The Larson Research Center

In his book, Lee Smolin identifies five, unsolved, problems of theoretical physics, but all of them are aspects of the one real problem: the unification of the continuous and discrete aspects of nature. The problem with string theorists’ enthusiasm for their approach, Smolin asserts, is that, while they don’t even know what it is, they believe that there is no alternative to string theory for solving this fundamental problem. In this view, we can either regard the fundamental entities of the standard model (SM) as 0D point particles, in which case we can’t unify the discrete domain of quantum physics and the continuous domain of general relativity, or we can regard them as 1D lengths, in which case unification of the discrete and continuous is possible, resulting in a unification of all the SM’s elementary particles and forces, as 1D motions, or vibrations, because string theory:

1. Explains existing gauge fields as open ended 1D vibrations
2. Explains gravitons as closed 1D vibrations
3. Explains unification of particles and forces with supersymmetry “sparticles”
   

“Here then is the dream that string theory seemed to make possible,” writes Smolin, referring to what is now known as the first superstring revolution. Before this revolution, resulting from the work of John Schwarz and Michael Green in the mid eighties, “some people doubted that string theory could ever be made consistent with quantum mechanics at any level…,” but now it “promised what no other theory had before - a quantum theory of gravity that is also a genuine unification of forces and matter,” at least in ten spacetime dimensions.

What? Yep, that’s right. For supersymmetric string theory to work, it needs nine space dimensions to go along with its one time dimension, ten spacetime dimensions in all. It can’t exist in three space dimensions. It needs six extra space dimensions to work. Smolin compares this to buying new car “option packages.” We can get the unification “option” we want, but only if we take the extra dimensions “option” too. “Much followed from this,” writes Smolin. “If the theory was not to be ruled out right away, there had to be a way to hide the extra dimensions.”

Ironically, as it turned out, there was a way to hide them. In fact, there were an infinite number of ways to hide them, and the problem became not one of how to hide the extra dimensions, but one of finding some reason why a particular one of the innumerable ways was preferred over the others! In effect, each way of folding up the hidden dimensions is a different string theory, with more and different constants and more and different unobserved particles and forces.

But, of course, a second string theory revolution was looming in the mid-nineties that would unify the many string theories that unified the particles and forces. Again, the basic idea was to increase the number of space dimensions. This time by just one, but this increase led to 11D M-theory, which includes so-called “branes,” or multi-dimensional entities that interact with the 1D strings, such as 2D surfaces. However, for this set of multi-dimensional objects to work it has to contain five types of string theories, including all the different ways the 6 extra dimensions of their geometries can be hidden, forming the specific background which defines the motion of the 1D strings and nD branes in a given theory. However, this means that the theory can’t be built on any one 10D background, but must encompass all backgrounds. In other words, M-theory must be background independent, but still consistent with the background dependent quantum theory. Smolin writes:

This is an important issue, perhaps the most important open question in string theory. Unfortunately, not much progress has been made on it. There have been some fascinating hints, but we still do not know what M-theory is, or whether there is any theory deserving of the name…M-theory remains a tantalizing conjecture. It’s tempting to believe it. At the same time, in absence of a real formulation, it is not really a theory - it is a conjecture about a theory we would love to believe in.

With the fundamentally different view of the nature of space and time that the RST provides, research at the LRC takes a completely different course than that taken by physicists in the LST community,  As we’ve been discussing in the New Math and New Physics blogs, instead of defining motion solely as a change of position against a fixed background geometry of space and time, as in quantum mechanics, or against a dynamic, background independent, geometry of spacetime, as in general relativity, the LRC’s researchers have three fundamentally different types of motion to work with, which promises to lead to an alternative to unification of the discrete and continuous faces of nature, which leading string theorists have no idea even exists.

There are many aspects to the new approach, but the one aspect that is most relevant to string theory is the dimensional aspect.  In string theory, the extra dimensions are necessary to get the various combinations of vibrations of strings that go into the calculations of n-dimensional magnitudes.  However, in addition to the problems with the huge variety of possible forms this approach can take, there is still the fact that there is no motivation for resorting to extra dimensions, other than that the mathematics of string theory requires it. Smolin quotes Nobel Prize winner Sheldon Glashow’s protest on this point:

But superstring physicists have not yet shown that their theory really works.  They cannot demonstrate that the standard theory is a logical outcome of string theory.  They cannot even be sure that their formalism includes a description of such things as protons and electrons.  And they have not made even one teeny-tiny experimental prediction.  Worst of all, superstring theory does not follow as a logical consequence of some appealing set of hypotheses about nature.  Why, you may ask, do the string theorists insists that space is nine-dimensional? Simply because string theory doesn’t make sense in other kind of space. 

In contrast, the RSt under development at the LRC is “a logical consequence of an appealing set of hypotheses.”  That set, known as the fundamental postulates of the system, assumes that everything in the universe consists of one component, motion, existing in three dimensions, in discrete units, with two reciprocal aspects, space and time.  However, In the chart of motion, we see that, mathematically, four dimensions of magnitude exist, the 0D point, the 1D line, the 2D area, and the 3D volume, and we see that there are three types of motion that can define each of those magnitudes in different ways.  The first way is via the change of position of the familiar vectorial motion, the only form of motion recognized by LST physics.  The second way is via the change of interval inherent in electromagnetic radiation, and the third way is via the change of scale inherent in a scalar expansion/contraction.

However, we know too that, mathematically, no new phenomena, beyond these four dimensions (0, 1, 2, 3) exists, thanks to Raul Bott’s proof of the periodicity theorem. So then the question is, “Why do string theorists need five more than this?”  Apparently, the reason is that in higher dimensions the quantum corrections (read “cheating” terms) in the wave equations have to cancel out, and this only happens if everything is symmetrical, which happens at d=10.  Nevertheless, the whole edifice depends on how motion is defined in gauge theory, and this involves complex numbers and the use of rotation in the complex plane; that is, a two-dimensional motion (rotation) is transformed into a one-dimensional motion using complex numbers.  Moving to this type of motion in two and three-dimensions, then, requires higher than four dimensions of space.

The chart of motion simplifies all this, but only if the concept of motion is understood to embrace much more than the M₂ motion, or change of position motion.  As we increase dimensions beyond three dimensions (four counting 0), the Bott periodicity theorem tells us that no new phenomena exists beyond these initial three (four) dimensions; that is, at dimension four (five) things change.  What we’ve discovered is that the nature of the change is that the tetraktys simply repeats.  That is to say that whereas 1, 2, 3, and 4, raised to the 0, 1, 2, and 3 powers successively, corresponds to the space magnitudes of point, line, area, and volume, the next four dimensions just repeat this.  Thus, for M₂ motion,

1. 2⁴ = point
   
2. 2⁵ = line
3. 2⁶ = area
4. 2⁷ = volume

However, now the point has the value of 2⁴ =16, and the volume has the value 2⁷ = 128, and so on.  This doesn’t make much difference in M2 motion, because nothing changes in terms of calculating change of position magnitudes, but, in M₄ motion, it is very significant, because it means calculating change of scale magnitudes in terms of higher densities, if you will; that is, a 4⁰ = 1 point, is much less than a 4⁴ = 256 point from a scalar magnitude point of view.

What this means, then, is that the higher dimensions, in the chart of motion, don’t have to be hidden, and that there is definitely only one way to get these magnitudes, not an infinite number of ways.