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A Hydrodynamics Formulation for a Nonlinear Dirac Equation

Joan Morrill i Gavarró, Michael Westdickenberg

Abstract

We derive a hydrodynamics formulation for a modified Dirac equation with a nonlinear mass term that preserves the homogeneity of the original Dirac equation. The nonlinear Dirac equation admits a symmetric split into the left and right-handed spinor components. It is formulated using Clifford algebra tools. We prove global existence for a regularized equation.

A Hydrodynamics Formulation for a Nonlinear Dirac Equation

Abstract

We derive a hydrodynamics formulation for a modified Dirac equation with a nonlinear mass term that preserves the homogeneity of the original Dirac equation. The nonlinear Dirac equation admits a symmetric split into the left and right-handed spinor components. It is formulated using Clifford algebra tools. We prove global existence for a regularized equation.
Paper Structure (16 sections, 10 theorems, 221 equations)

This paper contains 16 sections, 10 theorems, 221 equations.

Table of Contents

  1. Introduction
  2. Space algebra
  3. Involutions
  4. Inverse elements
  5. Scalar product
  6. Lorentz transforms
  7. Left-sided ideals
  8. Differential operators
  9. Matrix representation
  10. Nonlinear Dirac equation
  11. Plane waves
  12. Conservation laws
  13. Global existence
  14. Hydrodynamics formulation
  15. Hydrodynamic equations
  16. ...and 1 more sections

Key Result

Lemma 2.2

For all $\mathfrak{p} \in \mathrm{Cl}_{3}$ we have the expansion/basic Fierz identity Formula E:FIERZ also holds with $\mathfrak{e}_\mu$ replaced by $\mathfrak{e}^\mu$ and vice versa.

Theorems & Definitions (25)

  • Remark 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Remark 2.5
  • Remark 3.1
  • Lemma 4.1
  • ...and 15 more