Quantum causality,stochastics, trajectories and information

Abstract

A history of the discovery of `new' quantum mechanics and theparadoxes of its probabilistic interpretation are brieflyreviewed from the modern point of view of quantum probabilityand information. Modern quantum theory, which has beendeveloped during the last 20 years for the treatment of quantum opensystems including quantum noise, decoherence, quantum diffusionsand spontaneous jumps occurring under continuous in timeobservation, is not yet a part of the standard curriculum ofquantum physics. It is argued that the conventional formalism ofquantum mechanics is insufficient for the description of quantumevents, such as spontaneous decays say, and the new experimentalphenomena related to individual quantum measurements, but theyhave all received an adequate mathematical treatment in quantumstochastics of open systems.Moreover, the only reasonable probabilistic interpretation ofquantum mechanics put forward by Max Born was, in fact, inirreconcilable contradiction with traditional mechanical realityand causality. This led to numerous quantum paradoxes, some ofthem due to the great inventors of quantum theory such asEinstein and Schrödinger. They are reconsidered in thispaper from the modern point of view of quantum stochastics and information.The development of quantum measurement theory, initiated by vonNeumann, indicated a possibility for resolution of thisinterpretational crisis by divorcing the algebra of thedynamical generators and the algebra of the actual observables,or Bell's beables. It is shown that within this approachquantum causality can be rehabilitated in the form of asuperselection rule for compatibility of the actual historieswith the potential future. This rule, together with theself-compatibility of the measurements ensuring the consistencyof the histories, is called the nondemolition, or causalityprinciple in modern quantum theory. The application of this rulein the form of dynamical commutation relations leads to thederivation of the von Neumann projection postulate, and also tothe more general reductions, instantaneous, spontaneous, andeven continuous in time. This gives a dynamical solution, in theform of the quantum stochastic filtering equations, of thenotorious measurement problem which was tackled unsuccessfullyby many famous physicists starting with Schrödinger andBohr.It has been recently proved that the quantum stochastic modelfor the continuous in time measurements is equivalent to a Diractype boundary-value problem for the secondary quantized input`offer waves from future' in one extra dimension, and to areduction of the algebra of the consistent histories of pastevents to an Abelian subalgebra for the `trajectories of theoutput particles'. This supports the corpuscular-wave duality inthe form of the thesis that everything in the future arequantized waves, while everything in the past are trajectories of therecorded particles.