Applications of Lie Groups Related to Segal’s Cosmos. I. The Multi-Level Model as Chronometry’s Unification with the Standard Model

In this article, we introduce two Lie groups, $$G_{F}$$ and $$G_{E}$$ , as equivalent representations of $$SU(2,2)$$ . These representations are connected through conjugation by a specific matrix $$W$$ within $$Gl(4)$$ . The linear fractional action of $$G_{E}$$ on space-time $$U(2)$$ , denoted as $$E$$ (referred to as Segal’s compact cosmos), is well-known to exhibit conformal properties. It has played a pivotal role in the space-time bundle analysis conducted by Paneitz and Segal in the 1980s. Around 2016, Kon and Levichev conducted a study on the linear fractional action of $$G_{F}$$ on space-time $$F$$ , represented as $$U(1,1)$$ (known as Heraclitus’ cosmos). They established the conformal nature of this action, constructed a commutative diagram summarizing the relationships between the two actions, and derived a simple formula for the conformal coefficient, applicable to both $$G_{F}$$ and $$G_{E}$$ . Part of this review is dedicated to the Multi-Level Model (MLM) proposed by Levichev as a mathematical description of quarks and gluons. We revisit the terminology associated with MLM and compare its findings for quarks and leptons to the theoretical and experimental data acknowledged by the Standard Model (SM). MLM is founded on the use of $$E$$ and a sequence of embeddings: $$U(2)$$ into $$U(3)$$ , $$U(2)$$ into $$U(4)$$ , and so on. These groups are referred to as levels of matter, with $$U(2)$$ representing the 0th level, corresponding to our familiar mundane world, $$U(3)$$ as the 1st level, and $$U(4)$$ as the 2nd level, and so forth. This convention aligns with the SM quarks’ generational hierarchy, while each SM quark is MLM-perceived as ‘originating’ from the proton. Based on the 2020 discoveries by Jakobsen and Levichev, we propose that the MLM-proton is elementary and indestructible, eliminating the necessity for confinement. In terms of the concept of a ruling group, we mathematically define flavor and color for MLM-quarks. The number of colors and flavors is contingent on the level. We establish correspondence between levels $$U(3)$$ through $$U(5)$$ and the SM quarks, while also predicting the existence of three new SM-quarks. Utilizing the Han-Nambu scheme, we express the concept of quark’s electric charge in terms of color charges.