The Larson Research Center

The second post in the BAUT forum follows.  This and the last post are a continuation of the RST thread in the “Against the Mainstream” forum of BAUT.

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What Hestenes did was to take what is known as the operational interpretation of number, first recognized by Clifford, as opposed to the usual quantitative interpretation of number, and he used it to reinterpret the physical meaning of the imaginary number ‘i’.

As I think we discussed last year, Michael Atiyah has characterized the imaginary number as the “biggest single invention of the human mind.” He (and almost everyone else) is amazed by the fact that a pure invention, originally conceived to explain the negative roots of quadratic equations, ends up being essential to describe nature in quantum mechanics. It’s just part of a great mystery as to how this concept could be essential in the appropriate language for physics, if mathematics is no more than a formalism, which can have little to do with a common underlying reality shared with physics.

Wigner referred to the “unreasonable effectiveness of mathematics in physics,” as a “gift we neither understand nor deserve,” but Hestenes sees the advent of the imaginary number as part of a deliberate attempt, lasting centuries, to generalize the concept of number to the point that algebra could be used to express the n-dimensional concepts of geometry.

In the final analysis, it is the operational interpretation of an imaginary number that makes Hestenes achievement possible. As he writes in his New Foundations for Classical Mechanics,

The way the two approaches are brought together in the geometric product is through the use of orthogonality. In the geometric product, orthogonality is expressed in the duality of the inner and outer vector product, which is equivalent to a certain rotation through an angle, as well as the orthogonal directions of a vector in a plane. In this interpretation, since i^2 is -1, which represents a 180 degree rotation through the complex plane from 1 to -1, then a 90 degree rotation is represented by i^1, a 270 degree rotation is represented by i^3, and i^4 brings us back to the beginning, or to i^0, on the unit circle.

However, GA does more than provide us with a neat package for unifying the concepts of the vectors and spinors in one language. As impressive an achievement as this is, it is crucial to recognize that, in the process of giving geometric meaning to the ad hoc invention of ‘i’, GA also unifies the Clifford algebra, Cl3, with the concepts of vectorial motion and Euclidean geometry, in that it permits numbers to represent n-dimensional magnitudes with direction.

This long-sought generalization of number unifies the binomial expansion of n-dimensional numbers, the associated Clifford algebras, and the elements of Euclidean geometry, when it is realized that the expansion of the numbers in the associated algebra are isomorphic to the expansion of a zero-dimensional point into a one-dimensional line, a one-dimensional line into a two-dimensional plane, and a two-dimensional plane into a three-dimensional cube.

As the expansion grows, from 0 to 3 dimensions, the number of elements in the algebra grows as the number of elements in a cube; that is, at 2^0 = 1, we have only the concept of points in the geometry and only scalar numbers in the algebra, but at 2^1 = 2, we have both the concept of points and lines in the geometry, and scalar and vector numbers in the algebra. At 2^2 = 4, we have the point concept, two line concepts (orthogonal lines), and the concept of the plane in geometry, and scalar, vector, and bivector numbers in the algebra.

Finally, at the 2^3 = 8 dimensions, we find the culmination of all these in the maximum dimensions of the geometric cube: There are the concepts of points, lines, planes, and cubes in 3D geometry, and the corresponding concepts of scalar, vector, bivector, and trivector numbers in the 3D algebra. Moreover, thanks to the operational interpretation of number and Hestenes work, the scalar, vector, bivector, and trivector numbers of the algebra can also be thought of as scalar magnitudes, linear magnitudes, quadratic magnitudes, and octonic magnitudes, generated by rotation, i.e. vectorial motion.

Thus, Hestenes has unified the concepts of vectorial motion, mathematics, and geometry, in his GA. However, the significance of this achievement is not just academic, but has huge ramifications in physics that I don’t believe the mainstream physics community fully appreciates, at this point in time, preoccupied as they are with the string theory controversy, and the fundamental crisis in theoretical physics that gave rise to string theory, but which it seems only to be able to exacerbate.

Nevertheless, the mathematical and geometrical concepts clarified by GA, illuminate the central problem at the heart of the current crisis in theoretical physics: the issue of reconciling the concept of the continuum, upon which the fields of the general relativity theory of gravity are based, and the concept of the quantum, upon which the fields of the standard model theory of particle physics are based. The essential clue that GA provides is that while there is a fundamental disconnect (no pun intended) between the two concepts, they both suffer from the same ailment: the dreaded singularities, or infinities caused by the numerical/physical concept of zero.

To see how GA relates to the problem of infinities requires us to now turn our attention from the subject of mathematics to the subject of physics. In the previous posts of this thread last year, we discussed the ideas in Lee Smolin’s paper on background independence, where he identifies five major, unsolved, problems in theoretical physics, and how these are related to the famous debate between Newton and Leibnitz over the nature of space, which is reincarnated in the argument for the background dependence of quantum physics.

Since then, Smolin has published a book entitled, The Trouble With Physics: The Rise of String Theory, the Fall of a Science, and What Comes Next. In the first part of the book, he again reviews the five unsolved problems of theoretical physics and in this connection writes:

referring to the revolution of the last century that started with the discovery of the quantum. This revolution is unfinished, according to Smolin, in contrast to the previous revolution that started when the continuum secret of the Pythagoreans’ square root of 2 got out in the open, and finally culminated with Newton’s overthrow of the Aristotelian theories of space, time, motion and cosmology that competed with the ideas of the Copernican revolution.

The challenge of reconciling these two concepts, one discrete and the other continuous, did not seem unsurmountable in the early days of the unifinished revolution, but today it is more intractable than ever. The problem is not that we don’t have theories that work, but that the theories we have work too well, yet are incompatible. Smolin writes:

However, though this is the reason most cited for why we need a unified theory, it is not the only reason. There is more to it than that, and it has to do with the infinities that plague both the continuum based theory of gravity and the quantum based theory of particle physics. Regarding the former, Smolin writes:

Of course, this has lead to the effort to find a theory of quantum gravity, but the effort has been unsuccessful to date, and though quantum physics dealt with its own problems of infinities long ago, through the suspicious procedure of “renormalization,” that has effectively swept them under the rug, it is hoped that a unified theory can be found that will include gravity and, at the same time, tame the infinities of quantum theory without recourse to renormalization. Smolin continues:

Clearly, the problem of infinities is related to the continuum. Given Zeno’s paradox, there is no limit to how finely a continuum can be subdivided, whether it’s the continuum constituting 1D linear space, or it’s the continuum of a 3D volume of space. The reason string theory has been developed and highly touted is precisely because the nature of a string excludes the concept of zero, which corresponds to the concept of a point.

If particle physics can be described without recourse to the concept of point particles, or the infinite variables of a field, and the warping of a spacetime continuum, required to describe gravity, can be accomplished by the same means, then that way out of the impasse seems very promising.

However, the problem with the string theory approach brings us back again to the nature of space and time, but, in this case, the plague of the infinities of the continuum attack again, albeit from a different angle: string theory requires 10 dimensions of space, but there are only four dimensions of spacetime in current theory, leaving six dimensions that have to come from somewhere else. The thought is that these extra dimensions could be “compactified,” but there are essentially an infinite number of ways to compactify them!

Thus, we have a fundamental crisis in theoretical physics on our hands that is being caused by the fact that the dual of the point is infinity, and though nature knows a way to deal with this fact, evidently, we don’t.

In the meantime, Hestenes has found a way to unify the concepts of vectorial motion, algebra, and geometry. Could it be that this is a prerequisite to unifying the laws of physics? We’ll discuss it next time.