Authors: Matti Pitkänen
The focus of this book is the number theoretical vision about physics. This vision involves three loosely related parts.
Also a general vision about preferred extremals of Kähler action emerges. The basic idea is that imbedding space allows octonionic structure and that field equations in a given space-time region reduce to the associativity of the tangent space or normal space: space-time regions should be quaternionic or co-quaternionic. The first formulation is in terms of the octonionic representation of the imbedding space Clifford algebra and states that the octonionic gamma "matrices" span a complexified quaternionic sub-algebra. Another formulation is in terms of octonion real-analyticity. Octonion real-analytic function f is expressible as f=q1+Iq2, where qi are quaternions and I is an octonionic imaginary unit analogous to the ordinary imaginary unit. q2 (q1) would vanish for quaternionic (co-quaternionic) space-time regions. The local number field structure of the octonion real-analytic functions with composition of functions as additional operation would be realized as geometric operations for space-time surfaces. The conjecture is that these two formulations are equivalent.
Number theoretical vision suggests that infinite hyper-octonionic or -quaternionic primes could could correspond directly to the quantum numbers of elementary particles and a detailed proposal for this correspondence is made. Furthermore, the generalized eigenvalue spectrum of the Chern-Simons Dirac operator could be expressed in terms of hyper-complex primes in turn defining basic building bricks of infinite hyper-complex primes from which hyper-octonionic primes are obtained by dicrete SU(3) rotations performed for finite hyper-complex primes.
Besides this holy trinity I will discuss in the first part of the book loosely related topics such as the relationship between infinite primes and non-standard numbers.
Second part of the book is devoted to the mathematical formulation of the p-adic TGD. The p-adic counterpart of integration is certainly the basic mathematical challenge. Number theoretical universality and the notion of algebraic continuation from rationals to various continuous number fields is the basic idea behind the attempts to solve the problems. p-Adic integration is also a central problem of modern mathematics and the relationship of TGD approach to motivic integration and cohomology theories in p-adic numberfields is discussed.
The correspondence between real and p-adic numbers is second fundamental problem. The key problem is to understand whether and how this correspondence could be at the same time continuous and respect symmetries at least in discrete sense. The proposed explanation of Shnoll effect suggests that the notion of quantum rational number could tie together p-adic physics and quantum groups and could allow to define real-p-adic correspondence satisfying the basic conditions.
The third part is develoted to possible applications. Included are category theory in TGD framework; TGD inspired considerations related to Riemann hypothesis; topological quantum computation in TGD Universe; and TGD inspired approach to Langlands program.
Comments: 856 Pages. Updating
[v1] 3 Aug 2009
[v2] 14 Oct 2009
[v3] 2011-12-04 00:40:48
[v4] 2012-01-30 22:31:50
[v5] 2012-03-16 02:59:55
[v6] 2012-07-01 08:34:30
[v7] 2012-07-03 22:24:04
[v8] 2012-08-19 04:18:56
[v9] 2012-10-11 23:24:48
[vA] 2012-11-13 01:16:00
[vB] 2013-01-01 07:56:20
[vC] 2013-02-17 01:36:56
[vD] 2013-09-09 07:08:05
[vE] 2014-09-26 04:44:22
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