Levy flights and nonlocal quantum dynamics

Abstract

We develop a fully fledged theory of quantum dynamical patterns of behavior that are nonlocally induced. To this end we generalize the standard Laplacian-based framework of the Schrödinger picture quantum evolution to that employing nonlocal (pseudodifferential) operators. Special attention is paid to the Salpeter (here, $m\ge 0$) quasirelativistic equation and the evolution of various wave packets, in particular to their radial expansion in 3D. Foldy's synthesis of "covariant particle equations" is extended to encompass free Maxwell theory, which however is devoid of any "particle" content. Links with the photon wave mechanics are explored.

32 pages, 4 figures

References in corpus (7)

• Fractional Quantum Mechanics
• The role of the Riemann-Silberstein vector in classical and quantum theories of electromagnetism
• On the Consistency of the Solutions of the Space Fractional Schrödinger Equation
• Relativistic equations with fractional and pseudo-differential operators
• Modular Schrödinger equation and dynamical duality
• Schroedinger models for solutions of the Bethe-Salpeter equation in Minkowski space
• Indeterminacy relations in random dynamics

Cited by in corpus (18)

• Fractional Laplacians in bounded domains: Killed, reflected, censored and taboo Lévy flights
• Solving fractional Schroedinger-type spectral problems: Cauchy oscillator and Cauchy well
• Symmetry-breaking transitions in quiescent and moving solitons in fractional couplers
• Lévy flights in the infinite potential well as the hypersingular Fredholm problem
• Self-gravitating Interferometry and Intrinsic Decoherence
• Lévy flights confinement in a parabolic potential and fractional quantum oscillator
• Transition in the decay rates of stationary distributions of Lévy motion in an energy landscape
• Nonlocally-induced (fractional) bound states: Shape analysis in the infinite Cauchy well
• On the coherent states for a relativistic scalar particle
• Spectral properties of fractional Fokker-Plank operator for the Lévy flight in a harmonic potential
• General form of the covariant field equations of arbitrary spin and the relativistic canonical quantum mechanics
• Ultrarelativistic bound states in the spherical well
• Levy processes in bounded domains: Path-wise reflection scenarios and signatures of confinement
• Nonlocally-induced (quasirelativistic) bound states: Harmonic confinement and the finite well
• Fractional Laplacians and Levy flights in bounded domains
• Ultrarelativistic bound states in the shallow spherical well
• Relativistic Heat Equation via Lévy stable distributions: Exact Solutions
• Effective Approximation for a Nonlocal Stochastic Schrödinger Equation with Oscillating Potential