There are two basic approaches to the construction of quantum TGD. The
first approach relies on the vision of quantum physics as infinite-dimensional
Kahler geometry for the "world of classical worlds" identified as the space of
3-surfaces in in certain 8-dimensional space. Essentially a generalization of the
Einstein's geometrization of physics program is in question.
The second vision identifies physics as a generalized number theory and 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.
1. p-Adic physics and their fusion with real physics
The basic technical problems of the fusion of real physics and various p-adic
physics to single coherent whole 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 coming as = 2=pn 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.....
Keywords: Physics as generalized number theory, p-adic numbers,
ultrametricity, classical number fields, quaternions, octonions,
associativity, prime number, arithmetic quantum field theory.
