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Infinite Dimensional Analysis Quantum Probability and Related Topics

DEFORMED POISSON-LAWS AS CERTAIN TRANSFORM OF DEFORMED GAUSSIAN-LAWS

Ferenc Oravecz

Published Dec 1, 2002• 2 citations
Publisher

Abstract

We define a natural transformation of certain symmetric probability measures, and show that it brings the Gaussian-law as well as several deformed analogue of it to the Poisson-law or to the corresponding deformed Poisson-law, respectively. An important example is the q-Gaussian-law with its transform being the q-Poisson-law of Saitoh and Yoshida. By analogy we define an even more general class of deformed Poisson distributions, and show that, as in the classical and the q-case, these deformed Poisson-laws can be achieved as the distributions of certain operators.

DOI
10.1142/s0219025702000997

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We define a natural transformation of certain symmetric probability measures, and show that it brings the Gaussian-law as well as several deformed analogue of it to the Poisson-law or to the corresponding deformed Poisson-law, respectively. An important example is the q-Gaussian-law with its transform being the q-Poisson-law of Saitoh and Yoshida. By analogy we define an even more general class of deformed Poisson distributions, and show that, as in the classical and the q-case, these deformed Poisson-laws can be achieved as the distributions of certain operators.
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On (alpha,q)(alpha, q)(alpha,q)-Poisson Operators and Their Distributions (Mathematical aspects of quantum fields and related topics)
Generating functions for integrals of stochastic processes have been known in analytically closed form for just a handful of processes: namely, the Ornstein-Uhlenbeck, the Cox-Ingerssol-Ross (CIR) process and the exponential of Brownian motion. In virtue of their analytical tractability, these processes are extensively used in modelling applications. In the first part of the thesis, broad extensions of these process classes are constructed. The known models are shown to fit into a classification scheme for diffusion processes for which generating functions for integrals of the process and transition probability densities can be evaluated as integrals of hypergeometric functions against the spectral measure for certain self-adjoint operators. This scheme also extends to a class of finite-state Markov processes related to hypergeometric polynomials in the discrete series of the Askey classification tree. We also extend to quantum mechanics the technique of stochastic subordination, by means of which one can express any semi-martingale as a time-changed Brownian motion. Stochastic subordination preserves the Markov property of the process, whereas quantum subordination preserves quantum probability in the dynamics. As applications, we consider two versions of the q-deformed harmonic oscillator in both ordinary and imaginary time and show how the related q-deformed algebras can be understood as different patterns of time quantization rules. The second part of the thesis concerns the analogy between q -deformation and stochastic subordination. It is shown that the Poisson kernel of stochastic processes on chains of arbitrary length can be expressed in terms of the q-Racah polynomials, the most general q-deformed orthogonal polynomials in the discrete series of the Askey scheme. We give a new interpretation of this kernel as the transition probability density for a subordinated Markov process with only nearest neighbor hops. As an application, we give an elementary proof and extend a positivity result for a class of Poisson kernels which Gasper and Rahman established with direct methods.
The deformed wavelet transform is a new addition to the class of wavelet transforms that heavily rely on a pair of generalized translation and dilation operators governed by the well-known Dunkl transform. In this study, we adapt the symmetrical properties of the Dunkl Laplacian operator to prove a class of quantitative uncertainty principles associated with the deformed wavelet transform, including Heisenberg’s uncertainty principle, the Benedick–Amrein–Berthier uncertainty principle, and the logarithmic uncertainty inequalities. Moreover, using the symmetry between a square integrable function and its Dunkl transform, we establish certain local-type uncertainty principles involving the mean dispersion theorems for the deformed wavelet transform.
Abstract We derive a general operator transformation between any two functions and we show that the operator transformation of quantum quasi-distributions is a special case. We also show that our approach generalizes the Edgeworth series of standard probability theory, and results in a unified approach for the transformation of probability densities. Our approach generalizes the classical Edgeworth series in two ways. It relates two arbitrary distributions rather than one of them being a Gaussian and secondly we generalize to an arbitrary Hermitian operator rather than the differentiation operator which appears in the Edgeworth formulation. We also consider applying the transformation to the same distribution but at different times.
The discovery that any classical random variable with all moments gives rise to a full quantum theory (that in the Gaussian and Poisson cases coincides with the usual one) implies that a quantum-type formalism will enter into practically all applications of classical probability and statistics. The new challenge consists in finding the classical interpretation, for different types of classical contexts, of typical quantum notions such as entanglement, normal order, equilibrium states, etc. As an example, every classical symmetric random variable has a canonically associated conjugate momentum. In usual quantum mechanics (associated with Gaussian or Poisson classical random variables), the interpretation of the momentum operator was already clear to Heisenberg. How should we interpret the conjugate momentum operator associated with classical random variables outside the Gauss-Poisson class? The Introduction is intended to place in historical perspective the recent developments that are the main object of the present exposition.