Armen E. Allahverdyan, Roger Balian, Theo M. Nieuwenhuizen
The quantum measurement problem: why a unique outcome is obtained in each individual experiment, is currently tackled by solving models. After a general introduction, we review the many models proposed. Next, a flexible and realistic quantum model is introduced, describing the measurement of the z-component of a spin through interaction with a magnetic memory simulated by a Curie-Weiss magnet, including N >>1 spins coupled to a phonon bath. Initially prepared in a metastable paramagnetic state, it may transit to either its up or down ferromagnetic state, triggered by its coupling with the tested spin, so that its magnetization acts as a pointer. A solution of the dynamics is worked out. The first steps consist in the solution of the Hamiltonian dynamics for the spin-apparatus density matrix D(t). Its off-diagonal blocks rapidly decay owing to the many d.o.f. of the pointer. On a longer time scale, the dynamics produces a final state D(tf) that involves correlations between the system and the pointer, ensuring registration. The description of individual runs is approached within a specified version of the statistical interpretation. There exist many decompositions of D(tf), so one cannot infer the states that would describe small subsets of runs. This is overcome by suitable interactions within the apparatus, which produce a special combination of relaxation and decoherence associated with the broken invariance. Any subset of runs thus quickly reaches a stable state, which satisfies the same hierarchic property as in classical probability theory; the reduction of the state for each individual run follows. Standard quantum statistical mechanics alone appears sufficient to explain the occurrence of a unique answer in each run and the emergence of classicality in a measurement process. Pedagogical exercises are proposed, while the statistical interpretation is promoted for teaching.
View original:
http://arxiv.org/abs/1107.2138
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