group theoretic approach

[AntonioLao]

Group theoretic approach to physical reality has been the preferred choice since the emergence of abstract algebras in the 19th century. The first abstract structure to be introduced was called the group and many of the basic ideas of abstract group theory can be found implicitly and explicitly as far back as 1800. Now that various abstract theories exist, historians of mathematics are more interested in tracing back how many of these abstract ideas were preceded by the work of Gauss, Abel, Galois, Cauchy, and many others. However, the only significant point that bears mentioning is that, once the abstract notion was in place, it was relatively easy for latter mathematicians to obtain ideas and theorems by rephrasing these works of the past. Unfortunately, the advantage of obtaining results that might be useful in many specific areas of physical reality by considering abstract version was soon lost sight of, and the study of abstract structures and the derivation of their properties became an end in itself. Reference: Morris Kline, Mathematical Thought from Ancient to Modern Times, Oxford University Press, 1972.

This is truly unfortunate in light of recent experimentally unsuccessful extensions of group theory to include a quantum theory of gravity or for establishing the existence of Higgs boson, magnetic monopole, and other supersymmetric elementary particles. Fortunately, if this group theoretic approach is replaced by a ring theoretic approach with two matrix operations of addition and multiplication then a supercomposition theory of space-time charges for representing the quantum vacuum fluctuations of squares of zero-point energies has a greater advantage over the singular operation of group theory. Moreover, a switch to ring theoretic approach to physical reality could never compromise the group relation to symmetry as expressed in the theorem by Amalie (Emmy) Noether (1882-1935). She was the German mathematician who established theory of rings and non-commutative algebras in Göttingen circa 1933. Therefore, the ring relation to perfect symmetry can be formulated using the square symmetric singular Hadamard matrices of any order establishing the abstract algebras of zero charge matrices as well as the zero mass matrices.

[PoPpAScience]

I rarely miss anything you post Antonio. I feel someday you will say what I see in my mind. You are the mathematician that my family wanted me to be when I was young, but I chose to be a wandering philosopher instead. In school if I got 99 on a test in math it was because of a clerical mistake. So I envy your skills and think about maybe going to school in my 60's to see if I can get to your level of math thinking skills. Wanted to say this for awhile and thought I would.

I wish there was something I could say that could make you contemplate about the concept of a substance that is infinite in nature, with out boundaries. But this substance can be Attenuated and Compressed. I feel you are the one person that could describe this substance in Mathematical terms.

Thanks for all your great posts.

[Lloyd Gillespie]

Privileged character of 3+1 spacetime... LINK
There are two kinds of dimensions, spatial (bidirectional) and temporal (unidirectional). Let the number of spatial dimensions be N and the number of temporal dimensions be T. That N = 3 and T = 1, setting aside the compactified dimensions invoked by string theory and undetectable to date, can be explained by appealing to the physical consequences of letting N differ from 3 and T differ from 1. The argument is often of an anthropic character.

Immanuel Kant argued that 3-dimensional space was a consequence of the inverse square law of universal gravitation. While Kant's argument is historically important, John D. Barrow says that it "...gets the punch-line back to front: it is the three-dimensionality of space that explains why we see inverse-square force laws in Nature, not vice-versa." (Barrow 2002: 204). This is because the law of gravitation (or any other inverse-square law) follows from the concept of flux and the proportional relationship of flux density and the strength of field. If N = 3, then 3-dimensional solid objects have surface areas proportional to the square of their size in any selected spatial dimension. In particular, a sphere of radius r has area of 4πr². More generally, in a space of N dimensions, the strength of the gravitational attraction between two bodies separated by a distance of r would be inversely proportional to rN−1.

In 1920, Paul Ehrenfest showed that if we fix T = 1 and let N > 3, the orbit of a planet about its sun cannot remain stable. The same is true of a star's orbit around the center of its galaxy.[10] Ehrenfest also showed that if N is even, then the different parts of a wave impulse will travel at different speeds. If N > 3 and odd, then wave impulses become distorted. Only when N = 3 or 1 are both problems avoided. In 1922, Hermann Weyl showed that Maxwell's theory of electromagnetism works only when N = 3 and T = 1, writing that this fact "...not only leads to a deeper understanding of Maxwell's theory, but also of the fact that the world is four dimensional, which has hitherto always been accepted as merely 'accidental,' become intelligible through it."[11] Finally, Tangherlini[12] showed in 1963 that when N > 3, electron orbitals around nuclei cannot be stable; electrons would either fall into the nucleus or disperse.

Properties of n+m-dimensional spacetimes


Max Tegmark[13] expands on the preceding argument in the following anthropic manner. If T differs from 1, the behavior of physical systems could not be predicted reliably from knowledge of the relevant partial differential equations. In such a universe, intelligent life capable of manipulating technology could not emerge. Moreover, if T > 1, Tegmark maintains that protons and electrons would be unstable and could decay into particles having greater mass than themselves. (This is not a problem if the particles have a sufficiently low temperature.) If N > 3, Ehrenfest's argument above holds; atoms as we know them (and probably more complex structures as well) could not exist. If N < 3, gravitation of any kind becomes problematic, and the universe is probably too simple to contain observers. For example, when N < 3, nerves cannot cross without intersecting.

In general, it is not clear how physical law could function if T differed from 1. If T > 1, subatomic particles which decay after a fixed period would not behave predictably, because time-like geodesics would not be necessarily maximal.[14] N = 1 and T = 3 has the peculiar property that the speed of light in a vacuum is a lower bound on the velocity of matter; all matter consists of tachyons.

Hence anthropic and other arguments rule out all cases except N = 3 and T = 1—which happens to describe the world about us. Curiously, the cases N = 3 or 4 have the richest and most difficult geometry and topology. There are, for example, geometric statements whose truth or falsity is known for all N except one or both of 3 and 4.[citation needed] N = 3 was the last case of the Poincaré conjecture to be proved.

For an elementary treatment of the privileged status of N = 3 and T = 1, see chpt. 10 (esp. Fig. 10.12) of Barrow;[15] for deeper treatments, see §4.8 of Barrow and Tipler (1986) and Tegmark.[13] Barrow has repeatedly cited the work of Whitrow.[16]

Not that I gree with it all, but anyway...

[labelwench]

Not that I understand it all, either, yet such is sometimes helpful in preventing one from becoming too attached to any single theory, for as soon as one forms such attachment, it is very easy to loose the objective viewpoint that is required when examining the underpinnings of that which we currently believe, and which is ever evolving.

[AntonioLao]

the concept of a substance that is infinite in nature, with out boundaries. But this substance can be Attenuated and Compressed.

This substance would be the square of energy. It is without boundary in the topological sense, which means its volume is derived from the vector cross product of two directions. The volume is maximized if and only if these two coplanar vectors are orthogonal. Changing the angles between them gives the attenuation of square of energy and decreasing magnitude gives compression of square energy density. The math is simple vector analysis restricted to 3 dimensions not the more abstract and complicated multidimensional tensor analysis or complex imaginary spinor analysis.