure of what they represent; they have fewer extra mathematical hooks on which to hang the mental structure that we project onto phenomena.

Fortunately, in constructing an initial “formalism” of our RST-based theory, it appears quite clear that we have no superfluous structure, even though it seems that we too must incorporate a gauge principle in our theory. Whether or not this implies that the qauge theories of QFT have an actual physical meaning, we can’t say, because the motivation is not the same in both cases, but we can say that they most definitely have a physical meaning in our theory.

The motivation for introducing gauge theory into QFT has to do, naturally, with rotation.  Rotation, in LST theory, has to do with complex numbers, where the complex number represents a vector on the unit circle, a radius, if you will.  Hence, rotation in the complex plane consists of an infinite set of radii, which are specifiable in an infinite set of complex numbers, of “size one” (i.e. (a² + b²i)^1/2 = 1) and it is a set of one-dimensional numbers. Therefore, using complex numbers of “size one,” the circumference of the unit circle can be arbitrarily sub-divided, as an infinite continuum of complex numbers between 0 and 1. 

This is important, because the group properties of this set of numbers permits the binary operation of the group to be exploited to form combinations of rotations that are elements of the group, called U(1).  Since the binary operation of this group is multiplication, this means that the product of these complex numbers is actually a one-dimensional equivalent of successive rotations (a sum), in a two-dimensional coordinate system of real numbers, R(2).  In his book, Deep Down Things, Bruce A. Schumm describes the two equivalent concepts as follows:

The designation U(1) may be a bit obscure. For the set R(2) of rotations in two (real) dimensions, we read the “R” as “rotation in real space” and the “2” as “in two dimensions.”  In the case of the complex group U(1), however,  we saw that we admit the possibility of something mathematically equivalent to rotation with just a single complex number - a single complex dimension.”  Furthermore, these “rotations” involve themselves solely with the complex numbers of size one - of unit length. Thus, for U(1), we read the “U” as “the set of unit-length numbers” and the “1” as “in one complex dimension.”

The reason that the “size one” is important really comes to bear in the case of two-dimensional rotation, which requires two complex numbers, because there are three, “size one,” generators of two-dimensional rotations in the complex plane (SU(2)), which makes its algebra equivalent to the x, y, z rotations, in three, real, dimensions (R(3)).

All of this, like a particular path through any complex maze (no pun intended), has a rather convoluted explanation, but once understood, it is not that difficult to comprehend. The key to understanding it, in the context of the RST and its mathematical counterpart, the RSM, is the clarification of the confusion introduced by the historical failure to understand the fundamental nature of reciprocal numbers (RNs).

However, with that course correction under our belt (see the latest post in The New Math Blog here), the great reverence paid to the discovery of gauge theory, otherwise known as Yang-Mills theory, after the two guys who first suggested it, is understandable.

What it amounts to is that rotations with real numbers is only a group under addition, but, with complex numbers, that group can be transformed into an equivalent group under multiplication.  Both groups are infinite groups, which is what the LST physicists needed, but without the magic of complex numbers, the only set of real numbers that is a group under multiplication is the set of non-zero rationals, which is a finite group of a given order (actually an infinite set of finite groups), but this set is not understood in the LST community as a group under multiplication, but rather as a field under addition and multiplication, due to the failure to recognize the dual RN interpretations. 

Nevertheless, once it is understood that the duality of nature, reflected in the union of discrete and continuous magnitudes, that is so difficult to resolve in terms of LST concepts, is easily understood as two, reciprocal, interpretations of the ratio of RNs, things begin to fall into place much more easily.

In the RSM, there are no such things as complex numbers, in the usual sense that incorporates the ad hoc invention of imaginary numbers. Yet, the simple concepts of our RST-based theory are able to unravel the mysterious features of quantum mechanics, and explain in simple terms, what all the shouting is about, because there is a one-to-one correspondence that can be made between the physical and mathematical logic of the RST and the RSM.

In this sense, we are not trying to construct a mathematical “formalism” that cleanly reflects the physical structure of the universe, without introducing superfluous structure, but we are rather seeking to unravel the common basis of both!   In the next post, I’ll try to develop the details more.