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Register a new userComparison of group-theoretical strategies for unification of 3+1 structures in physics
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Robert A. Wilson
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One of the most important advances in our understanding of the physical world arose from the unification of 3-dimensional space with 1-dimensional time into a 4-dimensional spacetime. Many other physical concepts also arise in similar 3+1 relationships, and attempts have been made to unify some of these also. Examples in particle physics include the three intermediate vector bosons of the weak interaction, and the single photon of electromagnetism. The accepted unification in this case is the Glashow-Weinberg-Salam model of electro-weak interactions, which forms part of the standard model. Another example is the three colours of quarks and one of leptons. In this case, the Pati-Salam model attempts the unification, but is not currently part of the accepted standard model. I investigate these and other instances of 3+1=4 in fundamental physics, to see if a comparison between the successful and unsuccessful unifications can throw some light on why some succeed and others fail. In particular, I suggest that applying the group-theoretical methods of the more successful unifications to the less successful ones could potentially break the logjam in theoretical particle physics.
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This paper discovers geometric unification theory of the grand unification and gravitational interactions and their new physics according to the general fiber bundle theory, symmetry and so on. Consequently, the research of this paper is based on the exact scientific bases of mathematics and physics. The Lagrangians of the grand unification and gravitational interactions are unifiedly deduced from quantitative causal principle (QCP) and satisfy the gauge invariant principle of general gauge fields interacting with Fermion and/or boson fields. The geometry and physics meanings of gauge invariant property of different physical systems are revealed, and it is discovered that all the Lagrangians of the four fundamental physics interactions are composed of the invariant quantities in corresponding spacetime structures. The difficulties that fundamental physics interactions and Noether theorem are not able to be unifiedly given and investigated are overcome, the geometric unification theory and origins of the four fundamental physics interactions and Noether theorem are shown by QCP, their two-order general Euler-Lagrange Equations and corresponding Noether conservation currents are derived in general curved spacetime. This paper further deduces QCP from symmetric principle. Consequently, geometric unification theory of the grand unification and gravitation theories and Noether theorem based on symmetric principle and the new physics are given in this paper. This paper further gives the unification of QCP and symmetric principle. Thus, this paper opens a door to both study and give new developments of geometric unification theory of physics laws, and using the new geometric unification theory, a lot of research works about different branches of physics can be anew done and expressed simpler with different symmetric characters.
The group theoretical description of the periodic system of elements in the framework of the Rumer-Fet model is considered. We introduce the concept of a single quantum system, the generating core of which is an abstract $C^ast$-algebra. It is shown that various concrete implementations of the operator algebra depend on the structure of generators of the fundamental symmetry group attached to the energy operator. In the case of generators of the complex shell of a group algebra of a conformal group, the spectrum of states of a single quantum system is given in the framework of the basic representation of the Rumer-Fet group, which leads to a group-theoretic interpretation of the Mendeleevs periodic system of elements. A mass formula is introduced that allows to give the termwise mass splitting for the main multiplet of the Rumer-Fet group. The masses of elements of the Seaborg table (eight-periodic extension of the Mendeleev table) are calculated starting from the atomic number $Z=121$ to $Z=220$. The continuation of Seaborg homology between lanthanides and actinides is established to the group of superactinides. A 10-periodic extension of the periodic table is introduced in the framework of the group-theoretic approach. The multiplet structure of the extended table periods is considered in detail. It is shown that the period lengths of the system of elements are determined by the structure of the basic representation of the Rumer-Fet group. The theoretical masses of the elements of 10th and 11th periods are calculated starting from $Z=221$ to $Z=364$. The concept of hypertwistor is introduced.
Any group $G$ gives rise to a 2-group of inner automorphisms, $mathrm{INN}(G)$. It is an old result by Segal that the nerve of this is the universal $G$-bundle. We discuss that, similarly, for every 2-group $G_{(2)}$ there is a 3-group $mathrm{INN}(G_{(2)})$ and a slightly smaller 3-group $mathrm{INN}_0(G_{(2)})$ of inner automorphisms. We describe these for $G_{(2)}$ any strict 2-group, discuss how $mathrm{INN}_0(G_{(2)})$ can be understood as arising from the mapping cone of the identity on $G_{(2)}$ and show that its underlying 2-groupoid structure fits into a short exact sequence $G_{(2)} to mathrm{INN}_0(G_{(2)}) to Sigma G_{(2)}$. As a consequence, $mathrm{INN}_0(G_{(2)})$ encodes the properties of the universal $G_{(2)}$ 2-bundle.
In this article we describe the summit sets in B_3, the smallest element in a summit set and we compute the Hilbert series corresponding to conjugacy classes.The results will be related to Birman-Menesco classification of knots with braid index three or less than three.
In a recent paper, Klaere et al. modeled the impact of substitutions on arbitrary branches of a phylogenetic tree on an alignment site by the so-called One Step Mutation (OSM) matrix. By utilizing the concept of the OSM matrix for the four-state nucleotide alphabet, Nguyen et al. presented an efficient procedure to compute the minimal number of substitutions needed to translate one alignment site into another.The present paper delivers a proof for this computation.Moreover, we provide several mathematical insights into the generalization of the OSM matrix to multistate alphabets.The construction of the OSM matrix is only possible if the matrices representing the substitution types acting on the character states and the identity matrix form a commutative group with respect to matrix multiplication. We illustrate a means to establish such a group for the twenty-state amino acid alphabet and critically discuss its biological usefulness.
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