**Authors:** Matti Pitkänen

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 spacesleads 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 Δφ = 2π/p^{n} 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.
If the Kähler geometry of WCW is expressible in terms of rational or algebraic functions, it
can in principle be continued the p-adic context. One can however consider the possibility that
that the integrals over partonic 2-surfaces defining
ux Hamiltonians exist p-adically as Riemann
sums. This requires that the geometries of the partonic 2-surfaces effectively reduce to finite
sub-manifold geometries in the discretized version of δM_{+}^{4}. If Kähler action is required
to exist p-adically same kind of condition applies to the space-time surfaces themselves. These
strong conditions might make sense in the intersection of the real and p-adic worlds assumed to
characterized living matter.

**Comments:** 51 Pages.

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[v1] 16 Jun 2010

[v2] 2012-01-30 21:58:07

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