Lévy 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 ≥ 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. I. INTRODUCTIONThe standard unitary quantum dynamics and the Schrödinger semigroup-driven random motion, [1,2], are examples of dual evolution scenarios that may be mapped among each other by means of a suitable analytic continuation in time procedure. This is an offspring [3][4][5] of Euclidean quantum field theory methods, albeit reduced to the purely quantum mechanical level. Both evolutions are generated by means of a common local Hamiltonian operator. Our departure point for subsequent analysis is an observation that a complete spectral resolution of the corresponding Hamiltonian actually determines a classical space-time homogeneous diffusion-type Markov process in R n .Within the general theory of so called infnitely divisible probability laws (see below) the familiar Laplacian (Wiener noise or Brownian motion generator) is known to be one isolated member of a surprisingly rich family of non-Gaussian Lévy noise generators. All of them stem from the fundamental Lévy-Khintchine formula, and typically refer to probability distributions of spatial jumps and the resultant jump-type Markov processes. That needs to be contrasted with the traditional diffusion imagery (Wiener noise and process) associated with the Laplacian, [6].The emergent Lévy generators are manifestly nonlocal (pseudo-differential) operators and, while being additively perturbed by a suitable external potential, give rise (via a canonical quantization procedure described subsequently) to Lévy-Schrödinger semigroups. The dual image of such semigroups comprises unitary dynamics scenarios which can be viewed as signatures of a nonlocal quantum behavior. As we discuss in below this dynamical nonlocality extends to the very concept of photons, within so-called photon wave mechanics, [7].Quite apart from "Euclidean" vs "real" time connotations, the considered dual dynamical systems refer to real time labels and clocks. Both of them drive probability density functions (pdfs) which, in the course of time evolution (t ∈ R + ) either maintain or develop asymptotic heavy-tails and typically have a finite number of moments in existence. This needs to be contrasted with the Gaussian standards of thinking (all pdf moments in existence, rapid decay at infinities etc.) that pervade the Laplacian-based quantum theory.The major goal of the present paper is to set on solid grounds the quantization programme that completely avoi...

Solving fractional Schrödinger-type spectral problems: Cauchy oscillator and Cauchy well

“…A particular mass zero version (Cauchy generator) of the relativistic Hamiltonian may be linked with the photon wave mechanics and Maxwell fields [1]. Likewise, other fractional generators seem to have more in common with fields than with particles, regardless of their classical or quantum connotations.…”

“…Except for so-called relativistic Hamiltonians (c.f. the spinless Salpeter equation, with or without external local potentials [1][2][3]), generic nonlocal energy generators are not supported by a classical mechanical intuition of massive particles in motion.…”

“…The emergent Lévy generators are manifestly nonlocal (pseudo-differential) operators and give rise (via a canonical quantization procedure, [1]) to Lévy-Schrödinger semigroups and affiliated nonlocally-induced jump-type processes. The dual (Euclidean) image of such semigroups comprises unitary dynamics scenarios which give rise to a nonlocal quantum behavior.…”

“…[1]. The pertinent motion scenario stems from generalizing the standard Laplacian-based framework of the Schrödinger picture quantum evolution to that employing nonlocal (pseudo-differential) operators.…”

“…[1] where basic tenets of the nonlocally induced random and quantum dynamics were analyzed. A number of mentions was maid with respect to various inconsistencies and faulty statements omnipresent in the literature devoted to so-called fractional quantum mechanics spectral problems.…”