Wolfgang Freudenberg

Characterization of states of infinite boson systems. I. On the construction of states of boson systems

In a previous paper [11] it was shown that to each locally normal state of a boson system one can associate a point process that can be interpreted as the position distribution of the state. In the present paper the so-called conditional reduced density matrix of a normal or locally normal state is introduced. The whole state is determined completely by its position distribution and this function. There are given sufficient conditions on a point processQ and a functionk ensuring the existence of a state such thatQ is its position distribution andk its conditional reduced density matrix. Several examples will show that these conditions represent effective and useful criteria to construct locally normal states of boson systems. Especially, we will sketch an approach to equilibrium states of infinite boson systems. Further, we consider a class of operators on the Fock space representing certain combinations of position measurements and local measurements (observables related to bounded areas). The corresponding expectations can be expressed by the position distribution and the conditional reduced density matrix. This class serves as an important tool for the construction of states of (finite and infinite) boson systems. Especially, operators of second quantization, creation and annihilation operators are of this type. So, independently of the applications in the above context this class of operators may be of some interest.

Point processes and the position distribution of infinite boson systems

It is shown that to each locally normal state of a boson system one can associate a point process that can be interpreted as the position distribution of the state. The point process contains all information one can get by position measurements and is determined by the latter. On the other hand, to each so-called Σc-point processQ we relate a locally normal state with position distributionQ.

Characterization of states of infinite boson systems. II. On the existence of the conditional reduced density matrix

In the present paper we deal with the problem of existence and uniqueness of the conditional reduced density matrix (c.r.d.m.) corresponding to a locally normal state of a boson system. The c.r.d.m. was introduced in [3] (Part I of the present series of papers). In order to characterize the class of states possessing a c.r.d.m. we will introduce the family of conditional states of a locally normal state, and we will discuss the relation between the conditional states, the c.r.d.m. and the conditional distribution of the position distribution of the state.

Time Evolution and Invariance of Boson Systems Given by Beam Splittings

Based on a model for general beam splittings we search for states of boson systems which are invariant under the combination of the evolution given by the splitting procedure and some inherent evolution. It turns out that for finite systems only trivial invariant normal states may appear. However, for locally normal states on a related quasilocal algebra representing states of infinite boson systems, one can find examples of nontrivial invariant states. We consider as example a beam splitting combined with a contraction compensating the loss of intensity caused by the splitting process. In general, we observe interesting connections between the splitting procedure and certain thinning operations in classical probability theory. Several applications to physics seem to be natural since these beam splitting models are used to describe measuring procedures on electromagentic fields.

On a Mathematical Model of Brain Activities

The procedure of recognition can be described as follows: There is a set of complex signals stored in the memory. Choosing one of these signals may be interpreted as generating a hypothesis concerning an “expexted view of the world”. Then the brain compares a signal arising from our senses with the signal chosen from the memory leading to a change of the state of both signals. Furthermore, measurements of that procedure like EEG or MEG are based on the fact that recognition of signals causes a certain loss of excited neurons, i.e. the neurons change their state from “excited” to “nonexcited”. For that reason a statistical model of the recognition process should reflect both—the change of the signals and the loss of excited neurons. A first attempt to explain the process of recognition in terms of quantum statistics was given in [1]. In the present note it is not possible to present this approach in detail. In lieu we will sketch roughly a few of the basic ideas and structures of the proposed model of the ...

On a Quantum‐Like Model of the Recognition Process

We deal with a quantum‐like model of the recognition process. The model was developed in the past years in a series of papers (cf. [1, 3, 4, 5, 6]). The present paper which is a continuation of [7] is focussed on a relation between recognition and a certain process of self‐collapses. It is shown that the process of self‐collapses can be described by a classical homogenous discrete‐time Markov chain. Finally we pass over to a continuous‐time Markov chain that seems to give a more appropriate and reasonable characterisation of the process of recognition.

Teleportation schemes in infinite dimensional Hilbert spaces

The success of quantum mechanics is due to the discovery that nature is described in infinite dimension Hilbert spaces, so that it is desirable to demonstrate the quantum teleportation process in a certain infinite dimensional Hilbert space. We describe the teleportation process in an infinite dimensional Hilbert space by giving simple examples.

NetzCope: A Tool for Displaying and Analyzing Complex Networks

Networks are a natural and popular mechanism for the representation and investigation of a broad class of systems. But extracting information from a network can present significant challenges. We present NetzCope, a software application for the display and analysis of networks. Its key features include the visualization of networks in two or three dimensions, the organization of vertices to reveal structural similarity, and the detection and visualization of network communities by modularity maximization.

Nonindependent splittings and Gibbs states

We discuss a nonindependent (beam) splitting for which the related thinning leaves the class of equilibrium states for a one mode electromagnetic field invariant. The thinning affects only the parameters of the state, showing a nonlinear loss of energy. After the splitting, the energy values of both split parts are independent. This independence is a characteristic property of the geometric distribution, the distribution of energy values in the equilibrium state. Also, we observe that the class of states where the full states of the split parts are independent is formed by the so-called phase states.