p-Adic TGD: Mathematical Ideas

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The mathematical basis of p-adic Higgs mechanism discussed in papers hep-th@xxx.lanl.gov 9410058-62 is considered in this paper. The basic properties of p-adic numbers, of their algebraic extensions and the so called canonical identification between positive real numbers and p-adic numbers are described. Canonical identification induces p-adic topology and differentiable structure on real axis and allows definition of definite integral with physically desired properties. p-Adic numbers together with canonical identification provide analytic tool to produce fractals. Canonical identification makes it possible to generalize probability concept, Hilbert space concept, Riemannian metric and Lie groups to p-adic context. Conformal invariance generalizes to arbitrary dimensions since p-adic numbers allow algebraic extensions of arbitrary dimension. The central theme of all developments is the existence of square root, which forces unique algebraic extension with dimension D=4 and D=8 for p>2 and p=2 respectively. This in turn implies that the dimensions of p-adic Riemann spaces are multiples of 4 in p>2 case and of 8 in p=2 case.