Non-commutative Lévy processes for generalized (particularly anyon) statistics

Abstract

Let $T={\mathbb{R}}^d$. Let a function $Q:T^2\to \mathbb{C}$ satisfy $Q(s,t)=\overline{Q(t,s)}$ and $|Q(s,t)|=1$. A generalized statistics is described by creation operators ${\partial }_t^{†}$ and annihilation operators ${\partial }_t$, $t\in T$, which satisfy the $Q$-commutation relations. From the point of view of physics, the most important case of a generalized statistics is the anyon statistics, for which $Q(s,t)$ is equal to $q$ if $s<t$, and to $\bar{q}$ if $s>t$. Here $q\in \mathbb{C}$, $|q|=1$. We start the paper with a detailed discussion of a $Q$-Fock space and operators $({\partial }_t^{†},{\partial }_t{)}_{t\in T}$ in it, which satisfy the $Q$-commutation relations. Next, we consider a noncommutative stochastic process (white noise) $\omega (t)={\partial }_t^{†}+{\partial }_t+\lambda {\partial }_t^{†}{\partial }_t$, $t\in T$. Here $\lambda \in \mathbb{R}$ is a fixed parameter. The case $\lambda =0$ corresponds to a $Q$-analog of Brownian motion, while $\lambda \ne 0$ corresponds to a (centered) $Q$-Poisson process. We study $Q$-Hermite ($Q$-Charlier respectively) polynomials of infinitely many noncommutatative variables $(\omega (t){)}_{t\in T}$. The main aim of the paper is to explain the notion of independence for a generalized statistics, and to derive corresponding Lévy processes. To this end, we recursively define $Q$-cumulants of a field $(\xi (t){)}_{t\in T}$. This allows us to define a $Q$-Lévy process as a field $(\xi (t){)}_{t\in T}$ whose values at different points of $T$ are $Q$-independent and which possesses a stationarity of increments (in a certain sense). We present an explicit construction of a $Q$-Lévy process, and derive a Nualart-Schoutens-type chaotic decomposition for such a process.

References in corpus (6)

• Non-Abelian Anyons and Topological Quantum Computation
• Anyons and the quantum Hall effect - a pedagogical review
• Cumulants in noncommutative probability II. Generalized Gaussian random variables
• On the Fock space for nonrelativistic anyon fields and braided tensor products
• Meixner class of non-commutative generalized stochastic processes with freely independent values I. A characterization
• The Theory of Quantum Levy Processes

Cited by in corpus (10)

• Fock space associated to Coxeter group of type B
• Stirling operators in spatial combinatorics
• Bose-Einstein condensation and condensation of $q$-particles in equilibrium and non equilibrium thermodynamics: a new approach
• Fock representations of $Q$-deformed commutation relations
• Fock representations of multicomponent (particularly non-Abelian anyon) commutation relations
• Weakly Monotone Fock Space and Monotone Convolution of the Wigner Law
• Gauge-invriant quasi-free states on the algebra of the anyon commutation relations
• On the Thermodynamics of the q-Particles
• Some results on continuous deformed free group factors
• Quasi-free states on a class of algebras of multicomponent commutation relations