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Fortschritte der Physik/Progress of Physics

Stochastic Quantum Scattering and State Reduction at Measurement

Frank Kaschluhn

Published Jan 1, 1991• 0 citations
Publisher

Abstract

A stochastic modification of quantum scattering theory is proposed. It states that the final scattering state reduces alternatively to substates corresponding to transitions with or without energy-momentum transfer. The state reduction at the measuring process comes out as to require.
DOI
10.1002/prop.2190390804

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arXiv:quant-ph/0107153•Journal of Physics A Mathematical and General•Oct 15, 2001•117 citations
Abstract A statistical law for interaction processes in quantum theory is presented. It states that the final state of the scattered systems, evaluated on the basis of quantum theory, reduces alternatively to different substates provided correlation states are produced by the interaction. Such processes also underlie the measuring process (interaction system‐apparatus). To test the modified scattering theory experiments are proposed. The consequences for the description of the interaction between a system and a macroscopic body, in particular a measuring apparatus, are discussed. The reduction of the wave function needed here comes out quite naturally.
A small quantum scattering system (the microsystem) is studied in interaction with a large system (the macrosystem) described by unknown stochastic variables. The interaction between the two systems is diagonal for the microsystem in a certain orthonormal basis, and the interaction gives an imprint on the macrosystem. Moreover, the interaction is assumed to involve only small transfers of energy and momentum between the two systems (as compared to typical energies/momenta within the microsystem). The analysis is carried out within scattering theory. Calculated in the conventional way, the transition amplitude for the whole system factorizes. The interaction taking place within the macrosystem is assumed to depend on the stochastic variables in such a way that, on the average, no particular basis vector state of the microsystem is favoured. The density matrix is studied in a formalism which includes generation of the ingoing state and absorption of the final state. Then the dependence of the final state on the conventional scattering amplitude for the microsystem is highly non-linear. In the thermodynamic limit of the macrosystem, the density matrix of the ensemble (of microsystem plus macrosystem) develops into a final state which involves a set of macroscopically distinguishable states, each with the microsystem in one of the basis vector states and the macrosystem in an entangled state. For an element of the ensemble, i.e., for a single measurement, the result is instead a random walk, where the microsystem ends up in one of the basis vector states (reduction of the wave packet).
The measurement process of quantum mechanics is analysed in the scattering theory of quantum field theory. A matrix of bilinear forms of the scattering amplitudes (the R-matrix) is used as the basic descriptive tool. The measurement process is viewed as a final-state interaction described through a series of linear stochastic mappings of the R-matrix, not changing the observable to be measured. The unknown details of the measurement apparatus enter through the stochasticity of the mappings. Although linear in terms of the R-matrix, the mappings are nonlinear in the density matrix, which is obtainable from the R-matrix through normalization. The eigenstates of the observable are the attractors of the mapping process. This result, known from previous generalizations of quantum mechanics, is obtained here within linear quantum mechanics. The conclusion is that the measurement process can be understood within relativistic quantum field theory itself without any generalization or metatheory.
A small quantum scattering system (the microsystem) is studied in interaction with a large quantum system (the macrosystem) described by unknown stochastic variables. The interaction between the two systems is diagonal for the microsystem in a certain basis, and it leads to an imprint on the macrosystem. Moreover, the interaction is assumed to involve only small transfers of energy and momentum between the two systems (as compared to typical energies/momenta within the microsystem). This makes it suitable to carry out the analysis in scattering theory, where the transition amplitude for the whole system factorizes. The interaction taking place within the macrosystem is assumed to depend on the stochastic variables in such a way that, on the average, no particular channel is favoured. The result is then, in the thermodynamic limit of the macrosystem, that the whole system bifurcates and the microsystem ends up in a state described by one of the basis vectors (in the mentioned basis). The macrosystem ends up in an entangled state tied to this basis vector. For the ensemble of macrosystems, the interaction with the microsystem leads, on the average, to the usual decoherence and diagonal density matrix for the microsystem. The macrosystem can be interpreted as representing a measurement device for performing a measurement on the microsystem. The whole discussion is carried out within quantum mechanics itself without any modification or generalization.
Stochastic models for quantum state reduction give rise to statistical laws that are in most respects in agreement with those of quantum measurement theory. Here we examine the correspondence of the two theories in detail, making a systematic use of the methods of martingale theory. An analysis is carried out to determine the magnitude of the fluctuations experienced by the expectation of the observable during the course of the reduction process and an upper bound is established for the ensemble average of the greatest fluctuations incurred. We consider the general projection postulate of Lüders applicable in the case of a possibly degenerate eigenvalue spectrum, and derive this result rigorously from the underlying stochastic dynamics for state reduction in the case of both a pure and a mixed initial state. We also analyse the associated Lindblad equation for the evolution of the density matrix, and obtain an exact time-dependent solution for the state reduction that explicitly exhibits the transition from a general initial density matrix to the Lüders density matrix. Finally, we apply Girsanov's theorem to derive a set of simple formulae for the dynamics of the state in terms of a family of geometric Brownian motions, thereby constructing an explicit unravelling of the Lindblad equation.