Dirichlet forms in quantum theory

Karwowski, W.; Streit, Ludwig

doi:10.1142/9789814596534_0015

Cited by 1 publication

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“…We are somewhat guided by basic tenets of that theory, but our considerations will refer directly to the unitary (quantum) dynamics and therefore our Hilbert space contains complex valued L 2 functions, the real ones forming merely a subset in L 2 and not a linear subspace, [73].…”

Section: A Mathematical Prerequisitesmentioning

confidence: 99%

Lévy flights and nonlocal quantum dynamics

Garbaczewski

Stephanovich

2013

Journal of Mathematical Physics

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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...

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Section: A Mathematical Prerequisitesmentioning

confidence: 99%