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q-Gaussian Processes: Non-Commutative And Classical Aspects

Marek Bozejko, Burkhard Kümmerer, Roland Speicher

Published Jan 1, 1995• 225 citations
OpenAlex

Abstract

. We examine, for \\Gamma1 ! q ! 1, q-Gaussian processes, i.e. families of operators (non-commutative random variables) X t = a t + a t -- where the a t fulfill the q-commutation relations asa t \\Gamma qa t as = c(s; t) \\Delta 1 for some covariance function c(\\Delta; \\Delta) -- equipped with the vacuum expectation state. We show that there is a q- analogue of the Gaussian functor of second quantization behind these processes and that this structure can be used to translate questions on q-Gaussian processes into corresponding (and much simpler) questions in the underlying Hilbert space. In particular, we use this idea to show that a large class of q-Gaussian processes possess a non-commutative kind of Markov property, which ensures that there exist classical versions of these non-commutative processes. This answers an old question of Frisch and Bourret [FB]. Introduction What we are going to call q-Gaussian processes was essentially introduced in a remarkable paper by Frisch ...

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. We examine, for \Gamma1 ! q ! 1, q-Gaussian processes, i.e. families of operators (non-commutative random variables) X t = a t + a t -- where the a t fulfill the q-commutation relations asa t \Gamma qa t as = c(s; t) \Delta 1 for some covariance function c(\Delta; \Delta) -- equipped with the vacuum expectation state. We show that there is a q- analogue of the Gaussian functor of second quantization behind these processes and that this structure can be used to translate questions on q-Gaussian processes into corresponding (and much simpler) questions in the underlying Hilbert space. In particular, we use this idea to show that a large class of q-Gaussian processes possess a non-commutative kind of Markov property, which ensures that there exist classical versions of these non-commutative processes. This answers an old question of Frisch and Bourret [FB]. Introduction What we are going to call q-Gaussian processes was essentially introduced in a remarkable paper by Frisch ...
We survey some results obtained in the last 25 years to illustrate the following probabilistic quantization statement: quantum probability is not a generalization of classical probability but a deeper level of it. Generalizing the classical theory of orthogonal polynomials, one can show that any random field XXX admitting all the moments can be represented as the sum of three operators, which are natural extensions of the creation, annihilation, and preservation operators in the usual boson Fock quantum theory. These operators generate a noncommutative ∗*∗-algebra on which the quantum extension of the expected value relative to the probability distribution of the field XXX induces a quantum state. So, any classical algebraic probability space generates a quantum space with its own commutation relations. The Heisenberg commutation relation characterizes the classical fields, while the new type of commutation relations (of type II) appears in non-Gaussian cases. The same machinery, but applied to Bernoulli fields, leads to Fermi--Dirac anticommutation relations (see the introduction).
We survey some results obtained in the last 25 years to illustrate the following probabilistic quantization statement: quantum probability is not a generalization of classical probability but a deeper level of it. Generalizing the classical theory of orthogonal polynomials, one can show that any random field XXX admitting all the moments can be represented as the sum of three operators, which are natural extensions of the creation, annihilation, and preservation operators in the usual boson Fock quantum theory. These operators generate a noncommutative ∗*∗-algebra on which the quantum extension of the expected value relative to the probability distribution of the field XXX induces a quantum state. So, any classical algebraic probability space generates a quantum space with its own commutation relations. The Heisenberg commutation relation characterizes the classical fields, while the new type of commutation relations (of type II) appears in non-Gaussian cases. The same machinery, but applied to Bernoulli fields, leads to Fermi-Dirac anticommutation relations (see the introduction).
We review the basic concepts of quantum probability and revise the classical and quantum stochastic (QS) calculus using the universal Itô B*-algebra approach. A non-commutative generalization of Lévy process is defined in Fock–Guichardet space in terms of the underlying B*-Itô modular algebra. The main notions and results of classical and QS analysis are reformulated in terms of the non-commutative but associative stochastic covariation of quantum integrals defined as sesquilinear forms adapted to a quantum Lévy process in Fock space. The general semi-Markov QS dynamics is defined in terms of a hemigroup of semi-morphic transformations as adjoints to a semiunitary hemigroup resolving a QS Schrödinger equation in Fock space. The corresponding semi-Markov hemigroup of completely positive contractions resolving a quantum master equation with generalized Lindblad generator as a non-commutative analogue of Feller–Kolmogorov equation is derived by taking vacuum conditional expectation in Fock space.
Starting from the notion of algebraic hidden processes introduced in Ref. 9 , we propose a definition of quantum hidden Markov process which extends the notion of quantum Markov chain (QMC) in the same sense in which classical hidden Markov processes (HMPs) extend classical Markov chains. We give several structure theorems for this new family of quantum states which shares with the QMCs the unique property that the finite dimensional joint expectations are explicitly given without these states being linearly equivalent to product states (as the Gaussian ones). The special feature of this new class of quantum states is best exemplified by looking at their restrictions to diagonal sub-algebras: we prove that they lead to a new family of classical stochastic processes that generalize in a non-trivial way the classical HMPs.
The structure of the projection-operator theory of Mori for quantum Brownian motion is reconsidered, emphasizing the nature of underlying quantum processes. Quantum-mechanical assumptions of the theory are extracted to sets of axioms. The Hilbert space scrV formed with relevant observables for quantum generalized Langevin equations (QGLE's) includes, in general, elements that cannot be operators on state vectors of the original system. The definition of Liouville operator L, as a mapping from one operator to another, thus requires its reconstruction in scrV. The work shows that the mentioned axioms not only ensure a unified reconstruction of L (hence of QGLE's) in systems in stationary states that may be nonequilibrium as well as equilibrium, but also extend intrinsically the domain of applicability of QGLE's. Illustrations of the setting for these extended QGLE's are presented by concrete examples of Hilbert spaces formed with quantum-mechanical Gaussian and non-Gaussian random variables, showing specifically the significance of a response-function-type inner product that assigns \ensuremath{\delta} correlation to the quantized Gaussian white noise which is an operator-valued distributional process with Kubo-Martin-Schwinger property.
Introduction Quantum probability is a generalization of both classical probability theory and quantum mechanics that allows to describe the probabilistic aspects of quantum mechanics. This generalization is formulated in two steps. First, the theory is reformulated in terms of algebras of functions on probability spaces. Therefore, the notion of a probability space (Ω, ℱ , P ) is replaced by the pair ( L ∞ (Ω), E (.) = ∫ Ω . d P ) consisting of the commutative von Neumann algebra of bounded random variables and the expectation functional. Then, the commutativity condition is dropped. In this way we arrive at the notion of a (von Neumann) algebraic probability space ( N ,Φ) consisting of a von Neumann algebra N and a normal (faithful tracial) state Φ. As we have seen this includes classical probability spaces in the form ( L ∞ (Ω), E ), it also includes quantum mechanical systems modeled by a Hilbert space H and a pure state ψ ∊ H (or a mixed state ρ ∊ S ( H )), if we take N = B ( H ) and Ф the state defined by Ф( X ) = 〈 ψ, X ψ 〉 (or Ф ( X ) = tr(ρ X )) for X ∊ B ( H ). Note that in this course we shall relax the conditions on N and Φ and work with involutive algebras and positive normalized functionals, that is, *-algebraic probability spaces. A striking feature of quantum probability (also called noncommutative probability) is the existence of several notions of independence. This is the starting point of this course, which intends to give an introduction to the theory of quantum stochastic processes with independent increments. However, before we come to these processes, we will give a general introduction to quantum probability. In Section 1.2, we recall the basic definitions of quantum probability and discuss some fundamental differences between classical probability and quantum probability. In Section 1.3 we address the question ‘Why do we need Quantum Probability?’ We discuss the EPR experiment and the Kochen–Specker Theorem, which show that we cannot model quantum physics with classical probability spaces because the values of observable quantities do not exist unless we specify which measurement we will carry out and which quantities we will determine. In this sense quantum physics requires a more radical description of chance.