Related papers: Entwined Paths, Difference Equations and the Dirac…
Numerical methods for the 1-D Dirac equation based on operator splitting and on the quantum lattice Boltzmann (QLB) schemes are reviewed. It is shown that these discretizations fall within the class of quantum walks, i.e. discrete maps for…
Quantum Dirac constraints in generic constrained system are solved by directly calculating in the one-loop approximation the path integral with relativistic gauge fixing procedure. The calculations are based on the reduction algorithms for…
A two-type continuous-state branching process in varying environments is constructed as the pathwise unique solution of a system of stochastic equations driven by time-space noises, where the pathwise uniqueness is derived from a comparison…
One-dimensional discrete-time quantum walks show a rich spectrum of topological phases that have so far been exclusively analysed in momentum space. In this work we introduce an alternative approach to topology which is based on the…
The relationship between the Dirac and reduced phase space quantizations is investigated for spin models belonging to the class of Hamiltonian systems having no gauge conditions. It is traced out that the two quantization methods may give…
We study several numerical discretization techniques for the one-space plus one-time dimensional Dirac equation, including finite difference and space-time finite element methods. Two finite difference schemes and several space-time finite…
Quantum walks represent paradigmatic quantum evolutions, enabling powerful applications in the context of topological physics and quantum computation. They have been implemented in diverse photonic architectures, but the realization of a…
Some problems founds in teaching physics related to curved paths that are unfortunately only described as illustration. A simple way to introduce the path is presented, which can help students to test their concept numerically. The…
Many time series are effectively generated by a combination of deterministic continuous flows along with discrete jumps sparked by stochastic events. However, we usually do not have the equation of motion describing the flows, or how they…
This paper is a continuation of the author's preceding one. In the preceding paper the author has rigorously constructed the Feynman path integral for the Dirac equation in the form of the sum-over-histories, satisfying the superposition…
A path integral representation of the evolution operator for the four-dimensional Dirac equation is proposed. A quadratic form of the canonical momenta regularizes the original representation of the path integral in the electron phase…
A path integral representation is given for the solutions of the 3+1 dimensional Dirac equation. The regularity of the trajectories, the non-relativistic limit and the semiclassical approximation are briefly mentioned.
In this paper, we present a detailed study on discrete-time Dirac quantum walks (DQWs) on triangular and honeycomb lattices. At the continuous limit, these DQWs coincide with the Dirac equation. Their differences in the discrete regime are…
A discretisation of differential geometry using the Whitney forms of algebraic topology is consistently extended via the introduction of a pairing on the space of chains. This pairing of chains enables us to give a definition of the…
Based on the weak existence and weak uniqueness, we study the pathwise uniqueness of the solutions for a class of one-dimensional stochastic differential equations driven by pure jump processes. By using Tanaka's formula and the local time…
We investigate pathwise turnpike behavior of discrete-time stochastic linear-quadratic optimal control problems. Our analysis is based on a novel strict dissipativity notion for such problems, in which a stationary stochastic process…
This paper extends the results of Ma, Wu, Zhang, Zhang [11] to the context of path-dependent multidimensional forward-backward stochastic differential equations (FBSDE). By path-dependent we mean that the coefficients of the…
For time travel to be consistent with the known laws of physics, the resulting paradoxes must be resolved. It has been suggested that parallel timelines (a.k.a. multiple histories) may provide a resolution. However, so far, a concrete…
We present a quantum cellular automaton model in one space-dimension which has the Dirac equation as emergent. This model, a discrete-time and causal unitary evolution of a lattice of quantum systems, is derived from the assumptions of…
It has been shown that the last passage time in certain symmetrized models of directed percolation can be written in terms of averages over random matrices from the classical groups $U(l)$, $Sp(2l)$ and $O(l)$. We present a theory of such…