Quantum stochastic processes

We first define a class of processes which we call regular quantum Markov processes. We next prove some basic results concerning such processes. A method is given for constructing quantum Markov processes using transition amplitude kernels. Finally we show that the Feynman path integral formalism can be clarified by approximating it with a quantum stochastic process.     

Quantum stochastic processes

In order to describe rigorously certain measurement procedures, where observations of the arrival of quanta at a counter are made throughout an interval of time, it is necessary to introduce the concept of a quantum stochastic process. While fully quantum mechanical in nature, these have a great deal of similarity with classical stochastic processes and can be characterized by and constructed from their infinitesimal generators. The infinitestimal generators are naturally obtained from certain fields which we prove must be of the boson or fermion type.This work was supported by a National Science Foundation grant GP-7952X.     

Fermionic and supersymmetric stochastic processes

The fermionic Fock space is represented by the Wiener chaos. This identification allows one to define fermionic Brownian motion with a probability measure. In the underlying geometrical picture this Brownian motion evolves in the linear space of the generators of the Grassmann algebra which spans the Fock space. More general stochastic processes can be derived with the help of stochastic differential equations. The generalization to supersymmetric processes is based on the Wiener-Grassmann product of Le Jan, an algebraic structure which is adequate to investigate differential operators on Wiener spaces.     

Integrable models and stochastic processes

A general way for constructing square lattice systems with certain Lie algebraic or quantum Lie algebraic symmetries is presented. These models give rise to series of integrable (stochastic) systems. As examples theAn-symmetric chain models and theSU(2)-invariant ladder models are investigated. Presented at the 10th Colloquium on Quantum Groups: “Quantum Groups and Intergrable Systems”, Prague, 21–23 June, 2001 SFB 256; BiBoS; CERFIM(Locarno); Acc. Arch.; USI(Mendriso)     

Fractal and lacunary stochastic processes

Discrete-time random walks simulate diffusion if the single-step probability density function (jump distribution) generating the walk is sufficiently shortranged. In contrast, walks with long-ranged jump distributions considered in this paper simulate Lévy or stable processes. A one-dimensional walk with a selfsimilar jump distribution (the Weierstrass random walk) and its higherdimensional generalizations generate fractal trajectories if certain transience criteria are met and lead to simple analogs of deep results on the Hausdorff-Besicovitch dimension of stable processes. The Weierstrass random walk is lacunary (has gaps in the set of allowed steps) and its characteristic function is Weierstrass' non-differentiable function. Other lacunary random walks with characteristic functions related to Riemann's zeta function and certain numbertheoretic functions have very interesting analytic structure.     

Quantum stochastic processes III

We construct an example of a quantum stochastic process with a non-zero, linear, time-independent source, for a massive scalar Boson field in four space-time dimensions. Also we study in detail a similar process with only a single degree of freedom.Work supported by A.F.O.S.R. contract no. F44620-67-C-0029     

Quantum stochastic processes II

We investigate properties of a class of quantum stochastic processes subject to a condition of irreducibility. These processes must be recurrent or transient and an equilibrium state can only exist in the former case. Every finite dimensional process is recurrent and it is possible to establish convergence in time to a unique equilibrium state. We study particularly the class of transition processes, which describe photon emissions of simple quantum mechanical systems in excited states.Work supported by U.S.A.F. contract number F 44620-67-C-0029.     

Transition probabilities and stochastic equations for the mean field ising model

The microscopic transition rate is briefly calculated from quantum principles to derive the microscopic master equation. By introducing _p, the phenomenological time, and coarse graining W_p, the transition rate, a complete normalized phenomenological transition rate is obtained. The Langer form is then approximately obtained.Supported in part by the Robert A. Welch Foundation.On leave of absence from the Institute of Theoretical Physics, Academia Sinica, Beijing, China.     

Granger causality and contiguity between stochastic processes

Although according to many econometricians the definition of causality proposed by Granger differs from other definitions of causation in the philosophy of science, in this Letter we argue that it is not completely lacking in philosophical legitimacy. We attempt to shed new light on the nexus between Granger causality and the concept of contiguity. In particular, we prove that the existence of a Granger causal link between two stochastic processes requires that these be “contiguous” or that there exist a chain of processes, one contiguous to the next, which link the two processes.     

Non-Markovian quantum stochastic processes and their entropy

A definition of a quantum stochastic process (QSP) in discrete time capable of describing non-Markovian effects is introduced. The formalism is based directly on the physically relevant correlation functions. The notion of complete positivity is used as the main mathematical tool. Two different but equivalent canonical representations of a QSP in terms of completely positive maps are derived. A quantum generalization of the Kolmogorov-Sinai entropy is proved to exist.     

Dynamical and Hamiltonian dilations of stochastic processes

This is a study of the problem, which stochastic processes could arise from dynamical systems by loss of information. The notions of “dilation” and “approximate dilation” of a stochastic process are introduced to give exact definitions of this particular relationship. It is shown that every generalized stochastic process is approximately dilatable by a sequence of dynamical systems, but for stochastic processes in full generality one needs nets.     

Computation in finitary stochastic and quantum processes

We introduce stochastic and quantum finite-state transducers as computation-theoretic models of classical stochastic and quantum finitary processes. Formal process languages, representing the distribution over a process’ behaviors, are recognized and generated by suitable specializations. We characterize and compare deterministic and nondeterministic versions, summarizing their relative computational power in a hierarchy of finitary process languages. Quantum finite-state transducers and generators are a first step toward a computation-theoretic analysis of individual, repeatedly measured quantum dynamical systems. They are explored via several physical systems, including an iterated-beam-splitter, an atom in a magnetic field, and atoms in an ion trap—a special case of which implements the Deutsch quantum algorithm. We show that these systems’ behaviors, and so their information processing capacity, depends sensitively on the measurement protocol.