The Macroscopic Level of Quantum Mechanics [1972a and b] (with C. George and I. Prigogine)
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.
KeywordsDensity Operator Atomic System Liouville Equation Macroscopic Level Macroscopic Observation
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