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On Totality of Certain Sets of Exponential Vectors

Michael Skeide

Published Jan 1, 2015• 0 citations
OpenAlex

Abstract

The result that exponential vectors to indicator functions are total in the Fock space (one degree of freedom) due to Parthasarathy and Sunder, is an important tool. In this talk we illustrate our proof of this result in the languange of Arveson systems and explain how it can be used to prove representation theorems for quantum Lévy processes.

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The result that exponential vectors to indicator functions are total in the Fock space (one degree of freedom) due to Parthasarathy and Sunder, is an important tool. In this talk we illustrate our proof of this result in the languange of Arveson systems and explain how it can be used to prove representation theorems for quantum Lévy processes.
As I type these words, I have in front of me a collection of notes from a seminar in Quantum Probability run by Luigi Accardi in Roma in 1984-5 where I was at the time based as a research fellow. In those days I was rather meticulous in ordering my jottings and on turning to page 131 in my file I find twenty pages entitled “Representations of Current Groups” by K.R. Parthasarathy. I remember being fascinated by these lectures (I think there were two) in which Partha gave an account of how the Lévy-Khinchine formula arises, naturally from the construction of continuous tensor products of unitary group representations (see e.g. [St], [Ar], [PS]) and then revealed the beautiful Fock space decomposition of infinitely divisible random variables as sums of creation, conservation and annihilation operators. This is in fact one of the many perspectives through which the historian can see the “birth” of quantum probability — as a student of Robin Hudson, I had seen the subject evolve from a different point of view, through the search for quantum central limit theorems and non-commuting analogues of Brownian motion. The marriage of these two viewpoints turned out to be a wonderful thing, leading as it did to the creation of quantum stochastic calculus.
An exponential Hilbert space, which is an abstraction of the familiar Fock space for bosons, provides a natural framework to discuss a wide class of field-operator representations. This framework is especially convenient when wide invariance groups, such as a unique translationally invariant state, are involved. In this paper, we develop the theory of exponential Hilbert spaces in a functional fashion suitable to discuss representations of field operators enjoying such invariance features. Representations of both current algebras and canonical field operators are discussed, and it is shown that these representations are natural generalizations of those characterizing infinitely divisible random processes. Questions of reducibility and equivalence are treated, and we prove that our construction gives rise to infinitely many unitarily inequivalent representations. Nevertheless, an extremely simple expression, bilinear in annihilation and creation operators, abstractly characterizes the operators of both the current algebras and canonical fields. Dynamical applications to quantum field theory will be treated in subsequent papers.
We continue the investigation of the Levy processes on a q-deformed full Fock space started in a previous paper. First, we show that the vacuum vector is cyclic and separating for the algebra generated by such a process. Next, we describe a chaotic representation property in terms of multiple integrals with respect to diagonal measures, in the style of Nualart and Schoutens. We define stochastic integration with respect to these processes, and calculate their combinatorial stochastic measures. Finally, we show that they generate infinite von Neumann algebras.
We continue the investigation of the Levy processes on a q-deformed full Fock space started in a previous paper. First, we show that the vacuum vector is cyclic and separating for the algebra generated by such a process. Next, we describe a chaotic representation property in terms of multiple integrals with respect to diagonal measures, in the style of Nualart and Schoutens. We define stochastic integration with respect to these processes, and calculate their combinatorial stochastic measures. Finally, we show that they generate infinite von Neumann algebras.
We continue the investigation of the Levy processes on a q-deformed full Fock space started in a previous paper. First, we show that the vacuum vector is cyclic and separating for the algebra generated by such a process. Next, we describe a chaotic representation property in terms of multiple integrals with respect to diagonal measures, in the style of Nualart and Schoutens. We define stochastic integration with respect to these processes, and calculate their combinatorial stochastic measures. Finally, we show that they generate infinite von Neumann algebras.