Octonions could make Einstein’s unfulfilled dream come true: a theory of everything
An eight-dimensional number system, known as the octonions or octonions, could help physicists find a single mathematical framework that describes the entire Universe. Mathematicians believe that translating reality into the language of octonions could solve some of the deepest problems in physics and pave the way for a “grand unified theory” that can describe the Universe and answer the big questions.
A new study by scientists Nichol Furey, of Humboldt University of Berlin, in Germany, and Mia Hughes, of Imperial College London, in the United Kingdom, explores the intriguing world of an eight-dimensional number system, the octonions or octonionsto try to achieve an integration of all the theories of physics, a unified theory that may be able to explain the deepest mysteries of nature and the cosmos.
a broken dream
The “theory of everything” or unified theory was the dream that Albert Einstein could not fulfill in life, although he revealed it for a long time. It is about reaching a definitive theory, an integrating equation that serves to explain all the physical phenomena that we know, and provides answers to the deepest questions about the nature of reality and the Universe. How was the cosmos created? What originated it? Does it have an end? How many “flats” or dimensions does reality have? For believers, this equation would be something like being in front of God, looking him in the face…
In the case of scientists, the search is more rational and less mystical. Einstein, who knew that God does not play dice, tried to unite his general relativity theory with the quantum mechanics in a failed theory of everything, designed to explain the laws that govern both the world of visible things and the mysterious realm of subatomic reality. He didn’t make it, for he lit a spark that many others are still trying to fan into a huge fire.
imaginary numbers
According to an article published in New Scientist, the way of the octonions could be more fruitful in terms of this effort than anything achieved to date. To explain it, it is necessary to understand that mathematicians have understood for centuries that there are other numbers than those we can count on our fingers: For example, there is no concrete answer to the result of the square root of -1, so it is known as i.
Something similar happens with many other mathematical operations, showing that there are “imaginary numbers”. When combined with real numbers, the so-called complex numbers arise, forming a system of numbers that could be defined as two-dimensional, because they integrate two different logics. It is not a merely theoretical exercise: electronics and computer science are based on complex numbers, while quantum physics would be unfeasible without this numerical universe.
Over time, scientists took things further: they incorporated two more sets of imaginary numbers called j and k, and the numbers were born. quaternions, a set of four-dimensional numbers. This theory, elaborated in the 19th century by William Rowan Hamilton, was further enriched by John Graves, who devised another system with eight dimensions, called octonions.
Layers and dimensions: from mathematics to reality
In the new study, published recently in the journal Physics Letters B, Furey and Hughes showed for the first time that division algebrasincluding octonions, could provide a link between the different physical theories and arrive at the long-sought unified theory, or at least take a step forward in that direction.
In principle, “they translated” all the mathematical symmetries and particle descriptions of various models of physics into the language of division algebras. In other words, led all these problems to the language of the octonions: for that, they used the Dixon algebra, a set of numbers that allowed them to combine real, complex, quaternion and octonion mathematics.
As a result, they got an original system that works on “layers” of symmetry, like many of the systems that exist in reality: a set of octonions is explained by quaternions, the quaternions are explained by complex numbers and these, in turn, are specified by a set of real numbers. Something similar to what we could appreciate if we take an object and describe its “layers”, from the visible to the subatomic structure.
chain breaks
But the most attractive thing that scientists managed to discover is that the relationships between the “layers & rdquor; do not follow a strict logic: the “symmetries & rdquor; they break before certain variants. For example, in an exchange from positive to negative, numbers that exist at one level disappear at another, and perhaps reappear at the next. Undoubtedly, very similar to what happens when the interaction between particles is observed at a quantum level and compared with other levels of reality.
In short, this behavior of the mathematical models and the “chain break” of the symmetries that support each level could be indicating the same structure that we find in the real physical universe and, perhaps, showing the path towards the grand unified theorylong sought after and still a beautiful utopia.
Reference
Division algebraic symmetry breaking. N. Furey and MJ Hughes. Physics Letters B (2022). DOI:https://doi.org/10.1016/j.physletb.2022.137186