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On non-Archimedean quantum mathematics and non-Archimedean (quantum) computation

Nikolaj Glazunov

TL;DR

This survey links non-Archimedean mathematics with quantum mathematics and computation, weaving together $p$-adic numbers, valuations, and adelic structures with quantum concepts such as qubits, spin, and algebraic quantum circuits. It surveys both the mathematical foundations (analyticity, path integrals, resurgence, noncommutative geometry) and the arithmetic-quantum interface ($p$-adic codes, adeles, $S$-algebraic groups, Weil cohomology) to outline a non-Archimedean framework for quantum theory. Key threads include the translation of local-to-global principles via adeles, the algebraic structuring of quantum circuits through logic and modular lattices, and the incorporation of Connes–Consani–Moscovici ideas of semilocal adelic operators and spectral interpretations of zeta zeros. The work emphasizes potential applications in arithmetic physics, quantum computation, and the broader synthesis of number theory with quantum mathematics, highlighting resurgence and absolute algebraic geometry as mathematical bridges.

Abstract

We consider selected aspects of (non-Archimedean) quantum mathematics and non-Archimedean (quantum) computation.

On non-Archimedean quantum mathematics and non-Archimedean (quantum) computation

TL;DR

This survey links non-Archimedean mathematics with quantum mathematics and computation, weaving together -adic numbers, valuations, and adelic structures with quantum concepts such as qubits, spin, and algebraic quantum circuits. It surveys both the mathematical foundations (analyticity, path integrals, resurgence, noncommutative geometry) and the arithmetic-quantum interface (-adic codes, adeles, -algebraic groups, Weil cohomology) to outline a non-Archimedean framework for quantum theory. Key threads include the translation of local-to-global principles via adeles, the algebraic structuring of quantum circuits through logic and modular lattices, and the incorporation of Connes–Consani–Moscovici ideas of semilocal adelic operators and spectral interpretations of zeta zeros. The work emphasizes potential applications in arithmetic physics, quantum computation, and the broader synthesis of number theory with quantum mathematics, highlighting resurgence and absolute algebraic geometry as mathematical bridges.

Abstract

We consider selected aspects of (non-Archimedean) quantum mathematics and non-Archimedean (quantum) computation.
Paper Structure (24 sections, 5 equations)

This paper contains 24 sections, 5 equations.

Table of Contents

  1. $p$-adic numbers
  2. $p$-adic is not Archimedean
  3. $p$-adic absolut value
  4. Functional case
  5. Product formula
  6. Adeles
  7. $p$-adic codes and their implementation
  8. Classical quantum mechanics
  9. Spin in classical quantum mechanics and its properties
  10. Spintronics
  11. Algebraic quantum circuits and exponential sums
  12. On quantum logic
  13. Modular structure
  14. On quantum circuits
  15. Analytification of the field of $p$-adic numbers
  16. ...and 9 more sections

Theorems & Definitions (6)

  • Example 1
  • Definition 1
  • Example 2
  • Remark 1
  • Example 3
  • Remark 2