The Macroscopic Level of Quantum Mechanics [1972a and b] (with C. George and I. Prigogine)

Synopsis

A synthetic account is given of a general treatment of large quantal systems, allowing of a clear-cut characterization of the macroscopic level of description of such systems. The time-evolution of the density operator is given by a Liouville equation, which is written down in a superspace formed by the direct product of the Hilbert space with itself. It is shown how to construct a projector Π in this superspace such that the subspace it defines contains the asymptotic time-evolution of the density operator for time intervals very large compared with those typical for atomic processes: this asymptotic subdynamics shows the characteristic features of macroscopic behaviour.

A quantitative criterion is formulated in superspace for the existence of a well-defined and unique macroscopic level in the sense just outlined of a separate subdynamics governing the asymptotic behaviour of the system. This ‘condition of dissipativity’ can be directly tested on the Hamiltonian of any given system.

In general, the subdynamics can only be formulated in superspace: it is not possible to return from the Π subspace to a Hilbert space description of the system in terms of state-vectors. Thus, the scope of the latter description is clearly limited, and a precise formulation is obtained of the complementarity between the dynamical account of the system on the atomic scale and its description at the level of macroscopic observation.

The epistemological problems of quantum mechanics receive from the present point of view an especially transparent treatment. In particular, the consistency of the use of classical concepts for the account of quantal phenomena is obvious, since the macroscopic description operates directly with probabilities, all quantal interference effects being eliminated from the Π subspace; thus, the rule of ‘reduction’ of the state-vector following a measurement performed upon an atomic system appears as an immediate consequence of the macroscopic character of the measuring process.

Editorial Note: we print here as one text two papers which had the same title: The Macroscopic Level of Quantum Mechanics. They were the work of Rosenfeld in collaboration with I. Prigogine, and C. George, of the Université Libre in Brussels and the Center for Statistical Mechanics and Thermodynamics. University of Texas, Austin. We are grateful to Professor Prigogine for kindly approving of this reprint, and of the arrangement. The papers were published in Nature 240 (1972).25–27, and Det kgl. Danske Videnskabernes Selskab. mat.-fys. Meddelelser 38 no. 12 (1972), 1–44. Here Part 1 and Parts 3–4–5 come from the publication of the Royal Danish Academy, while our Part 2 is the complete text of the Nature article which contains a summary of the technical Part II of the Danish paper.