Infinite Dimensional Analysis, Quantum Probability and Related Topics

We construct a family of classical deterministic dynamical systems (triples formed by a state space, an initial distribution, a dynamics) parametrized by pairs of vectors (a,b) in the unit circle in ℝ². The systems describe pairs of particles and the dynamics is strictly local, i.e. the dynamics of particle j=1,2 depends only on one of the two unit vectors, but not the other. To each particle one associates a family of ± 1-valued observables (j = 1,2), also parametrized by vectors a in the unit circle in ℝ². Moreover we assume that, if observable is measured on particle j=1,2, then the dynamics of this particle will be (chameleon effect).

Using these ingredients we prove that the dynamics and the initial distributions of the given systems can be chosen in such a way that, if the observable is measured on particle j=1,2, then the EPR correlations -cos(a₁-a₂) are reproduced.

This theoretical construction is then used to realize the following experiment: a central computer S (source) sends the same signal to two local computers A and B. After receiving the signal A (resp. B), chooses one vector a (resp. b) in the unit circle in ℝ² and computes the value of the corresponding observable (resp. ). These values are uniquely defined by the deterministic dynamics and the choices are independent of one another.

The local computers send back to the central one the results of the evaluation of these ±1-valued functions and the central computer evaluates the empirical correlations according to the usual statistical rules.

As a result the EPR correlations are reproduced with very good approximation and the Bell inequality is violated by a classical, macroscopic deterministic system performing purely local choices.

The program to run the experiment is available from http://volterra.mat.uniroma2.it

The compensated Poisson noise is expressed as a composite sum (splitting) of creation and annihilation operators, whose probabilistic interpretation relies on time changes. We construct an Itô table for this decomposition and obtain continuous and discrete time realizations of Lévy processes on the finite difference algebra and on , e.g. the space–time dual of the Poisson process (compensated gamma process), and the continuous binomial process.

Let H be a real separable Hilbert space and let E⊂H be a nuclear space with the chain {E_m: m=1,2,…} of Hilbert spaces such that E = ∩_m∈ℕE_m. Let E* and E_-m denote the dual spaces of E and E_m, respectively. For γ > 0, let be the collection of complex-valued continuous functions f defined on E* such that is finite for every m. Then is a complete countably normed space equipping with the family {‖·‖_m,γ : m = 1,2,…} of norms. Using a probabilistic approach, it is shown that every continuous linear functional T on can be represented uniquely by a complex Borel measure ν_T satisfying certain exponential integrability condition. The results generalize an infinite dimensional Riesz representation theorem given previously by the first author for the case γ = 2. As applications, we establish a Weierstrass approximation theorem on E* for γ≥1 and show that the space spanned by the class {exp[i(x,ξ)] : ξ ∈ E} is dense in for γ>0. In the course of the proof we give sufficient conditions for a function space on which every positive functional can be represented by a Borel measure on E*.

We develop systematically a new unifying approach to the analysis of linear stochastic, quantum stochastic and even deterministic equations in Banach spaces. Solutions to a wide class of these equations (in particular those describing the processes of continuous quantum measurements) are proved to coincide with the interaction representations of the solutions to certain Dirac type equations with boundary conditions in pseudo-Fock spaces. The latter are presented as the semiclassical limit of an appropriately dressed unitary evolutions corresponding to a boundary-value problem for rather general Schrödinger equations with bounded below Hamiltonians.

A notion of the heat kernel measure is introduced for the L² completion of a hyperfinite II₁-factor with respect to the trace. Some properties of this measure are derived from the corresponding stochastic differential equation. Then the Taylor map is studied for a space of holomorphic functions square integrable with respect to the heat kernel measure. We also define a skeleton map from this space to a Hilbert space of holomorphic functions on a certain Cameron–Martin group. This group is a subgroup of the group of invertible elements of the II₁-factor.

A notion of "quasi-universal product" for algebraic probability spaces is introduced as a generalization of Speicher's "universal product". It is proved that there exist only five quasi-universal products, namely, tensor product, free product, Boolean product, monotone product and anti-monotone product. This result means that, in a sense, there exist only five independences which have nice properties of "associativity" and "(quasi-)universality".