The Dirac equation as a quantum walk: higher dimensions, observational convergence

@article{Arrighi2014TheDE,
  title={The Dirac equation as a quantum walk: higher dimensions, observational convergence},
  author={Pablo Arrighi and M. Forets and Vincent Nesme},
  journal={Journal of Physics A},
  year={2014},
  volume={47},
  pages={465302}
}
The Dirac equation can be modelled as a quantum walk (QW), whose main features are being: discrete in time and space (i.e. a unitary evolution of the wave-function of a particle on a lattice); homogeneous (i.e. translation-invariant and time-independent) and causal (i.e. information propagates at a bounded speed, in a strict sense). This link, which was proposed already by Succi and Benzi, Bialynicki-Birula and Meyer, is shown to hold for Bargmann–Wigner equations and symmetric hyperbolic… 
Quantum walking in curved spacetime: discrete metric
A discrete-time quantum walk (QW) is essentially a unitary operator driving the evolution of a single particle on the lattice. Some QWs have familiar physics PDEs as their continuum limit. Some
Quantum walking in curved spacetime
TLDR
The continuum limit of a wide class of QWs is studied and it is shown that it leads to an entire class of PDEs, encompassing the Hamiltonian form of the massive Dirac equation in ($$1+1$$1-1) curved spacetime.
Weyl, Dirac and Maxwell Quantum Cellular Automata
Recent advances on quantum foundations achieved the derivation of free quantum field theory from general principles, without referring to mechanical notions and relativistic invariance. From the
Quantum walks, limits and transport equations
TLDR
Plastic QWs are those ones admitting both continuous time-discrete space and continuous spacetime time limit, and it is shown that such QW-based quantum simulators can be used to quantum simulate a large class of physical phenomena described by transport equations.
Quantum Walks, Weyl Equation and the Lorentz Group
Quantum cellular automata and quantum walks provide a framework for the foundations of quantum field theory, since the equations of motion of free relativistic quantum fields can be derived as the
Discrete spacetime, quantum walks, and relativistic wave equations
It has been observed that quantum walks on regular lattices can give rise to wave equations for relativistic particles in the continuum limit. In this paper we define the 3D walk as a product of
Discrete-time quantum walks as fermions of lattice gauge theory
It is shown that discrete-time quantum walks can be used to digitize, i.e., to time discretize fermionic models of continuous-time lattice gauge theory. The resulting discrete-time dynamics is thus
Path-sum solution of the Weyl quantum walk in 3 + 1 dimensions
We consider the Weyl quantum walk in 3+1 dimensions, that is a discrete-time walk describing a particle with two internal degrees of freedom moving on a Cayley graph of the group , which in an
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 64 REFERENCES
Relationship between quantum walks and relativistic quantum mechanics
Quantum walk models have been used as an algorithmic tool for quantum computation and to describe various physical processes. This article revisits the relationship between relativistic quantum
From Dirac to Diffusion: Decoherence in Quantum Lattice Gases
TLDR
A model for the interaction of the internal (spin) degree of freedom of a quantum lattice-gas particle with an environmental bath is described, which produces a model having both classical and quantum discrete random walks as different limits.
Free-Dirac-particle evolution as a quantum random walk
t is known that any positive-energy state of a free Dirac particle that is initially highly localized evolves in time by spreading at speeds close to the speed of light. As recently indicated by
A Quantum Lattice-Gas Model for the Many-Particle Schroedinger Equation
We consider a general class of discrete unitary dynamical models on the lattice. We show that generically such models give rise to a wavefunction satisfying a Schroedinger equation in the continuum
Alternate two-dimensional quantum walk with a single-qubit coin
We have recently proposed a two-dimensional quantum walk where the requirement of a higher dimensionality of the coin space is substituted with the alternance of the directions in which the walker
Weyl, Dirac, and Maxwell equations on a lattice as unitary cellular automata.
TLDR
The case of the spin-1/2 particle is studied in detail and it is shown that every local and unitary automaton on a cubic lattice leads in the continuum limit to the Weyl equation.
A causal net approach to relativistic quantum mechanics
In this paper we discuss a causal network approach to describing relativistic quantum mechanics. Each vertex on the causal net represents a possible point event or particle observation. By
The Dirac Equation
Ever since its invention in 1929 the Dirac equation has played a fundamental role in various areas of modern physics and mathematics. Its applications are so widespread that a description of all
Dissipative quantum Church-Turing theorem.
We show that the time evolution of an open quantum system, described by a possibly time dependent Liouvillian, can be simulated by a unitary quantum circuit of a size scaling polynomially in the
Convergence of a three-dimensional quantum lattice Boltzmann scheme towards solutions of the Dirac equation
  • D. Lapitski, P. Dellar
  • Physics, Mathematics
    Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
  • 2011
TLDR
First-order convergence towards solutions of the Dirac equation is demonstrated by an independent numerical method based on fast Fourier transforms and matrix exponentiation.
...
1
2
3
4
5
...