Modified Theories of Gravity include spin dependence in General Relativity, to account for additional sources of gravity instead of dark matter/energy approach. The spin-spin interaction is already included in the eff...Modified Theories of Gravity include spin dependence in General Relativity, to account for additional sources of gravity instead of dark matter/energy approach. The spin-spin interaction is already included in the effective nuclear force potential, and theoretical considerations and experimental evidence hint to the hypothesis that Gravity originates from such an interaction, under an averaging process over spin directions. This invites to continue the line of theory initiated by Einstein and Cartan, based on tetrads and spin effects modeled by connections with torsion. As a first step in this direction, the article considers a new modified Coulomb/Newton Law accounting for the spin-spin interaction. The physical potential is geometrized through specific affine connections and specific semi-Riemannian metrics, canonically associated to it, acting on a manifold or at the level of its tangent bundle. Freely falling particles in these “toy Universes” are determined, showing an interesting behavior and unexpected patterns.展开更多
A canonical p-adic Frobenius lift is defined in the context of p-adic numbers, viewed as deformations of the corresponding finite field. Applications to p-adic periods are considered, including to the classical Euler ...A canonical p-adic Frobenius lift is defined in the context of p-adic numbers, viewed as deformations of the corresponding finite field. Applications to p-adic periods are considered, including to the classical Euler gamma and beta functions and their p-adic analogues, from a cohomological point of view. Connections between various methods for computing scattering amplitudes are related to the moduli space problem and period domains.展开更多
Periods are algebraic integrals, extending the class of algebraic numbers, and playing a central, dual role in modern Mathematical-Physics: scattering amplitudes and coefficients of de Rham isomorphism. The Theory of ...Periods are algebraic integrals, extending the class of algebraic numbers, and playing a central, dual role in modern Mathematical-Physics: scattering amplitudes and coefficients of de Rham isomorphism. The Theory of Periods in Mathematics, with their appearance as scattering amplitudes in Physics, is discussed in connection with the Theory of Motives, which in turn is related to Conformal Field Theory (CFT) and Topological Quantum Field Theory (TQFT), on the physics side. There are three main contributions. First, building a bridge between the Theory of Algebraic Numbers and Theory of Periods, will help guide the developments of the later. This suggests a relation between the Betti-de Rham theory of periods and Grothendieck’s Anabelian Geometry, towards perhaps an algebraic analog of Hurwitz Theorem, relating the algebraic de Rham cohomology and algebraic fundamental group, both pioneered by A. Grothendieck. Second, a homotopy-homology refinement of the Theory of Periods will help explain the connections with quantum amplitudes. The novel approach of Yves Andre to Motives via representations of categories of diagrams, relates from a physical point of view to generalized TQFTs. Finally, the known “universality” of Galois Theory, as how symmetries “grow”, controlling the structure of the objects of study, is discussed, in relation to the above several areas of research, together with ensuing further insight into the Mathematical-Physics symbiosis. To better understand and investigate Kontsevich-Zagier conjecture on abstract periods, the article ponders on the case of algebraic Riemann Surfaces representable by Belyi maps. Reformulation of cohomology of cyclic groups as a discrete analog of de Rham cohomology and the Arithmetic Galois Theory will provide a purely algebraic toy-model of the said algebraic homology/homotopy group theory of Grothendieck as part of Anabelian Geometry. The corresponding Platonic Trinity 5,7,11/TOI/E678 leads to connections with ADE-correspondence, and beyond, e.g. Theory of Everything (TOE) and ADEX-Theory. In perspective of the “Ultimate Physics Theory”, quantizing “everything”, i.e. cyclotomic quantum phase and finite Platonic-Hurwitz geometry of qubit frames as baryons, could perhaps be “The Eightfold (Petrie polygon) Way” to finally understand what quark flavors and fermion generations really are.展开更多
Understanding the role of muons in Particle Physics is an important step understanding generations and the origin of mass as an expression of “internal structure”. A possible connection between muonic atoms and cycl...Understanding the role of muons in Particle Physics is an important step understanding generations and the origin of mass as an expression of “internal structure”. A possible connection between muonic atoms and cycloatoms is used as a pretext to speculate on the above core issue of the Standard Model.展开更多
文摘Modified Theories of Gravity include spin dependence in General Relativity, to account for additional sources of gravity instead of dark matter/energy approach. The spin-spin interaction is already included in the effective nuclear force potential, and theoretical considerations and experimental evidence hint to the hypothesis that Gravity originates from such an interaction, under an averaging process over spin directions. This invites to continue the line of theory initiated by Einstein and Cartan, based on tetrads and spin effects modeled by connections with torsion. As a first step in this direction, the article considers a new modified Coulomb/Newton Law accounting for the spin-spin interaction. The physical potential is geometrized through specific affine connections and specific semi-Riemannian metrics, canonically associated to it, acting on a manifold or at the level of its tangent bundle. Freely falling particles in these “toy Universes” are determined, showing an interesting behavior and unexpected patterns.
文摘A canonical p-adic Frobenius lift is defined in the context of p-adic numbers, viewed as deformations of the corresponding finite field. Applications to p-adic periods are considered, including to the classical Euler gamma and beta functions and their p-adic analogues, from a cohomological point of view. Connections between various methods for computing scattering amplitudes are related to the moduli space problem and period domains.
文摘Periods are algebraic integrals, extending the class of algebraic numbers, and playing a central, dual role in modern Mathematical-Physics: scattering amplitudes and coefficients of de Rham isomorphism. The Theory of Periods in Mathematics, with their appearance as scattering amplitudes in Physics, is discussed in connection with the Theory of Motives, which in turn is related to Conformal Field Theory (CFT) and Topological Quantum Field Theory (TQFT), on the physics side. There are three main contributions. First, building a bridge between the Theory of Algebraic Numbers and Theory of Periods, will help guide the developments of the later. This suggests a relation between the Betti-de Rham theory of periods and Grothendieck’s Anabelian Geometry, towards perhaps an algebraic analog of Hurwitz Theorem, relating the algebraic de Rham cohomology and algebraic fundamental group, both pioneered by A. Grothendieck. Second, a homotopy-homology refinement of the Theory of Periods will help explain the connections with quantum amplitudes. The novel approach of Yves Andre to Motives via representations of categories of diagrams, relates from a physical point of view to generalized TQFTs. Finally, the known “universality” of Galois Theory, as how symmetries “grow”, controlling the structure of the objects of study, is discussed, in relation to the above several areas of research, together with ensuing further insight into the Mathematical-Physics symbiosis. To better understand and investigate Kontsevich-Zagier conjecture on abstract periods, the article ponders on the case of algebraic Riemann Surfaces representable by Belyi maps. Reformulation of cohomology of cyclic groups as a discrete analog of de Rham cohomology and the Arithmetic Galois Theory will provide a purely algebraic toy-model of the said algebraic homology/homotopy group theory of Grothendieck as part of Anabelian Geometry. The corresponding Platonic Trinity 5,7,11/TOI/E678 leads to connections with ADE-correspondence, and beyond, e.g. Theory of Everything (TOE) and ADEX-Theory. In perspective of the “Ultimate Physics Theory”, quantizing “everything”, i.e. cyclotomic quantum phase and finite Platonic-Hurwitz geometry of qubit frames as baryons, could perhaps be “The Eightfold (Petrie polygon) Way” to finally understand what quark flavors and fermion generations really are.
文摘Understanding the role of muons in Particle Physics is an important step understanding generations and the origin of mass as an expression of “internal structure”. A possible connection between muonic atoms and cycloatoms is used as a pretext to speculate on the above core issue of the Standard Model.
