Related papers: Entwined Paths, Difference Equations and the Dirac…
We derive a semiclassical time evolution kernel and a trace formula for the Dirac equation. The classical trajectories that enter the expressions are determined by the dynamics of relativistic point particles. We carefully investigate the…
Localized solutions of the Dirac equation for an electron moving in free space and electromagnetic field lattices with periodic dependence on space-time coordinates (electromagnetic space-time crystals) are treated using the expansions in…
Quantum walks are standard tools for searching graphs for marked vertices, and they often yield quadratic speedups over a classical random walk's hitting time. In some exceptional cases, however, the system only evolves by sign flips,…
A process-theoretic approach to electrodynamics based on persistent Kac-type stochastic processes is developed. Finite-velocity stochastic propagation is taken as primary, while relativistic wave equations arise as emergent descriptions…
In the directed setting, the spaces of directed paths between fixed initial and terminal points are the defining feature for distinguishing different directed spaces. The simplest case is when the space of directed paths is homotopy…
We relate the combinatorics of periodic generalized Dyck and Motzkin paths to the cluster coefficients of particles obeying generalized exclusion statistics, and obtain explicit expressions for the counting of paths with a fixed number of…
We study two constrained scalar models. While there seems to be equivalence when the partially integrated Feynman path integral is expanded graphically, the dynamical behaviour of the two models are different when quantization is done using…
Rational Dyck paths are the rational generalization of classical Dyck paths. They play an important role in Catalan combinatorics, and have multiple applications in algebra and geometry. Two statistics over rational Dyck paths called run…
Topologically protected wave motion has attracted considerable interest due to its novel properties and potential applications in many different fields. In this work, we study edge modes and traveling edge states via the linear Dirac…
The set of discrete lattice paths from (0, 0) to (n, n) with North and East steps (i.e. words w $\in$ { x, y } * such that |w| x = |w| y = n) has a canonical monoid structure inherited from the bijection with the set of join-continuous maps…
We summarize a recent work on the subject title. The Dirac equation in a curved spacetime depends on a field of coefficients (essentially the Dirac matrices), for which a continuum of different choices are possible. We study the conditions…
General discrete one-dimensional stochastic models to describe the transport of single molecules along coupled parallel lattices with period $N$ are developed. Theoretical analysis that allows to calculate explicitly the steady-state…
Thermodynamics is independent of a description at a microscopic level consequently statistical thermodynamics must produce results independent of the coordinate system used to describe the particles and their interactions. In the path…
In this paper we construct an analytical separation (diagonalization) of the full (minimal coupling) Dirac equation into particle and antiparticle components. The diagonalization is analytic in that it is achieved without transforming the…
Through a very careful analysis of Dirac's 1932 paper on the Lagrangian in Quantum Mechanics as well as the second and third editions of his classic book {\it The Principles of Quantum Mechanics}, I show that Dirac's contributions to the…
In this paper we show that the existence of a primarily discrete space-time may be a fruitful assumption from which we may develop a new approach of statistical thermodynamics in pre-relativistic conditions. The discreetness of space-time…
The semiclassical solution of quantum Dirac constraints in generic constrained system is obtained by directly calculating in the one-loop approximation the gauge field path integral with relativistic gauge fixing procedure. The gauge…
Continuous-time random walks (CTRWs) on discrete state spaces, ranging from regular lattices to complex networks, are ubiquitous across physics, chemistry, and biology. Models with coarse-grained states, for example those employed in…
The discrete time quantum walk which is a quantum counterpart of random walk plays important roles in the theory of quantum information theory. In the present paper, we focus on discrete time quantum walks viewed as quantization of random…
Differential geomtrical methods for deriving the Dirac equation in Curved Spacetime are presented. Einstein's field equation is applied in a novel manner; in the most current standard reference, Birrell and Davies, 1994 [1], the suggestions…