The article is an overview of the role of graph complexes in the Feynman path integral quantizati... more The article is an overview of the role of graph complexes in the Feynman path integral quantization. The underlying mathematical language is that of PROPs and operads, and their representations. The sum over histories approach, the Feynman Legacy, is the bridge between quantum physics and quantum computing, pointing towards a deeper understanding of the fundamental concepts of space, time and information. 1
Complex periods are algebraic integrals over complex algebraic domains, also appearing as Feynman... more Complex periods are algebraic integrals over complex algebraic domains, also appearing as Feynman integrals and multiple zeta values. The Grothendieck-de Rham period isomorphisms for p-adic algebraic varieties defined via Monski-Washnitzer cohomology, is briefly reviewed. The relation to various p-adic analogues of periods are considered, and their relation to Buium-Manin arithmetic differential equations.
We review several known categorification procedures, and introduce a functorial categorification ... more We review several known categorification procedures, and introduce a functorial categorification of group extensions with applications to non-abelian group cohomology. Categorification of acyclic models and of topological spaces are briefly mentioned.
The Betti-de Rham period isomorphism ("Abelian Geometry") is related to algebraic funda... more The Betti-de Rham period isomorphism ("Abelian Geometry") is related to algebraic fundamental group (Anabelian Geometry), in analogy with the classical context of Hurewicz Theorem. To investigate this idea, the article considers an "Abstract Galois Theory", as a separated abstract structure from, yet compatible with, the Theory of Schemes, which has its historical origin in Commutative Algebra and motivation in the early stages of Algebraic Topology. The approach to Motives via Deformation Theory was suggested by Kontsevich as early as 1999, and suggests Formal Manifolds, with local models formal pointed manifolds, as the source of motives, and perhaps a substitute for a "universal Weil cohomology". The proposed research aims to gain additional understanding of periods via a concrete project, the discrete algebraic de Rham cohomology, a follow-up of author's previous work. The connection with Arithmetic Gauge Theory should provide additional intuiti...
Periods are numbers represented as integrals of rational functions over algebraic domains. A surv... more Periods are numbers represented as integrals of rational functions over algebraic domains. A survey of their elementary properties is provided. Examples of periods includes Feynman Integrals from Quantum Physics and Multiple Zeta Values from Number Theory. But what about finite characteristic, via the global-to-local principle? We include some considerations regarding periods and Jacobi sums, the analog of Veneziano amplitudes in String Theory.
Recently, GUTs based on the exceptional Lie algebras attempt unification of interactions of the S... more Recently, GUTs based on the exceptional Lie algebras attempt unification of interactions of the Standard Model as a gauge field theory, e.g. Garrett Lisi's E8-TOE. But the modern growing trend in quantum physics is based on the Quantum Information Processing paradigm (QIP). The present proposal will develop the Qubit Model, a QIP analog of the Quark Model within the SM framework. The natural principle that "quantum interactions should be discrete", technically meaning the reduction of the gauge group to finite subgroups of SO(3)/SU(2), implies that qubit-frames (3D-pixels), playing the role of baryons, have the Platonic symmetries as their Klein Geometry (Three generations of flavors): T,O,I, and hence their "doubles", the binary point groups are the root systems E6,7,8 of the exceptional Lie algebras, which control their Quantum Dynamics. The Qubit Model conceptually reinterprets the experimental heritage modeled into the SM, and has clear prospects of expla...
We review Pólya vector fields associated to holomorphic functions as an important pedagogical too... more We review Pólya vector fields associated to holomorphic functions as an important pedagogical tool for making the complex integral understandable to the students, briefly mentioning its use in other dimensions. Techniques of differential geometry are then used to refine the study of holomorphic functions from a metric (Riemannian), affine differential or differential viewpoint. We prove that the only nontrivial holomorphic functions, whose Pólya vector field is torse-forming in the cannonical geometry of the plane, are the special Möbius transformations of the form f(z)=b(z+d)−1. We define and characterize several types of affine connections, related to the parallelism of Pólya vector fields. We suggest a program for the classification of holomorphic functions, via these connections, based on the various indices of nullity of their curvature and torsion tensor fields.
uantum phenomena reflect resonance at the very fundamental level of a network processing Quantum ... more uantum phenomena reflect resonance at the very fundamental level of a network processing Quantum Information. The term “particle” is a hindrance preventing the shift to a new paradigm. A top-level mental-visual model for “elementary particles” and some other quantum experiments is needed, and provided: a foamy Riemann Surface is interpreted as a Quantum Chip, having both fermionic parameters (sources as integrals), and bosonic character (propagating energy-momentum).
The generic relativistic version of a particle-field theory, with non-isotropic sources, includes... more The generic relativistic version of a particle-field theory, with non-isotropic sources, includes a Gravity force perturbation of Coulombian Force, with the usual Magnetic Force resulting from Lorentz transformations. The quark model of Standard Model, with fractional charge structure of nucleons enveloped by electronic clouds, mandates such nonisotropic charges. Dynamic Nuclear Orientation (DNO), via electronic spin and LScoupling, allows to invert the population of low energy Gravitational attraction states, and achieve Gravity Control. The 1994 scientific experiment of Dr. Frederick Alzofon has confirmed Gravity Control can be achieved via DNO. Other researchers have contributed in the same general direction of unifying Electromagnetism and Gravity, supporting the non-isotropic charge concept, including Paul LaViolette, author of Subquantum Kinetics.
Kontsevich recently generalized Tian-Todorov Theorem regarding the structure of the Kuranish spac... more Kontsevich recently generalized Tian-Todorov Theorem regarding the structure of the Kuranish space of deformations of a Kahler manifold with trivial canonical bundle. An alternative proof was given using a general result regarding the smoothness of moduli space of formal deformations, based on BV-algebra resolutions. From this, various other generalizations ensue and a conjecture relating the dimension of the tangent space of formal deformations and the first non-trivial Hodge number h(n− 1, 1). Additional details are provided, together with a proposed explanation regarding the above conjecture. Related considerations regarding mirror symmetry and motives of Calabi-Yau manifolds are included, based on the idea of complexifying TQFTs, modeled after Chow pure motives.
Gravity is not a fundamental force. Alzofon’s Thermodynamic Gravity Theory is derived from Qubit ... more Gravity is not a fundamental force. Alzofon’s Thermodynamic Gravity Theory is derived from Qubit Model, an upgrade of the Quark Model within the Standard Model. Alzofon’s experiment is discussed: pros and cons. At the level of the Standard Model, Gravity is a result of the structure of the electric charges of quarks in nucleons, subject to Platonic symmetry and lack of parity invariance, related to CP-violation, due to the dihedral group as the Quantum Mirror Symmetry group. The AGNUE experiment performed at Hathaway Research International is briefly explained. It is designed to test Alzofon’s Theory. A glimpse of Gravity Control and what inertial mass is are presented. Further R&D will be funded under the upcoming Kickstarter Gravity [1].
To better understand and investigate Kontsevich-Zagier conjecture on abstract periods, we conside... more To better understand and investigate Kontsevich-Zagier conjecture on abstract periods, we consider the case of algebraic Riemann Surfaces representable by Belyi maps. The category of morphisms of Belyi ramified maps and Dessins D'Enfant, will be investigated in search of an analog for periods, of the Ramification Theory for decomposition of primes in field extensions, controlled by theirs respective algebraic Galois groups. This suggests a relation between the theory of (cohomological, Betti-de Rham) periods and Grothendieck's Anabelian Geometry [1] (homotopical/ local systems), towards perhaps an algebraic analog of Hurwitz Theorem, relating the the algebraic de Rham cohomology and algebraic fundamental group, both pioneered by A. Grothendieck. There seem to be good prospects of better understanding the role of absolute Galois group in the physics context of scattering amplitudes and Multiple Zeta Values, with their incarnation as Chen integrals on moduli spaces, as studied by Francis Brown, since the latter are a homotopical analog of de Rham Theory. The research will be placed in the larger context of the ADE-correspondence, since, for example, orbifolds of finite groups of rotations have crepant resolutions relevant in String Theory, while via Cartan-Killing Theory and exceptional Lie algebras, they relate to TOEs. Relations with the author's reformulation of cohomology of cyclic groups as a discrete analog of de Rham cohomology [2] and the arithmetic Galois Theory [3] will provide a purely algebraic toy-model of the said algebraic homology/homotopy group theory of Grothendieck as part of Anabelian Geometry. It will allow an elementary investigation of the main concepts defining periods and algebraic fundamental group, together with their conceptual relation to algebraic numbers and Galois groups. The Riemann surfaces with Platonic tessellations, especially the Hurwitz surfaces, are related to the finite Hopf sub-bundles with symmetries the "exceptional" Galois groups P SL 2 (F p), p = 5, 7, 11. The corresponding Platonic Trinity 5, 7, 11/T OI/E678 leads to connections with ADE-correspondence, and beyond, e.g. TOEs and ADEX-Theory [4]. Quantizing "everything" (cyclotomic quantum phase and finite Platonic-Hurwitz geometry of qubits/baryons) could perhaps be "The Eightfold (Petrie polygon) Way" to finally understand what quark flavors and fermion generations really are.
Journal of High Energy Physics, Gravitation and Cosmology, 2020
Understanding the role of muons in Particle Physics is an important step understanding generation... more 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.
A canonical p-adic Frobenius lift is defined in the context of p-adic numbers, viewed as deformat... more 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.
Is "Gravity" a deformation of "Electromagnetism"? G N m 2 e k C e 2 ≈ 10 −54 ↔ e −1/α ≈ 10 −59. T... more Is "Gravity" a deformation of "Electromagnetism"? G N m 2 e k C e 2 ≈ 10 −54 ↔ e −1/α ≈ 10 −59. Thus "Gravity" emerges already "quantum", in the discrete framework of QID, based on the quantized complex harmonic oscillator: the quantized qubit. All looks promising, but will the details backup this "grand design scheme"? Contents
The current research regarding the Riemann zeros suggests the existence of a non-trivial algebrai... more The current research regarding the Riemann zeros suggests the existence of a non-trivial algebraic/analytic structure on the set of Riemann zeros [1, 2, 3]. The duality between primes and Riemann zeta function zeros suggests some new goals and aspects to be studied: adelic duality and the POSet of prime numbers. The article presents computational evidence of the structure of the imaginary parts t of the non-trivial zeros of the Riemann zeta function ρ = 1/2 + it, called in this article the Riemann Spectrum, using the study of their distribution, as in [3]. The novelty represents in considering the associated characters p it , towards an algebraic point of view, than rather in the sense of Analytic Number Theory. This structure is tentatively interpreted in terms of adelic characters, and the duality of the rationals. Second, the POSet structure of prime numbers studied in [4], is tentatively mirrored via duality in the Riemann spectrum. A direct study of the convergence of their Fourier series, along Pratt trees, is proposed. Further considerations, relating the Riemann Spectrum, adelic characters and distributions, in terms of Hecke idelic characters, local zeta integrals (Mellin transform) and ω-eigen-distributions, are explored following [11].
A natural partial order on the set of prime numbers was derived by the author from the internal s... more A natural partial order on the set of prime numbers was derived by the author from the internal symmetries of the primary finite fields [1], independently of [2], who investigated Pratt trees [3] used for primality tests. It leads to a correspondence with the Hopf algebra of rooted trees, and as an application, to an alternative approach to the Prime Number Theorem.
International Journal of Mathematics and Mathematical Sciences, 2003
A nonassociative algebra endowed with a Lie bracket, called atorsion algebra, is viewed as an alg... more A nonassociative algebra endowed with a Lie bracket, called atorsion algebra, is viewed as an algebraic analog of a manifold with an affine connection. Its elements are interpreted as vector fields and its multiplication is interpreted as a connection. This provides a framework for differential geometry on a formal manifold with a formal connection. A torsion algebra is a natural generalization of pre-Lie algebras which appear as the torsionless case. The starting point is the observation that the associator of a nonassociative algebra is essentially the curvature of the corresponding Hochschild quasicomplex. It is a cocycle, and the corresponding equation is interpreted as Bianchi identity. The curvature-associator-monoidal structure relationships are discussed. Conditions on torsion algebras allowing to construct an algebra of functions, whose algebra of derivations is the initial Lie algebra, are considered. The main example of a torsion algebra is provided by the pre-Lie algeb...
Page 1. arXiv:submit/0309981 [math-ph] 31 Aug 2011 REMARKS ON PHYSICS AS NUMBER THEORY LM IONESCU... more Page 1. arXiv:submit/0309981 [math-ph] 31 Aug 2011 REMARKS ON PHYSICS AS NUMBER THEORY LM IONESCU Abstract. There are numerous indications that Physics, at its foundations, is algebraic Number Theory. The ...
The article is an overview of the role of graph complexes in the Feynman path integral quantizati... more The article is an overview of the role of graph complexes in the Feynman path integral quantization. The underlying mathematical language is that of PROPs and operads, and their representations. The sum over histories approach, the Feynman Legacy, is the bridge between quantum physics and quantum computing, pointing towards a deeper understanding of the fundamental concepts of space, time and information. 1
Complex periods are algebraic integrals over complex algebraic domains, also appearing as Feynman... more Complex periods are algebraic integrals over complex algebraic domains, also appearing as Feynman integrals and multiple zeta values. The Grothendieck-de Rham period isomorphisms for p-adic algebraic varieties defined via Monski-Washnitzer cohomology, is briefly reviewed. The relation to various p-adic analogues of periods are considered, and their relation to Buium-Manin arithmetic differential equations.
We review several known categorification procedures, and introduce a functorial categorification ... more We review several known categorification procedures, and introduce a functorial categorification of group extensions with applications to non-abelian group cohomology. Categorification of acyclic models and of topological spaces are briefly mentioned.
The Betti-de Rham period isomorphism ("Abelian Geometry") is related to algebraic funda... more The Betti-de Rham period isomorphism ("Abelian Geometry") is related to algebraic fundamental group (Anabelian Geometry), in analogy with the classical context of Hurewicz Theorem. To investigate this idea, the article considers an "Abstract Galois Theory", as a separated abstract structure from, yet compatible with, the Theory of Schemes, which has its historical origin in Commutative Algebra and motivation in the early stages of Algebraic Topology. The approach to Motives via Deformation Theory was suggested by Kontsevich as early as 1999, and suggests Formal Manifolds, with local models formal pointed manifolds, as the source of motives, and perhaps a substitute for a "universal Weil cohomology". The proposed research aims to gain additional understanding of periods via a concrete project, the discrete algebraic de Rham cohomology, a follow-up of author's previous work. The connection with Arithmetic Gauge Theory should provide additional intuiti...
Periods are numbers represented as integrals of rational functions over algebraic domains. A surv... more Periods are numbers represented as integrals of rational functions over algebraic domains. A survey of their elementary properties is provided. Examples of periods includes Feynman Integrals from Quantum Physics and Multiple Zeta Values from Number Theory. But what about finite characteristic, via the global-to-local principle? We include some considerations regarding periods and Jacobi sums, the analog of Veneziano amplitudes in String Theory.
Recently, GUTs based on the exceptional Lie algebras attempt unification of interactions of the S... more Recently, GUTs based on the exceptional Lie algebras attempt unification of interactions of the Standard Model as a gauge field theory, e.g. Garrett Lisi's E8-TOE. But the modern growing trend in quantum physics is based on the Quantum Information Processing paradigm (QIP). The present proposal will develop the Qubit Model, a QIP analog of the Quark Model within the SM framework. The natural principle that "quantum interactions should be discrete", technically meaning the reduction of the gauge group to finite subgroups of SO(3)/SU(2), implies that qubit-frames (3D-pixels), playing the role of baryons, have the Platonic symmetries as their Klein Geometry (Three generations of flavors): T,O,I, and hence their "doubles", the binary point groups are the root systems E6,7,8 of the exceptional Lie algebras, which control their Quantum Dynamics. The Qubit Model conceptually reinterprets the experimental heritage modeled into the SM, and has clear prospects of expla...
We review Pólya vector fields associated to holomorphic functions as an important pedagogical too... more We review Pólya vector fields associated to holomorphic functions as an important pedagogical tool for making the complex integral understandable to the students, briefly mentioning its use in other dimensions. Techniques of differential geometry are then used to refine the study of holomorphic functions from a metric (Riemannian), affine differential or differential viewpoint. We prove that the only nontrivial holomorphic functions, whose Pólya vector field is torse-forming in the cannonical geometry of the plane, are the special Möbius transformations of the form f(z)=b(z+d)−1. We define and characterize several types of affine connections, related to the parallelism of Pólya vector fields. We suggest a program for the classification of holomorphic functions, via these connections, based on the various indices of nullity of their curvature and torsion tensor fields.
uantum phenomena reflect resonance at the very fundamental level of a network processing Quantum ... more uantum phenomena reflect resonance at the very fundamental level of a network processing Quantum Information. The term “particle” is a hindrance preventing the shift to a new paradigm. A top-level mental-visual model for “elementary particles” and some other quantum experiments is needed, and provided: a foamy Riemann Surface is interpreted as a Quantum Chip, having both fermionic parameters (sources as integrals), and bosonic character (propagating energy-momentum).
The generic relativistic version of a particle-field theory, with non-isotropic sources, includes... more The generic relativistic version of a particle-field theory, with non-isotropic sources, includes a Gravity force perturbation of Coulombian Force, with the usual Magnetic Force resulting from Lorentz transformations. The quark model of Standard Model, with fractional charge structure of nucleons enveloped by electronic clouds, mandates such nonisotropic charges. Dynamic Nuclear Orientation (DNO), via electronic spin and LScoupling, allows to invert the population of low energy Gravitational attraction states, and achieve Gravity Control. The 1994 scientific experiment of Dr. Frederick Alzofon has confirmed Gravity Control can be achieved via DNO. Other researchers have contributed in the same general direction of unifying Electromagnetism and Gravity, supporting the non-isotropic charge concept, including Paul LaViolette, author of Subquantum Kinetics.
Kontsevich recently generalized Tian-Todorov Theorem regarding the structure of the Kuranish spac... more Kontsevich recently generalized Tian-Todorov Theorem regarding the structure of the Kuranish space of deformations of a Kahler manifold with trivial canonical bundle. An alternative proof was given using a general result regarding the smoothness of moduli space of formal deformations, based on BV-algebra resolutions. From this, various other generalizations ensue and a conjecture relating the dimension of the tangent space of formal deformations and the first non-trivial Hodge number h(n− 1, 1). Additional details are provided, together with a proposed explanation regarding the above conjecture. Related considerations regarding mirror symmetry and motives of Calabi-Yau manifolds are included, based on the idea of complexifying TQFTs, modeled after Chow pure motives.
Gravity is not a fundamental force. Alzofon’s Thermodynamic Gravity Theory is derived from Qubit ... more Gravity is not a fundamental force. Alzofon’s Thermodynamic Gravity Theory is derived from Qubit Model, an upgrade of the Quark Model within the Standard Model. Alzofon’s experiment is discussed: pros and cons. At the level of the Standard Model, Gravity is a result of the structure of the electric charges of quarks in nucleons, subject to Platonic symmetry and lack of parity invariance, related to CP-violation, due to the dihedral group as the Quantum Mirror Symmetry group. The AGNUE experiment performed at Hathaway Research International is briefly explained. It is designed to test Alzofon’s Theory. A glimpse of Gravity Control and what inertial mass is are presented. Further R&D will be funded under the upcoming Kickstarter Gravity [1].
To better understand and investigate Kontsevich-Zagier conjecture on abstract periods, we conside... more To better understand and investigate Kontsevich-Zagier conjecture on abstract periods, we consider the case of algebraic Riemann Surfaces representable by Belyi maps. The category of morphisms of Belyi ramified maps and Dessins D'Enfant, will be investigated in search of an analog for periods, of the Ramification Theory for decomposition of primes in field extensions, controlled by theirs respective algebraic Galois groups. This suggests a relation between the theory of (cohomological, Betti-de Rham) periods and Grothendieck's Anabelian Geometry [1] (homotopical/ local systems), towards perhaps an algebraic analog of Hurwitz Theorem, relating the the algebraic de Rham cohomology and algebraic fundamental group, both pioneered by A. Grothendieck. There seem to be good prospects of better understanding the role of absolute Galois group in the physics context of scattering amplitudes and Multiple Zeta Values, with their incarnation as Chen integrals on moduli spaces, as studied by Francis Brown, since the latter are a homotopical analog of de Rham Theory. The research will be placed in the larger context of the ADE-correspondence, since, for example, orbifolds of finite groups of rotations have crepant resolutions relevant in String Theory, while via Cartan-Killing Theory and exceptional Lie algebras, they relate to TOEs. Relations with the author's reformulation of cohomology of cyclic groups as a discrete analog of de Rham cohomology [2] and the arithmetic Galois Theory [3] will provide a purely algebraic toy-model of the said algebraic homology/homotopy group theory of Grothendieck as part of Anabelian Geometry. It will allow an elementary investigation of the main concepts defining periods and algebraic fundamental group, together with their conceptual relation to algebraic numbers and Galois groups. The Riemann surfaces with Platonic tessellations, especially the Hurwitz surfaces, are related to the finite Hopf sub-bundles with symmetries the "exceptional" Galois groups P SL 2 (F p), p = 5, 7, 11. The corresponding Platonic Trinity 5, 7, 11/T OI/E678 leads to connections with ADE-correspondence, and beyond, e.g. TOEs and ADEX-Theory [4]. Quantizing "everything" (cyclotomic quantum phase and finite Platonic-Hurwitz geometry of qubits/baryons) could perhaps be "The Eightfold (Petrie polygon) Way" to finally understand what quark flavors and fermion generations really are.
Journal of High Energy Physics, Gravitation and Cosmology, 2020
Understanding the role of muons in Particle Physics is an important step understanding generation... more 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.
A canonical p-adic Frobenius lift is defined in the context of p-adic numbers, viewed as deformat... more 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.
Is "Gravity" a deformation of "Electromagnetism"? G N m 2 e k C e 2 ≈ 10 −54 ↔ e −1/α ≈ 10 −59. T... more Is "Gravity" a deformation of "Electromagnetism"? G N m 2 e k C e 2 ≈ 10 −54 ↔ e −1/α ≈ 10 −59. Thus "Gravity" emerges already "quantum", in the discrete framework of QID, based on the quantized complex harmonic oscillator: the quantized qubit. All looks promising, but will the details backup this "grand design scheme"? Contents
The current research regarding the Riemann zeros suggests the existence of a non-trivial algebrai... more The current research regarding the Riemann zeros suggests the existence of a non-trivial algebraic/analytic structure on the set of Riemann zeros [1, 2, 3]. The duality between primes and Riemann zeta function zeros suggests some new goals and aspects to be studied: adelic duality and the POSet of prime numbers. The article presents computational evidence of the structure of the imaginary parts t of the non-trivial zeros of the Riemann zeta function ρ = 1/2 + it, called in this article the Riemann Spectrum, using the study of their distribution, as in [3]. The novelty represents in considering the associated characters p it , towards an algebraic point of view, than rather in the sense of Analytic Number Theory. This structure is tentatively interpreted in terms of adelic characters, and the duality of the rationals. Second, the POSet structure of prime numbers studied in [4], is tentatively mirrored via duality in the Riemann spectrum. A direct study of the convergence of their Fourier series, along Pratt trees, is proposed. Further considerations, relating the Riemann Spectrum, adelic characters and distributions, in terms of Hecke idelic characters, local zeta integrals (Mellin transform) and ω-eigen-distributions, are explored following [11].
A natural partial order on the set of prime numbers was derived by the author from the internal s... more A natural partial order on the set of prime numbers was derived by the author from the internal symmetries of the primary finite fields [1], independently of [2], who investigated Pratt trees [3] used for primality tests. It leads to a correspondence with the Hopf algebra of rooted trees, and as an application, to an alternative approach to the Prime Number Theorem.
International Journal of Mathematics and Mathematical Sciences, 2003
A nonassociative algebra endowed with a Lie bracket, called atorsion algebra, is viewed as an alg... more A nonassociative algebra endowed with a Lie bracket, called atorsion algebra, is viewed as an algebraic analog of a manifold with an affine connection. Its elements are interpreted as vector fields and its multiplication is interpreted as a connection. This provides a framework for differential geometry on a formal manifold with a formal connection. A torsion algebra is a natural generalization of pre-Lie algebras which appear as the torsionless case. The starting point is the observation that the associator of a nonassociative algebra is essentially the curvature of the corresponding Hochschild quasicomplex. It is a cocycle, and the corresponding equation is interpreted as Bianchi identity. The curvature-associator-monoidal structure relationships are discussed. Conditions on torsion algebras allowing to construct an algebra of functions, whose algebra of derivations is the initial Lie algebra, are considered. The main example of a torsion algebra is provided by the pre-Lie algeb...
Page 1. arXiv:submit/0309981 [math-ph] 31 Aug 2011 REMARKS ON PHYSICS AS NUMBER THEORY LM IONESCU... more Page 1. arXiv:submit/0309981 [math-ph] 31 Aug 2011 REMARKS ON PHYSICS AS NUMBER THEORY LM IONESCU Abstract. There are numerous indications that Physics, at its foundations, is algebraic Number Theory. The ...
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