Infinite Dimensional Analysis, Quantum Probability and Related Topics

A derivation operator and a divergence operator are defined on the algebra of bounded operators on the symmetric Fock space over the complexification of a real Hilbert space and it is shown that they satisfy similar properties as the derivation and divergence operator on the Wiener space over . The derivation operator is then used to give sufficient conditions for the existence of smooth Wigner densities for pairs of operators satisfying the canonical commutation relations. For , the divergence operator is shown to coincide with the Hudson–Parthasarathy quantum stochastic integral for adapted integrable processes and with the noncausal quantum stochastic integrals defined by Lindsay and Belavkin for integrable processes.

A notion of "monotonic independence" is formulated in the setting of C*-probability space. Based on this independence, a noncommutative central limit theorem and a noncommutative law of small numbers are given.

In this paper we will develop a systematic method to answer the questions (Q1) (Q2) (Q3) (Q4) (stated in Sec. 1) with complete generality. As a result, we can solve the difficulties (D1) (D2) (discussed in Sec. 1) without uncertainty. For these purposes we will introduce certain classes of growth functions u and apply the Legendre transform to obtain a sequence which leads to the weight sequence {α(n)} first studied by Cochran et al.⁶ The notion of (nearly) equivalent functions, (nearly) equivalent sequences and dual Legendre functions will be defined in a very natural way. An application to the growth order of holomorphic functions on ℰ_c will also be discussed.

In analogy to classical systems, quantum K-systems and quantum Anosov systems are defined. Their relation especially for modular systems is discussed as well as the consequences on clustering properties. Examples for such systems in the framework of quantum field theory and one-dimensional theories are offered.