[Paper Review] p-Adic and Adelic Superanalysis

Бранко Драгович, Andrei Khrennikov|ArXiv.org|Dec 27, 2005

advanced mathematical theories10 references3 citations

TL;DR

This paper introduces p-adic and adelic superanalysis by extending real, p-adic, and adelic structures to supersymmetry, using a Grassmann algebra with two anticommuting generators. It constructs v-adic superalgebras and superspaces over Q_v (v = ∞ or p), defines norms for superfields, and establishes the foundation for adelic supersymmetry and p-adic/M-theory models.

ABSTRACT

After a brief review of p-adic numbers, adeles and their functions, we consider real, p-adic and adelic superalgebras, superspaces and superanalyses. A concrete illustration is given by means of the Grassmann algebra generated by two anticommuting elements.

Motivation & Objective

• To extend supersymmetry to p-adic and adelic number fields, motivated by the adelic symmetry principle in quantum gravity and string theory.
• To develop a rigorous mathematical framework for p-adic and adelic superanalysis based on Z2-graded Banach superalgebras.
• To construct explicit models of v-adic superspaces using Grassmann algebras with two anticommuting generators over Q_v.
• To define norms on superfields and points in v-adic superspaces, ensuring compatibility with adelic structures.
• To lay the groundwork for future development of p-adic and adelic superstring and M-theory models.

Proposed method

• Constructs the Grassmann algebra G(Q_v : η₁, η₂) over p-adic and real numbers, with η₁, η₂ satisfying η_iη_j = -η_jη_i and η_i² = 0.
• Represents superfields as x = u + vη₁η₂ ∈ Λ₀(Q_v) and θ = αη₁ + βη₂ ∈ Λ₁(Q_v), with u, v, α, β ∈ Q_v.
• Defines v-adic norms ||x||_v and ||θ||_v using |·|_∞ for real and |·|_p for p-adic absolute values.
• Establishes Banach superalgebras Λ(Q_v) by equipping them with the defined norms, ensuring completeness.
• Constructs v-adic superspace points X^(v) = (x^(v), θ^(v)) with norm ||X^(v)||_v combining real and p-adic components.
• Imposes the adelic condition that max{|u_i|_p, |v_i|_p, |α_j|_p, |β_j|_p} ≤ 1 for all but finitely many primes p to ensure adelic compatibility.

Experimental results

Research questions

• RQ1 How can supersymmetry be consistently extended to p-adic and adelic number fields?
• RQ2 What is the structure of a v-adic superalgebra over Q_v (v = ∞ or p) generated by two anticommuting Grassmann variables?
• RQ3 How can norms be defined on superfields and superspace points in p-adic and real settings to ensure completeness?
• RQ4 What conditions ensure adelic compatibility in the construction of superspaces?
• RQ5 How can this framework be generalized to complex-valued superfields and applied to p-adic and adelic superstring theories?

Key findings

• The Grassmann algebra over Q_v with two generators forms a Z2-graded Banach superalgebra under the defined v-adic norms.
• Real and p-adic norms for superfields are explicitly constructed: ||x||_v = |u|_∞ + |v|_∞ for v = ∞, and max{|u|_p, |v|_p} for v = p.
• The norm of a Grassmann superfield θ = αη₁ + βη₂ is ||θ||_v = |α|_∞ + |β|_∞ for v = ∞, and max{|α|_p, |β|_p} for v = p.
• The norm of a v-adic superspace point X^(v) combines real and p-adic components: ||X^(v)||_v = Σ_i(|u_i|_∞ + |v_i|_∞) + Σ_j(|α_j|_∞ + |β_j|_∞) for v = ∞, and max_{i,j} of the absolute values for v = p.
• Adelic superspace requires that the p-adic norms of all coordinates are ≤ 1 for all but finitely many primes, ensuring convergence in the adelic product.
• The framework provides a consistent foundation for p-adic and adelic superanalysis, enabling future development of p-adic and adelic superstring and M-theory models.