Luigi Accardi

Quantum theory and its stochastic limit

1. Notations and Statement of the Problem.- 2. Quantum Fields.- 3. Those Kinds of Fields We Call Noises.- 4. Open Systems.- 5. Spin-Boson Systems.- 6. Measurements and Filtering Theory.- 7. Idea of the Proof and Causal Normal Order.- 8. Chronological Product Approach to the Stochastic Limit.- 9. Functional Integral Approach to the Stochastic Limit.- 10. Low-Density Limit: The Basic Idea.- 11. Six Basic Principles of the Stochastic Limit.- 12. Particles Interacting with a Boson Field.- 13. The Anderson Model.- 14. Field-Field Interactions.- 15. Analytical Theory of Feynman Diagrams.- 16. Term-by-Term Convergence.- References.

Conditional Expectations in von Neumann Algebras and a Theorem of Takesaki

Conditional expectations play an important role in classical probability theory. In the general context of von Neumann algebras they were impliciteiy used by von Neumann [41, Chap. II] and by Dixmier [14]. Nakamura and Turumaru [27] and Umegaki [36-391 introduced an axiomatic definition of the concept of conditional expectation in the framework of von Neumann (or C*-) algebras and established many properties of these objects especially in the context of von Neumann algebras with a finite trace. Their starting point was the characterization, given by Moy [26], of the classical conditional expectations as operators on spaces of measurable functions. Tomiyama showed [33 ] that conditional expectations, in the sense of the above mentioned authors, can be characterized as norm one projection in C*algebras. The importance of norm one projection in the classification problem of von Neumann algebras was recognized by Hakeda and Tomiyama [23] and subsequent research on this argument confirmed the usefulness of these objects. This line of thought culminated in the fundamental work of Connes [ 121 in which approximately finite von 245 0022.1236/82/020245-29

INTERACTING FOCK SPACES AND GAUSSIANIZATION OF PROBABILITY MEASURES

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The weak coupling limit as a quantum functional central limit

We prove that any probability measure on ℝ, with moments of all orders, is the vacuum distribution, in an appropriate interacting Fock space, of the field operator plus (in the nonsymmetric case) a function of the number operator. This follows from a canonical isomorphism between the L2-space of the measure and the interacting Fock space in which the number vectors go into the orthogonal polynomials of the measure and the modified field operator into the multiplication operator by the x-coordinate. A corollary of this is that all the momenta of such a measure are expressible in terms of the Szego–Jacobi parameters, associated to its orthogonal polynomials, by means of diagrams involving only noncrossing pair partitions (and singletons, in the nonsymmetric case). This means that, with our construction, the combinatorics of the momenta of any probability measure (with all moments) is reduced to that of a generalized Gaussian. This phenomenon we call Gaussianization. Finally we define, in terms of the Szego–...

Quantum independent increment processes on superalgebras

We show that, in the weak coupling limit, the laser model process converges weakly in the sense of the matrix elements to a quantum diffusion whose equation is explicitly obtained. We prove convergence, in the same sense, of the Heisenberg evolution of an observable of the system to the solution of a quantum Langevin equation. As a corollary of this result, via the quantum Feynman-Kac technique, one can recover previous results on the quantum master equation for reduced evolutions of open systems. When applied to some particular model (e.g. the free Boson gas) our results allow to interpret the Lamb shift as an Ito correction term and to express the pumping rates in terms of quantities related to the original Hamiltonian model.

A Representation Free Quantum Stochastic Calculus

On introduit la notion de processus quantique independant a increment stationnaire sur des superalgebres et on demontre un theoreme de reconstruction qui etablit une correspondance un-a-un entre ces processus et leurs generateurs infinitesimaux

THE WIGNER SEMI-CIRCLE LAW IN QUANTUM ELECTRO DYNAMICS

Abstract We develop a representation free stochastic calculus based on three inequalities (semimartingale inequality, scalar forward derivative inequality, scalar conditional variance inequality). We prove that our scheme includes all the previously developed stochastic calculi and some new examples. The abstract theory is applied to prove a Boson Levy martingale representation theorem in bounded form and a general existence, uniqueness, and unitarity theorem for quantum stochastic differential equations.

Nonrelativistic Quantum Mechanics as a Noncommutative Markof Process

In the present paper, the basic ideas of thestochastic limit of quantum theory are applied to quantum electro-dynamics. This naturally leads to the study of a new type of quantum stochastic calculus on aHilbert module. Our main result is that in the weak coupling limit of a system composed of a free particle (electron, atom,...) interacting, via the minimal coupling, with the quantum electromagnetic field, a new type of quantum noise arises, living on a Hilbert module rather than a Hilbert space. Moreover we prove that the vacuum distribution of the limiting field operator is not Gaussian, as usual, but a nonlinear deformation of the Wigner semi-circle law. A third new object arising from the present theory, is the so-calledinteracting Fock space. A kind of Fock space in which then quanta, in then-particle space, are not independent, but interact. The origin of all these new features is that we do not introduce the dipole approximation, but we keep the exponential response term, coupling the electron to the quantum electromagnetic field. This produces a nonlinear interaction among all the modes of the limit master field (quantum noise) whose explicit expression, that we find, can be considered as a nonlinear generalization of theFermi golden rule.

Some loopholes to save quantum nonlocality

It is well known that quantum mechanics presents many analogies with the theory of Markof processes: In both cases one is concerned with a statistical theory in which the states of a system undergo a deterministic evolution; the analogy between the Green function of the Schrodinger equation and the transition probabilities of a Markof process, together with the fact that a quantum mechanical system is determined by the assignment of a functional on a space of trajectories, are guiding ideas to Feynman’s approach to quantum mechanics [9]; the formal analogy between the diffusion equation and the Schrodinger equation has now become, through the systematic use of techniques of analytic continuation, a powerful tool in the treatment of the latter [18]; a one-to-one correspondence between wave functions of a large class of quantum systems and a class of Markof processes has been constructed in such a way that the corresponding statistical theories, at fixed times, coincide [19]; and, more recently, ideas and techniques of the theory of Markof processes have been used with success also in boson quantum field theory [20, 111. The connection between the two theories lies at a deep level: The fact that the evolution of quantum systems is described by a differential equation of first order in time expresses the locality of the correlation between observables at different times; and the most general way of expressing, in a statistical theory, a property of local correlation is given by the Markof (or, more generally, (d)-Markof [6]) property. The present work is concerned with the analysis of the property of “local statistical correlation” in the particular context of nonrelativistic quantum mechanics-as described by the axiomatics of von Neumann-

Notions of independence related to the free group

The EPR‐chameleon experiment has closed a long standing debate between the supporters of quantum nonlocality and the thesis of quantum probability according to which the essence of the quantum pecularity is non Kolmogorovianity rather than non locality.The theory of adaptive systems (symbolized by the chameleon effect) provides a natural intuition for the emergence of non‐Kolmogorovian statistics from classical deterministic dynamical systems. These developments are quickly reviewed and in conclusion some comments are introduced on recent attempts to “reconstruct history” on the lines described by Orwell in “1984”.