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45
From Physics to Number theory via Noncommutative Geometry, II  Chapter 2: Renormalization, The RiemannHilbert correspondence, and . . .
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Renormalization and motivic Galois theory
 International Math. Research Notices
"... Abstract. We investigate the nature of divergences in quantum field theory, showing that they are organized in the structure of a certain “ motivic Galois group ” U ∗ , which is uniquely determined and universal with respect to the set of physical theories. The renormalization group can be identifie ..."
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Cited by 24 (12 self)
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Abstract. We investigate the nature of divergences in quantum field theory, showing that they are organized in the structure of a certain “ motivic Galois group ” U ∗ , which is uniquely determined and universal with respect to the set of physical theories. The renormalization group can be identified canonically with a one parameter subgroup of U ∗. The group U ∗ arises through a Riemann–Hilbert correspondence. Its representations classify equisingular flat vector bundles, where the equisingularity condition is a geometric formulation of the fact that in quantum field theory the counterterms are independent of the choice of a unit of mass. As an algebraic group scheme, U ∗ is a semidirect product by the multiplicative group Gm of a prounipotent group scheme whose Lie algebra is freely generated by one generator in each positive integer degree. There is a universal singular frame in which all divergences disappear. When computed as iterated integrals, its coefficients are certain rational numbers that appear in the local index formula of Connes–Moscovici. When working with formal Laurent series over Q, the data of equisingular flat vector bundles
Picard groups in Poisson geometry
 MOSCOW MATH. J
, 2003
"... We study isomorphism classes of symplectic dual pairs P ← S → P, where P is an integrable Poisson manifold, S is symplectic, and the two maps are complete, surjective Poisson submersions with connected and simplyconnected fibres. For fixed P, these Morita selfequivalences of P form a group Pic(P) ..."
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Cited by 12 (2 self)
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We study isomorphism classes of symplectic dual pairs P ← S → P, where P is an integrable Poisson manifold, S is symplectic, and the two maps are complete, surjective Poisson submersions with connected and simplyconnected fibres. For fixed P, these Morita selfequivalences of P form a group Pic(P) under a natural “tensor product ” operation. Variants of this construction are also studied, for rings (the origin of the notion of Picard group), Lie groupoids, and symplectic groupoids.
Quantum fields and motives
 J. Geom. Phys
, 2006
"... The main idea of renormalization is to correct the original Lagrangian of a quantum field theory by an infinite series of counterterms, labelled by the Feynman graphs that encode the combinatorics of the perturbative expansion of the theory. These counterterms have the effect of cancelling the ultra ..."
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Cited by 11 (2 self)
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The main idea of renormalization is to correct the original Lagrangian of a quantum field theory by an infinite series of counterterms, labelled by the Feynman graphs that encode the combinatorics of the perturbative expansion of the theory. These counterterms have the effect of cancelling the ultraviolet divergences. Thus, in the procedure of perturbative renormalization, one introduces a counterterm C(Γ) in the initial Lagrangian for every divergent one particle irreducible (1PI) Feynman diagram Γ. In the case of a renormalizable theory, all the necessary counterterms C(Γ) can be obtained by modifying the numerical parameters that appear in the original Lagrangian. It is possible to modify these parameters and replace them by (divergent) series, since they are not observable, unlike actual physical quantities that have to be finite. One of the fundamental difficulties with any renormalization procedure is a systematic treatment of nested and overlapping divergences in multiloop diagrams. Dimensional regularization and minimal subtraction. One of the most effective renormalization techniques in quantum field theory is
Meeting Descartes and Klein somewhere in a noncommutative space
 Highlights of Mathematical Physics
"... Abstract. We combine the coordinate method and Erlangen program in the framework of noncommutative geometry through an investigation of symmetries of noncommutative coordinate algebras. As the model we use the coherent states construction and the wavelet transform in functional spaces. New examples ..."
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Cited by 9 (7 self)
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Abstract. We combine the coordinate method and Erlangen program in the framework of noncommutative geometry through an investigation of symmetries of noncommutative coordinate algebras. As the model we use the coherent states construction and the wavelet transform in functional spaces. New examples are a three dimensional spectrum of a nonnormal matrix and a quantisation procedure from symplectomorphisms.
Category Theory: an abstract setting for analogy and comparison
 In: What is Category Theory? Advanced Studies in Mathematics and Logic, Polimetrica Publisher
, 2006
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The Clifford Algebra Approach to Quantum Mechanics A: The Schrödinger and Pauli Particles
"... PACS numbers: Using a method based on Clifford algebras taken over the reals, we present here a fully relativistic version of the Bohm model for the Dirac particle. This model is different from the one originally proposed by Bohm and Hiley and by Doran and Lasenby. We obtain exact expressions for th ..."
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Cited by 5 (4 self)
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PACS numbers: Using a method based on Clifford algebras taken over the reals, we present here a fully relativistic version of the Bohm model for the Dirac particle. This model is different from the one originally proposed by Bohm and Hiley and by Doran and Lasenby. We obtain exact expressions for the Bohm energymomentum density, a relativistic quantum HamiltonJacobi for the conservation of energy which includes an expression for the quantum potential and a relativistic time development equation for the spin vectors of the particle. We then show that these reduce to the corresponding nonrelativistic expressions for the Pauli particle which have already been derived by Bohm, Schiller and Tiomno and in more general form by Hiley and Callaghan. In contrast to the original presentations, there is no need to appeal to classical mechanics at any stage of the development of the formalism. All the results for the Dirac, Pauli and Schrödinger cases are shown to emerge respectively from the hierarchy of Clifford algebras C13, C30, C01 taken over the reals as Hestenes has already argued. Thus quantum mechanics is emerging from one mathematical structure with no need to appeal to an external Hilbert space with wave functions.
Relativity without tears
 Acta Phys.Polon. B
, 2008
"... Special relativity is no longer a new revolutionary theory but a firmly established cornerstone of modern physics. The teaching of special relativity, however, still follows its presentation as it unfolded historically, trying to convince the audience of this teaching that Newtonian physics is natur ..."
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Cited by 4 (3 self)
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Special relativity is no longer a new revolutionary theory but a firmly established cornerstone of modern physics. The teaching of special relativity, however, still follows its presentation as it unfolded historically, trying to convince the audience of this teaching that Newtonian physics is natural but incorrect and special relativity is its paradoxical but correct amendment. I argue in this article in favor of logical instead of historical trend in teaching of relativity and that special relativity is neither paradoxical nor correct (in the absolute sense of the nineteenth century) but the most natural and expected description of the real spacetime around us valid for all practical purposes. This last circumstance constitutes a profound mystery of modern physics better known as the cosmological constant problem. PACS numbers: 03.30.+p Preface “To the few who love me and whom I love – to those who feel rather than to those who think – to the dreamers and those who put faith in dreams as in the only realities – I offer this Book of Truths, not in its character of TruthTeller, but for the Beauty that abounds in its Truth; constituting it true. To these I present the composition as an ArtProduct alone; let us say as a Romance; or, if I be not urging too lofty a claim, as a Poem. What I here propound is true: – therefore it cannot die: – or if by any means it be now trodden down so that it die, it will rise again ”to the Life Everlasting.” Nevertheless, it is a poem only that I wish this work to be judged after I am dead ” [1]. 1.
The Dirichlet Hopf algebra of arithmetics
 JOURNAL OF KNOT THEORY AND ITS RAMIFICATIUONS
, 2006
"... Many constructs in mathematical physics entail notational complexities, deriving from the manipulation of various types of index sets which often can be reduced to labelling by various multisets of integers. In this work, we develop systematically the “Dirichlet Hopf algebra of arithmetics ” by dual ..."
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Cited by 4 (3 self)
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Many constructs in mathematical physics entail notational complexities, deriving from the manipulation of various types of index sets which often can be reduced to labelling by various multisets of integers. In this work, we develop systematically the “Dirichlet Hopf algebra of arithmetics ” by dualizing the addition and multiplication maps. Then we study the additive and multiplicative antipodal convolutions which fail to give rise to Hopf algebra structures, but form only a weaker Hopf gebra obeying a weakened homomorphism axiom. A careful identification of the algebraic structures involved is done featuring subtraction, division and derivations derived from coproducts and chochains using branching operators. The consequences of the weakened structure of a Hopf gebra on cohomology are explored, showing this has major impact on number theory. This features multiplicativity versus complete multiplicativity of number theoretic arithmetic functions. The deficiency of not being a Hopf algebra is then cured by introducing an ‘unrenormalized’ coproduct and an ‘unrenormalized ’ pairing. It is then argued that exactly the failure of the homomorphism property (complete multiplicativity) for noncoprime integers is a blueprint for the problems in quantum field theory (QFT) leading to the need for renormalization. Renormalization turns out to be the morphism from the algebraically sound Hopf algebra to the physical and number