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Quantum Cellular Automaton Model for the Generalized Dirac Equation

Xingyou Song

TL;DR

This work constructs a unitary quantum cellular automaton that converges to the (1+1)-dimensional Generalized Dirac Equation with a chiral mass term. By deriving the exact dispersion relation and providing a path-integral interpretation, the authors validate the continuum limit and expose the internal spin dynamics arising from the chiral parameter $\rho$. Numerical simulations show that the mass controls group velocity and Zitterbewegung, while $\rho$ reshapes local spin polarization without altering total probability density, yielding a clear visualization of relativistic spinor dynamics on a lattice. This approach offers a robust framework for quantum simulation of relativistic fermions with tunable chiral properties and highlights the deep connection between quantum walks, unitarity, and relativistic quantum field theory.

Abstract

This study presents a unitary quantum cellular automaton (QCA) that, in the continuum limit, converges to the (1+1)-dimensional Generalized Dirac Equation (GDE). We outline the construction of the unitary, discrete-time evolution and derive the model's exact dispersion relation from its eigenvalue spectrum, showing it possesses the correct relativistic limit. We then explore its dynamics through numerical simulations. While the total probability density is shown to be insensitive to the chiral angle parameter $(ρ)$ of the GDE, we demonstrate that this parameter has a profound and directly observable effect on the particle's local spin polarization. By measuring this quantity, we reveal the hidden internal dynamics of the spinor, providing a clear, visual confirmation of the physical consequences of the chiral mass term.

Quantum Cellular Automaton Model for the Generalized Dirac Equation

TL;DR

This work constructs a unitary quantum cellular automaton that converges to the (1+1)-dimensional Generalized Dirac Equation with a chiral mass term. By deriving the exact dispersion relation and providing a path-integral interpretation, the authors validate the continuum limit and expose the internal spin dynamics arising from the chiral parameter . Numerical simulations show that the mass controls group velocity and Zitterbewegung, while reshapes local spin polarization without altering total probability density, yielding a clear visualization of relativistic spinor dynamics on a lattice. This approach offers a robust framework for quantum simulation of relativistic fermions with tunable chiral properties and highlights the deep connection between quantum walks, unitarity, and relativistic quantum field theory.

Abstract

This study presents a unitary quantum cellular automaton (QCA) that, in the continuum limit, converges to the (1+1)-dimensional Generalized Dirac Equation (GDE). We outline the construction of the unitary, discrete-time evolution and derive the model's exact dispersion relation from its eigenvalue spectrum, showing it possesses the correct relativistic limit. We then explore its dynamics through numerical simulations. While the total probability density is shown to be insensitive to the chiral angle parameter of the GDE, we demonstrate that this parameter has a profound and directly observable effect on the particle's local spin polarization. By measuring this quantity, we reveal the hidden internal dynamics of the spinor, providing a clear, visual confirmation of the physical consequences of the chiral mass term.

Paper Structure

This paper contains 19 sections, 23 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Formalism of Quantum Walks on a Lattice
  3. The Hilbert Space of a Lattice Fermion
  4. Unitary Evolution via Operator Splitting
  5. The Coin Operator (C):
  6. The Shift Operator (S):
  7. Observables
  8. Generalized Dirac Equation
  9. From Schrödinger to Dirac: The Need for a Relativistic Equation
  10. The Standard Generalized Dirac Equation in 1+1 Dimensions
  11. Discretization and Unitary Evolution
  12. Eigenvalue Spectrum and the Dispersion Relation
  13. Path Integral Interpretation and Combinatorics of the Walk
  14. The Continuum Limit of the Path Integral.
  15. Numerical Simulations
  16. ...and 4 more sections

Figures (2)

  • Figure 1: (Top): Short-time evolution of a single Dirac particle for different mass parameters. From left to right ($R \in \{0.20, 0.80, 1.20\}$), increasing mass slows the propagation and narrows the light cone. The intricate internal pattern is the characteristic Zitterbewegung. (Bottom): Long-time evolution of the same systems on a periodic lattice. The wave packets are observed to wrap around the boundaries and interfere with themselves, creating a repeating, intricate pattern that demonstrates the effect of the lattice topology.
  • Figure 2: The effect of the chiral angle $\rho$ on the particle's spin polarization, with mass fixed at $R=0.20$. From left to right $\rho \in \{0, \pi/8, \pi/2\}$), increasing $\rho$ dramatically breaks the symmetry of the spin pattern, revealing the hidden dynamics of the GDE. Red indicates spin-up dominance; blue indicates spin-down dominance.