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Infinite Dimensional Analysis Quantum Probability and Related Topics

ON THE RELATION OF THE SQUARE OF WHITE NOISE AND THE FINITE DIFFERENCE ALGEBRA

Luigi Accardi, Michael Skeide

Published Mar 1, 2000• 21 citations
Publisher

Abstract

The algebra of square of white noise1 contains a subalgebra generated by elements fulfilling the relations of Feinsilver's finite difference algebra.6 Moreover, Boukas' representation space3 is the same as the representation space of the algebra of square of white noise discovered in Ref. 2. In other words, Boukas' representation extends to a representation of the algebra of square of white noise.

DOI
10.1142/s021902570000011x

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The algebra of square of white noise1 contains a subalgebra generated by elements fulfilling the relations of Feinsilver's finite difference algebra.6 Moreover, Boukas' representation space3 is the same as the representation space of the algebra of square of white noise discovered in Ref. 2. In other words, Boukas' representation extends to a representation of the algebra of square of white noise.
The abstract commutation relations of the algebra of the square of white noise of Accardi, Lu, and Volovich are shown to be realized by operator processes acting on the Fock space of Accardi and Skeide which is very closely related to the Finite Difference Fock space of Boukas and Feinsilver. The processes are shown to satisfy the necessary conditions for inclusion in the framework of the representation free quantum stochastic calculus of Accardi, Fagnola, and Quaegebeur. The connection between the Finite-Difference operators and the creation, annihilation, and conservation operators on usual symmetric Boson Fock space is further studied.
We describe the "no-go" theorems recently obtained by Accardi-Boukas-Franz in [\\cite{1}] for the Boson case, and by Accardi-Boukas in [\\cite{2}] for the qqq-deformed case, on the issue of the existence of a common Fock space representation of the renormalized powers of quantum white noise (RPWN).
# Part 1: # White Noise Functional Approach to Polymer Entanglements (C C Bernido & M V Carpio-Bernido) # White Noise Analysis, Quantum Field Theory, and Topology (A Hahn) # A Topic on Noncanonical Representations of Gaussian Processes (Y Hibino) # Integral Representation of Hilbert-Schimdt Operators on Boson Fock Space (U C Ji) # The Dawn of White Noise Analysis (I Kubo) # White Noise Stochastic Integration (H-H Kuo) # Connes-Hida Calculus and Bismut-Quillen Superconnections (R Leandre & H Ouerdiane) # A Quantum Decomposition of Levy Processes (Y-J Lee & H-H Shih) # Generalized Entanglement and Its Classification (T Matsuoka) # A White Noise Approach to Fractional Brownian Motion (D Nualart) # Adaptive Dynamics in Quantum Information and Chaos (M Ohya) # Micro-Macro Duality in Quantum Physics (I Ojima) # White Noise Measures Associated to the Solutions of Stochastic Differential Equations (H Ouerdiane) # A Remark on Sets of Infinite Dimensional Spaces with Full or Zero Capacity (J Ren & M Rockner) # An Infinite Dimensional Laplacian in White Noise Theory (K Saito) # Invariance of Poisson Noise (Si Si et al.) # Nonequilibrium Steady States with Bose-Einstein Condensates (S Tasaki & T Matsui) # Multidimensional Skew Reflected Diffusions (G Trutnau) # On Quantum Mutual Type Entropies and Quantum Capacity (N Watanabe) # Part 2: # White Noise Calculus and Stochastic Calculus (L Accardi & A Boukas)