Both Gauss's Law and Maxwell's approach are direct consequences of a point of view that seeks to explore symmetries and anti-symmetries in nature and mathematics, and the analogies between them.
Symmetry, Anti-Symmetry, and Symmetry Breaking VIII
Francis Bacon, Novum Organum
We could further explore the subject of symmetry, antisymmetry, and symmetry breaking for a few more years, through the mathematical development of vector algebra, without the fascination inherent in it eroding even a tiny portion. The deepening of this treatment could follow an upward trajectory towards the elucidation of the intimacy of matter and its relationship with the observable universe.
One way to realize this idea would be to follow Geoffrey M. Dixon's book, Division Algebras: Octonions, Quaternions, Complex Numbers and the Algebraic Design of Physics. There are only four division algebras: the real numbers, the complex numbers, the quaternions and the octonions. Dixon uses a recent mathematical structure called adjunct division algebra, in which the four division algebras appear in the role of spinor spaces.
Deciphering the meaning of these terms would consume a few more years of our lives. Dixon believes that this part of mathematics is sufficient to describe and unravel the structure of our physical reality and, to prove it right, deduces from these algebras part of the famous standard model, the leptons and quarks, fundamental particles of the nucleus of the atom. Understanding Dixon's reasoning is a noble motivation for human life. We will make a temporary change in our route to examine some of the other equally noble motivations that are now available.
Mostly mathematicians and physicists cannot complain about the lack of motivation to be alive at the present historical moment. In the world of mathematical physics there are a number of beautiful and profound theories, with an as yet unknown impact on human life and the ability of Homo sapiens sapiens to interfere with nature. Some even claim that this is the best time in human history to live because of the point reached by the development of science and technology, particularly mathematics and physics, beyond, without doubt, economics, and the fact that this point be a formidable technological turning point.
We would like to select two priceless news stories, especially those that like mathematics and cultivate it in some way. The first is about the extraordinary achievement of the French mathematician Alain Connes and his collaborators.
This news came out at Scientific American do Brasil (SCIAM) in September 2006. Connes is eagerly awaiting the start of the Large Hadron Collider in Geneva, Switzerland, to prove whether or not in this laboratory the existence of the Higgs particle. This particle has been known to physicists and mathematicians for a long time, but Connes deduced its existence from his noncommutative space model: “Instead of searching for new particles, we develop a subtler geometry, and refinements of that geometry generate the new ones. particles ”he told Brazilian SCIAM.
Connes had already won the Fields Medal, the highest honor in the field of mathematics that anyone can aspire to, for his work and especially for his theory called Noncommutative Geometry. He has elaborated over the past thirty years a conception of noncommutative space that contains all the algebras that extend the symmetry groups relevant to the Standard Model of elementary particles: “What really interests me is the complex calculations performed by physicists and experimentally tested. I spent twenty years trying to understand renormalization. Not that I didn't understand what physicists were doing, but I didn't understand the meaning of the math behind it, ”he said.
Connes and his colleague, physicist Dirk Kreimer, found that the important renormalization practiced by physicists a few years ago could be completely justified by solving one of Hilbert's famous 23 problems formulated in 1900 at the Paris Mathematics Congress. With this, these two skilled users of mathematics have taken a major step toward unifying the theory of relativity with quantum mechanics. It is undeniable that we live exciting times!
Along with physicist Carlo Rovelli, Connes has shown that time can naturally emerge from the noncommutativity of observable quantities of gravity. Could anyone imagine a prettier theorem than this? Time is not given beforehand, it is not prior to anything else. It simply appears as a consequence of the observation of gravity.
Would those who were unimpressed by the Higgs particle deduction, if rational beings, still continue to take a lethargic and indifferent stance in the face of this theorem about time?
Connes reports that his Noncommutative Theory of Physical Reality is different from the Superstring Theory. The latter cannot be directly tested by any laboratory, built or imagined by Homo sapiens sapiens, at this stage of science and technology. However, Connes predicted the Higgs particle mass: 160 billion Electron volts; and states that this prediction and renormalization could be tested on the Large Hadron Collider.
In fact, this is invaluable news. In the next few years we will be able to access publications that reasonably explain to lay people the reality of elementary particles, the unification of Einstein's Theory of Relativity with Quantum Mechanics, and the beautiful mathematics of the algebras that underlie all these symmetries, anti- symmetries and symmetry breaks.
The second priceless news, especially for mathematicians and physicists, is about economics, which may seem somewhat paradoxical, and which we will divide into two parts.
The first part is that, after a major development in the twentieth century, it can be seen mathematically that the way the economy produces wealth has changed radically.
Mathematics has contributed fundamentally to a rigorous and scientific treatment of economics, especially in the twentieth century. In October 1990, economist Paul Romer published an article presenting an extremely original and courageous mathematical model of the nature of economic growth. Romer has shown mathematically that, after two hundred years, the Knowledge Economy has emerged from informality and an uncomfortable position at the rear of Economic Theory.
If we allow our imagination to fly for a moment, then it will not be difficult to associate “Knowledge Economy” with “Accelerated Mathematical Development”. This type of association is not new. The appearance of Infinitesimal Calculus in the 16th and 17th Centuries accelerated and the observation of Nature became "Physics" or "Observation of Nature with Knowledge." The same could be said of "Chemistry" and many other branches of human knowledge.
In the late nineteenth century, British engineers struggling with difficult electro-technical problems arising from the construction of telegraph and electricity grids discovered that many of these problems could be formulated mathematically and solved by taking advantage of the developmental stage of “Accelerated Calculus”. ”And“ Accelerated Physics ”. In this broth of culture appeared Oliver Heaviside with his enigmatic and brilliant "Operational Calculus". Albert Einstein saw the Tensors of Differential Geometry with the help of his colleague and friend Marcel Grossman, a mathematics teacher at the Zurich Polytechnic School. There is a very rich history of acceleration of mathematical knowledge implying the acceleration of other knowledge.
Why would this phenomenon also not occur in relation to the economy?
If the intricate incentive system for the creation of new ideas is underdeveloped, then society suffers from the general lack of progress as much as when those incentives are too abundant or too restricted.
The phenomenon of waning return yielded its fundamental position to the phenomenon of increasing return or abundant return. The resource scarcity axiom has partly given way to the new resource abundance axiom. The space of economics is no longer the space of people and things, but now the space of people, things and ideas. Abundance of ideas tends to generate abundance of resources and goods. This third element, represented by the word "ideas", is the key to understanding a fundamental enigma of economics, that of faster and greater creation of wealth.
The idea that the economy creates wealth is growing faster and more intensely, and increasingly abundant, because its factors of production are no longer just land, capital and labor, but also people. , things and ideas. This is the first part of the second priceless news.
The second part of the second priceless news is that through a combination of Game Theory, a genuinely mathematical theory, Computer Theory, and the current Theory of Evolution in Biology, a view has been developed that economics has processes of innovation analogous to the processes that generate diversity in the biosphere and their dynamics evolve according to the laws of Darwinism.
From this perspective, the way economics creates wealth would be an evolutionary adaptive process. Here there seems to be a paradox that mathematics, in the field of equations, becomes useless since evolutionary processes are not equable. This is the topic we will study and share in our next columns with our readers.
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