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Physics as Generalized Number Theory I: p-Adic Physics and Number Theoretic Universality (by Matti Pitkänen)

Abstract: Physics as a generalized number theory program involves three threads: various p-adic physics and their fusion together with real number based physics to a larger structure, the attempt to understand basic physics in terms of classical number fields (in particular, identifying associativity condition as the basic dynamical principle), and infinite primes whose construction is formally analogous to a repeated second quantization of an arithmetic quantum field theory. In this article p-adic physics and the technical problems relates to the fusion of p-adic physics and real physics to a larger structure are discussed. The basic technical problems relate to the notion of definite integral both at space-time level, imbedding space level and the level of WCW (the "world of classical worlds"). The expressibility of WCW as a union of symmetric spaces leads to a proposal that harmonic analysis of symmetric spaces can be used to define various integrals as sums over Fourier components. This leads to the proposal the p-adic variant of symmetric space is obtained by a algebraic continuation through a common intersection of these spaces, which basically reduces to an algebraic variant of coset space involving algebraic extension of rationals by roots of unity. This brings in the notion of angle measurement resolution for given p-adic prime p. Also a proposal how one can complete the discrete version of symmetric space to a continuous p-adic versions emerges and means that each point is effectively replaced with the p-adic variant of the symmetric space identifiable as a p-adic counterpart of the real discretization volume so that a fractal p-adic variant of symmetric space results. http://prespacetime.com/index.php/pst/article/view/52

Physics as Generalized Number Theory II: Classical Number Fields (by Matti Pitkänen)

Abstract: Physics as a generalized number theory program involves three threads: various p-adic physics and their fusion together with real number based physics to a larger structure, the attempt to understand basic physics in terms of classical number fields discussed in this article, and infinite primes whose construction is formally analogous to a repeated second quantization of an arithmetic quantum field theory. In this article the connection between standard model symmetries and classical number fields is discussed. The basis vision is that the geometry of the infinite-dimensional WCW ("world of classical worlds") is unique from its mere existence. This leads to its identification as union of symmetric spaces whose Kähler geometries are fixed by generalized conformal symmetries. This fixes space-time dimension and the decomposition M4 X S and the idea is that the symmetries of the Kähler manifold S make it somehow unique. The motivating observations are that the dimensions of classical number fields are the dimensions of partonic 2-surfaces, space-time surfaces, and imbedding space and M8 can be identified as hyper-octonions- a sub-space of complexified octonions obtained by adding a commuting imaginary unit. This stimulates some questions. http://prespacetime.com/index.php/pst/article/view/53

Physics as Generalized Number Theory III: Innite Primes (by Matti Pitkänen)

Abstract: Physics as a generalized number theory program involves three threads: various p-adic physics and their fusion together with real number based physics to a larger structure, the attempt to understand basic physics in terms of classical number fields, and infinite primes discussed in this article. The construction of infinite primes is formally analogous to a repeated second quantization of an arithmetic quantum field theory by taking the many particle states of previous level elementary particles at the new level. Besides free many particle states also the analogs of bound states appear. In the representation in terms of polynomials the free states correspond to products of first order polynomials with rational zeros. Bound states correspond to n-th order polynomials with non-rational but algebraic zeros. The construction can be generalized to classical number fields and their complexifications obtained by adding a commuting imaginary unit. Special class corresponds to hyper-octonionic primes for which the imaginary part of ordinary octonion is multiplied by the commuting imaginary unit so that one obtains a sub-space M8 with Minkowski signature of metric. Also in this case the basic construction reduces to that for rational or complex rational primes and more complex primes are obtained by acting using elements of the octonionic automorphism group which preserve the complex octonionic integer property. http://prespacetime.com/index.php/pst/article/view/55

The Geometry of CP2 and its Relationship to Standard Model (by Matti Pitkänen)

Abstract: This appendix contains basic facts about CP2 as a symmetric space and Kähler manifold. The coding of the standard model symmetries to the geometry of CP2, the physical interpretation of the induced spinor connection in terms of electro-weak gauge potentials, and basic facts about induced gauge fields are discussed. http://prespacetime.com/index.php/pst/article/view/56