Quantum Measurement as a Final-State Interaction with a Macroscopic External System

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). Thus, the macrosystem can be interpreted as 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. 1 ar X iv :0 70 5. 16 49 v1 [ qu an tph ] 1 1 M ay 2 00 7