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p-Adic Dirac Equations and the Jackiw-Rebbi Model

W. A. Zúñiga-Galindo

Abstract

We present a new p-adic version of the Jackiw-Rebbi model. In the new model, the real numeric line is replaced by a p-adic line (the field of p-adic numbers Q_{p}), and the Dirac Hamiltonian is replaced by a non-local operator acting on complex-valued functions defined on Q_{p}. These Hamiltonians admit localized wavefunctions and allow long-range interactions, so spooky action at a distance is allowed. These features are not present in the original model. The new model gives the same predictions as the standard one. The p-adic line serves as a discrete model for the physical space; in this type of space, non-locality emerges naturally.

p-Adic Dirac Equations and the Jackiw-Rebbi Model

Abstract

We present a new p-adic version of the Jackiw-Rebbi model. In the new model, the real numeric line is replaced by a p-adic line (the field of p-adic numbers Q_{p}), and the Dirac Hamiltonian is replaced by a non-local operator acting on complex-valued functions defined on Q_{p}. These Hamiltonians admit localized wavefunctions and allow long-range interactions, so spooky action at a distance is allowed. These features are not present in the original model. The new model gives the same predictions as the standard one. The p-adic line serves as a discrete model for the physical space; in this type of space, non-locality emerges naturally.
Paper Structure (21 sections, 92 equations)

This paper contains 21 sections, 92 equations.

Table of Contents

  1. Introduction
  2. Basic facts on $p$-adic analysis
  3. The field of $p$-adic numbers
  4. Topology of $\mathbb{Q}_{p}$
  5. The Haar measure
  6. The Bruhat-Schwartz space
  7. The Fourier transform
  8. Multiplicative characters
  9. Twisted Taibleson-Vladimirov operator
  10. 1D $p$-Adic Jackiw-Rebbi model
  11. The wavefunction at the interface
  12. The wavefunction at the bulk
  13. 1D Jackiw-Rebbi model: Archimedean versus $p$-Adic
  14. Configuration space
  15. Hamiltonians
  16. ...and 6 more sections