Selected Papers of Léon Rosenfeld pp 571-598 | Cite as

# 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.

## Keywords

Density Operator Atomic System Liouville Equation Macroscopic Level Macroscopic Observation## Preview

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## Bibliography

- [1]N. Bohr., Dialectica 2 (1948), 312. N. Bohr, ‘Discussion with Einstein on Epistemological Problems in Atomic Physics’ in Albert Einstein: Philosopher-Scientist (Library of Living Philosophers, Evanston, III., 1949 ). N. Bohr, ‘Quantum Physics and Philosophy: Causality and Complementarity’ in Philosophy in the Mid-Century, a Survey, ed. R. Klibansky., ( Firenze 1958 ).Google Scholar
- [2]N. Bohr, and L. Rosenfeld., Dan. Vid. Selsk mat.-fys. Medd. 12, no. 8 (1933) (See this volume, p. 357 — Ed.]; Phys. Rev. 78 (1950), 794. [See this volume, p. 401 — Ed.]. L. Rosenfeld., ‘On Quantum Electrodynamics’ in Niels Bohr and the Development of Physics (Pergamon Press, London, 1955). [See this volume, p. 413 — Ed.].Google Scholar
- [3]N. Bohr, Faraday Lecture in J. Chem. Soc. (1932), 349. L. Rosenfeld, ‘Questions of Irreversibility and Ergodicity’ in Rendiconti Scuola Int. Fisica, Corso XIV, Varenna 1960 (Zanichelli, Bologna. 1962). (See this volume, p. 808 — Ed.)Google Scholar
- [4]A. Daneri, A. Loinger, and G. M. Prosperi., Nucl. Phys. 33 (1962), 297; Nuovo Cimento 44B (1966), 119. A. Loinger, Nucl. Phys. A108 (1968), 245.CrossRefGoogle Scholar
- [5]L. Rosenfeld, ‘The Measuring Process in Quantum Mechanics’ in Suppl. Progress Theor. Physics, Commemoration issue, 1965, p. 222. [See this volume, p. 536 — Ed.]Google Scholar
- [6]I. Prigogine, Non-Equilibrium Statistical Mechanics ( Interscience Publ., New York, 1962 ).Google Scholar
- [7]L. Rosenfeld, Acta Physica Polonica 14 (1955), 3. [See this volume, p. 762 — Ed.]Google Scholar
- [8]A. Loinger., ‘A Study of the Quantum Ergodic Problem’ in Rendiconti Scuola Int. Fisica, Corso XIV, Varenna 1960 (Zanichelli, Bologna, 1962). J. E. Farquhar, Ergodic Theory in Statistical Mechanics (Inlerscience Publ., New York, 1964), chap. VIII.Google Scholar
- [9]C. George, Physica 30 (1964), 1513; G. Severne, Physica 31 (1965), 877; also ref. [10].CrossRefGoogle Scholar
- [10]M. Baus, Bull. Cl. Sc., Acad. Roy. Belg. 53 (1967), 1291. 1332, 1352.Google Scholar
- [11]J. Rae, Bull. Cl. Sc., Acad. Roy. Belg. 55 (1969), 980, 1040.Google Scholar
- [12]R. Balescu, P. Clavin, P. Mandel, and J. W. Turner., Bull. Cl. Sc., Acad. Roy. Belg. 55 (1969), 1055.Google Scholar
- [13]I. Prigogine., ‘Dynamic Foundations of Thermodynamics and Statistical Mechanics’ in A Critical Review of Thermodynamics, ed. B. Stuart, B. Gal-Or, and A. J. Brainard., ( Mono Book Corp., 1970 ).Google Scholar
- [14]I. Prigogine, C. George, and F. Henin., Physica 45 (1969), 418.CrossRefGoogle Scholar
- [15]I. Prigogine, C. George, and F. Henin., Proc. Nat. Acad. U.S.A. 65 (1970), 789; the same with P. Mandel and J. W. Turner, ibid., 66 (1970), 709.Google Scholar
- [16]A. Grecos, Physica 51 (1970), 50.CrossRefGoogle Scholar
- [17]A. Grecos and I. Prigogine, Physica 59 (1972), 77.CrossRefGoogle Scholar
- [18]R. Balescu and J. Wallenborn, Physica 54 (1971), 477.CrossRefGoogle Scholar
- [19]C. George, I. Prigogine, and L. Rosenfeld., Det Kgl. Danske Videnskabernes Selska. mat.-fys. Meddelelser 38 (1972). 12. [See Editorial Note in this volume on p. 596 — Ed.)Google Scholar
- [20]R. Balescu, and L. Brenig., Physica 54 (1971), 504.CrossRefGoogle Scholar
- [21]A. Grews and I. Prigogine, Proc. US Natl. Acad. Sci. 69 (1972), 1629.Google Scholar
- [22]L. Lanz, L. A. Lugiato, and G. Ramella, Physica 54 (1971). 94; see also L. L., and G. Ramella, Physica 44 (1969), 499; L. Lanz, and L. A. Lugiato, Physica 44 (1969), 532 and 47 (1970), 345.CrossRefGoogle Scholar
- [23]L. Lanz, G. M. Prosperi, and A. Sabbadini, Nuovo Cimento 2B (1971), 184.CrossRefGoogle Scholar
- [24]E. Wigner, Z. Physik 133 (1952), 101.CrossRefGoogle Scholar
- [25]I. Prigogine, in Theoretical Physics and Biology (1967), ed. M. Marois ( North Holland Publ. Co., Amsterdam, 1969 ).Google Scholar
- [26]P. Glansdorff, and I. Prigogine, Stability, Structure and Fluctuations (Wiley Interscience, New York, 1971; French edition, Masson, 1971 ).Google Scholar
- [27]I. Prigogine, R. Lefever, A. Goldbeter, and M. Herschkowitz-Kaufman, Nature 223 (1969), 913.CrossRefGoogle Scholar
- [28]M. Eigen, ‘Self-Organization of Matter and the Evolution of Biological Macromolecules’, Naturwissenschaften 58 (1971), 465.CrossRefGoogle Scholar
- [29]P. Résibois, J. Stat. Phys. 2 (1970), 21.CrossRefGoogle Scholar