[Paper Review] Some p-Adic Aspects of Superanalysis

Бранко Драгович|ArXiv.org|Jan 8, 2004

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TL;DR

This paper introduces adelic superanalysis by unifying real and p-adic superspaces into a single framework using adeles, enabling a comprehensive approach to p-adic and adelic supersymmetric models. It constructs adelic superspaces and superalgebras via products over real and p-adic components, with functions satisfying adelic convergence conditions, laying a foundation for p-adic and adelic quantum field theories and superstring theory.

ABSTRACT

A brief review of a superanalysis over real and $p$-adic superspaces is presented. Adelic superspace is introduced and an adelic superanalysis, which contains real and $p$-adic superanalysis, is initiated.

Motivation & Objective

• To develop a consistent framework for p-adic superanalysis over p-adic numbers and their algebraic extensions.
• To extend real superspace structures to include p-adic and adelic completions of rational numbers.
• To initiate adelic superanalysis as a unifying theory incorporating both real and p-adic supersymmetric structures.
• To provide a mathematical foundation for p-adic and adelic supersymmetric quantum mechanics and field theories.
• To explore the role of algebraic extensions of p-adic numbers in enriching superanalysis beyond the complex extension of reals.

Proposed method

• Constructs adelic superspace as a restricted product of real and p-adic superspaces over a finite set of primes S.
• Defines adelic superalgebras via products of Grassmann algebras over R, Q_p, and Z_p, with componentwise operations.
• Imposes adelic convergence conditions: |F_p|_p ≤ 1 for all but finitely many primes p on superfields F.
• Uses the adelic ring A = R × ∏_{p∈S} Q_p × ∏_{p∉S} Z_p to define the underlying number system for superspace coordinates.
• Applies the ultrametric property of p-adic metrics to ensure non-Archimedean structure in the p-adic components.
• Extends the adelic product formula for string amplitudes to superstring models via superfields and graded algebras.

Experimental results

Research questions

• RQ1 How can p-adic superanalysis be consistently formulated over p-adic numbers and their algebraic extensions?
• RQ2 What is the structure of an adelic superspace that unifies real and p-adic supersymmetries?
• RQ3 How do adelic superfields behave under the adelic convergence condition |F_p|_p ≤ 1 for almost all p?
• RQ4 Can the adelic product formula for string amplitudes be generalized to superstring amplitudes using superfields?
• RQ5 What role do algebraic extensions of Q_p play in enriching the structure of p-adic superanalysis beyond the complexification of R?

Key findings

• Adelic superspace is defined as a restricted product of real and p-adic superspaces: A^{n,m}_{Λ(A)} = ∪_S (R^n × Λ_1^m(R)) × ∏_{p∈S} (Q_p^n × Λ_1^m(Q_p)) × ∏_{p∉S} (Z_p^n × Λ_1^m(Z_p)).
• Adelic superalgebras are constructed as products of real, p-adic, and p-adic integer Grassmann algebras, forming a commutative superalgebra.
• The adelic supercommutator is defined as a collection of real and p-adic supercommutators, preserving graded commutativity.
• Adelic superfields F(X) = (F_∞, F_2, ..., F_p, ...) satisfy |F_p|_p ≤ 1 for all but finitely many primes p, ensuring adelic convergence.
• The framework allows for the construction of p-adic supersymmetric quantum mechanics using algebraic extensions of Q_p, such as Q_p(√τ) for p ≠ 2 and p = 2.
• The approach generalizes the adelic product formula A_∞(a,b) × ∏_p A_p(a,b) = 1 to superamplitudes via superfields and graded algebras over adeles.