Abstract
A canonical p-adic Frobenius lift is defined in the context of p-adic numbers, viewed as deformations of the corresponding finite field. Applications to p-adic periods are considered, including to the classical Euler gamma and beta functions and their p-adic analogues, from a cohomological point of view. Connections between various methods for computing scattering amplitudes are related to the moduli space problem and period domains.
Highlights
Algebraic integrals called periods [1] [2] [3]1, are a new class of numbers extending algebraic numbers and bridging quantum physics and mathematics together, in a way yet to be better understood [5] [6] [7].Presenting p-adic numbers as deformations of finite fields allows a better understanding of Frobenius lifts and their connection with p-derivations in the sense of Buium [8]
That Deformation Theory is a natural generalization of Lie Theory from the framework of Lie algebras/Lie groups to quite general algebraic structure [35], including the modern mathematics of Quantum Groups, and should be enough incentive for “prefering” to advertise p-adic “analysis” as p-adic deformation theory
The advantage of the p-adics side, of being graded mathematics, is that numbers can be treated as functions: the historical connection with the integers may be broken, and p-adic “numbers” are just h-adic Laurent series, so that the corresponding fields Qq may be treated both analytically and as number fields
Summary
Algebraic integrals called periods [1] [2] [3]1, are a new class of numbers extending algebraic numbers and bridging quantum physics and mathematics together, in a way yet to be better understood [5] [6] [7]. Presenting p-adic numbers as deformations of finite fields allows a better understanding of Frobenius lifts and their connection with p-derivations in the sense of Buium [8]. In this way “numbers are functions”, as recognized before [9], allowing to view initial structure deformation problems as arithmetic differential equations as in [10], and providing a cohomological interpretation to Buium calculus via Hochschild cohomology which controls deformations of algebraic structures. A deformation quantization of the theory of finite fields should be beneficial for a brainstorming translation of the “good concepts” from “continuous mathematical-physics over complex numbers, to the discrete graded case of p-adics and adeles”. ( ) (a0, 0) ∗ (b0, 0) =a0 + b0, cp (a0,b0 ) , where we have dropped the subscript from + p
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