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Journal of Physics A Mathematical and General

Evolution equations and Lévy processes on quantum groups

Uwe Franz, René Schott

Published Feb 6, 1998• 12 citations
Publisher

Abstract

Evolution equations like the heat or diffusion equation or the Schrödinger equation can be associated with stochastic processes. In this paper we study the relation between equations of the form and Lévy processes (i.e. quantum stochastic processes with independent and stationary increments) on quantum groups and braided groups. Solutions of such equations are calculated as Appell systems. Wigner distributions of these processes are defined and it is proven that they satisfy a Fokker-Planck equation.

DOI
10.1088/0305-4470/31/5/009

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Evolution equations like the heat or diffusion equation or the Schrödinger equation can be associated with stochastic processes. In this paper we study the relation between equations of the form and Lévy processes (i.e. quantum stochastic processes with independent and stationary increments) on quantum groups and braided groups. Solutions of such equations are calculated as Appell systems. Wigner distributions of these processes are defined and it is proven that they satisfy a Fokker-Planck equation.
Starting from the forward and backward infinitesimal generators of bilateral, time-homogeneous Markov processes, the self-adjoint Hamiltonians of the generalized Schrödinger equations are first introduced by means of suitable Doob transformations. Then, by broadening with the aid of the Dirichlet forms, the results of the Nelson stochastic mechanics, we prove that it is possible to associate bilateral, and time-homogeneous Markov processes to the wave functions stationary solutions of our generalized Schrödinger equations. Particular attention is then paid to the special case of the Lévy-Schrödinger (LS) equations and to their associated Lévy-type Markov processes, and to a few examples of Cauchy background noise.
We analyze the time-dependent solutions of the pseudo-differential Lévy–Schrödinger wave equation in the free case, and compare them with the associated Lévy processes. We list the principal laws used to describe the time evolutions of both the Lévy process densities and the Lévy–Schrödinger wave packets. To have self-adjoint generators and unitary evolutions we will consider only absolutely continuous, infinitely divisible Lévy noises with laws symmetric under change of sign of the independent variable. We then show several examples of the characteristic behavior of the Lévy–Schrödinger wave packets, and in particular of the multimodality arising in their evolutions: a feature at variance with the typical diffusive unimodality of both the corresponding Lévy process densities and usual Schrödinger wavefunctions.
Many examples of quantum independent stationary increment processes are solutions of quantum stochastic differential equations. We give a common characterization of these examples by a quantum stochastic differential equation on an abstract *-bialgebra. Specializing this abstract *-bialgebra and the coefficients of the equation, we obtain the equations for the Unitary Noncommutative Stochastic processes of [12], the Quantum Wiener Process [2], the Azéma martingales [11] and for other examples. The existence and uniqueness of a solution of the general equation is shown. Assuming the boundedness of this solution, we prove that it is a continuous and stationary independent increment process.
The main purpose of the paper is an essentially probabilistic analysis of relativistic quantum mechanics. It is based on the assumption that whenever probability distributions arise, there exists a stochastic process that is either responsible for the temporal evolution of a given measure or preserves the measure in the stationary case. Our departure point is the so-called Schr\"odinger problem of probabilistic evolution, which provides for a unique Markov stochastic interpolation between any given pair of boundary probability densities for a process covering a fixed, finite duration of time, provided we have decided a priori what kind of primordial dynamical semigroup transition mechanism is involved. In the nonrelativistic theory, including quantum mechanics, Feynman-Kac-like kernels are the building blocks for suitable transition probability densities of the process. In the standard ``free'' case (Feynman-Kac potential equal to zero) the familiar Wiener noise is recovered. In the framework of the Schr\"odinger problem, the ``free noise'' can also be extended to any infinitely divisible probability law, as covered by the L\'evy-Khintchine formula. Since the relativistic Hamiltonians \ensuremath{\Vert}\ensuremath{\nabla}\ensuremath{\Vert} and \ensuremath{\surd}-\ensuremath{\Delta}+m2{\mathit{m}}^{2}m2 -m are known to generate such laws, we focus on them for the analysis of probabilistic phenomena, which are shown to be associated with the relativistic wave (D'Alembert) and matter-wave (Klein-Gordon) equations, respectively. We show that such stochastic processes exist and are spatial jump processes. In general, in the presence of external potentials, they do not share the Markov property, except for stationary situations. A concrete example of the pseudodifferential Cauchy-Schr\"odinger evolution is analyzed in detail. The relativistic covariance of related wave equations is exploited to demonstrate how the associated stochastic jump processes comply with the principles of special relativity.
: We perform a detailed analysis of the non stationary solutions of the evolution (Fokker-Planck) equations associated to either stationary or non stationary quantum states by the stochastic mechanics. For the excited stationary states of quantum systems with singular velocity fields we explicitely discuss the exact solutions for the HO case. Moreover the possibility of modifying the original potentials in order to implement arbitrary evolutions ruled by these equations is discussed with respect to both possible models for quantum measurements and applications to the control of particle beams in accelerators. 1. Introduction In a few papers [1] the analogy between diffusive classical systems and quantum systems has been reconsidered from the standpoint of the stochastic mechanics (SM) [2], [3], and particular attention was devoted there to the evolution of the classical systems associated to a quantum wave function when the conditions imposed by the stochastic variational principle are...
The authors consider a quantum system with a finite number N of states and show that a Markov process evolving in an 'extended' discrete phase space can be associated with the discrete Wigner function of the system. This Wigner function is built using the Weyl quantisation procedure on the group Z N *Z N . Moreover, the authors can use this process to compute the quantum mean values as probabilistic expectations of functions of this process. This probabilistic formulation can be seen as a stochastic mechanics in phase space.