The fact that every particle in the universe, at all times, in all places, and under all circumstances, obeys identical physical rules may be the most fascinating thing about the universe. We may describe nature by identifying the mathematical framework encapsulating the universal laws that nature operates under. A new physical framework is frequently created due to the discovery of a novel mathematical structure. When that framework adequately reflects the Universe, new physics can be derived. We now come to Pedro Teixeira's question about the relationship between octonions and one of the most exciting mathematical possibilities for our Universe:
Octonions, do they stand a possibility of being the explanation for how our existence works, or is this concept just hype?
Using complex numbers nearly everywhere and every time, there still are cases that we could not solve with them, that we need more to clear up, with quaternions: a + bi + cj + dk. Quaternions have a wide range of practical applications in addition to being quite helpful in mathematics. In contrast to how a complex number (which has a real axis and an imaginary axis) depicts points in a two-dimensional plane, a quaternion contains enough dimensions and degrees of freedom to describe points in three dimensions.
Quaternions have a wide range of practical applications in addition to being quite helpful in mathematics. In contrast to how a complex number (which has a real axis and an imaginary axis) depicts points in a two-dimensional plane, a quaternion contains enough dimensions and degrees of freedom to describe points in three dimensions.
So, you could ask, how far can you expand the quaternions? Is there another method of using mathematics that would allow for the possibility of a deeper structure?
The answer is yes, but with a price. Going from quaternions to our octonions, which contain eight components each, is the next step toward a more sophisticated mathematical structure, but it has a cost. Even if Q1 * Q2 is different from Q2 * Q1, quaternions are still associative, therefore the order of multiplication counts. Q1 * Q2 * Q3 = Q1 * (Q2 * Q3) if you have three quaternions (Q1, Q2, and Q3). Multiplication order matters, but it matters in a fundamentally different way when you have three octonions since they are non-commutative and non-associative.
The mathematics of octonions is expressive of operations that extend far beyond known physics, explaining phenomena that emerge in extensions such as Grand Unified Theories (GUTs) and string theory, in contrast to the mathematics of quaternions, which is connected to a variety of well-known physical theories.
There are several compelling reasons to be interested in octonions, although any applicability to physics is speculative. The octonions theoretically enlighten us on the range of spacetime dimensions required to build a supersymmetric quantum field theory. They are linked to the exceptional Lie groups that are used to construct GUTs and have an effect on superstring theories via the E(8) group.
Although the octonions alone will never be "the solution" to how reality functions, they do offer a powerful, generalized mathematical framework with several distinctive characteristics. In addition to real, complex, and quaternion mathematics, it also presents fundamentally distinct mathematical features that may be used to generate innovative, speculative, and up-to-now unproven predictions about physics.
Despite the lack of specific observables predicted by octonions themselves, they can help us choose which extensions to existing physics would be intriguing to consider and which ones could be less so. As the physics professor, Pierre Ramond used to remark, "octonions are to physics what the Sirens were to Ulysses." They are undoubtedly alluring, but if you dive in, they might trap you and take you to an unavoidable, mesmerizing death.
Works Cited:
Anon, 1958. Interesting results easily achieved using complex numbers. Mathematics Stack Exchange. Available at: https://math.stackexchange.com/questions/4961/interesting-results-easily-achieved-using-complex-numbers.
Brooks, M., 2022. Octonions: The strange maths that could unite the laws of nature. New Scientist. Available at: https://www.newscientist.com/article/0-octonions-the-strange-maths-that-could-unite-the-laws-of-nature/.
Siegel, E., 2019. Ask Ethan: Could octonions unlock how reality really works? Forbes. Available at: https://www.forbes.com/sites/startswithabang/2019/12/14/ask-ethan-could-octonions-unlock-how-reality-really-works/?sh=ee896647c1e5.
Wolchover, N. & Quanta Magazine moderates comments to facilitate an informed, substantive, 2020. The peculiar math that could underlie the laws of nature. Quanta Magazine. Available at: https://www.quantamagazine.org/the-octonion-math-that-could-underpin-physics-20180720/.