*A priori* probability and localized observers

- 10 Citations
- 61 Downloads

## Abstract

A physical and mathematical framework for the analysis of probabilities in quantum theory is proposed and developed. One purpose is to surmount the problem, crucial to any reconciliation between quantum theory and space-time physics, of requiring instantaneous “wave-packet collapse” across the entire universe. The physical starting point is the idea of an observer as an entity, localized in space-time, for whom any physical system can be described at any moment, by a set of (not necessarily pure) quantum states compatible with his observations of the system at that moment. The mathematical starting point is the theory of local algebras from axiomatic relativistic quantum field theory. A function defining the*a priori* probability of mistaking one local state for another is analysed. This function is shown to possess a broad range of appropriate properties and to be uniquely defined by a selection of them. Through a general model for observations, it is argued that the probabilities defined here are as compatible with experiment as the probabilities of conventional interpretations of quantum mechanics but are more likely to be compatible, not only with modern developments in mathematical physics, but also with a complete and consistent theory of measurement.

## Preview

Unable to display preview. Download preview PDF.

### References

- 1.M. J. Donald, “Quantum theory and the brain,”
*Proc. R. Soc. London A***427**, 43–93 (1990).Google Scholar - 2.
- 3.M. J. Donald, “Further results on the relative entropy,”
*Math. Proc. Cambridge Philos. Soc.***101**, 363–373 (1987).Google Scholar - 4.H. Everett, III, “The theory of the universal wave function,” in B. S. DeWitt and N. Graham,
*The Many-Worlds Interpretation of Quantum Mechanics*(Princeton University Press, Princeton, 1973).Google Scholar - 5.J. Glimm and A. Jaffe, “Field theory models,” in
*Statistical Mechanics and Quantum Field Theory*, C. DeWitt and R. Stora, eds. (Gordon & Breach, New York, 1971), pp. 1–108. Reprinted in Ref. 8.Google Scholar - 6.J. Glimm and A. Jaffe, “Boson quantum field models,” in
*Mathematics of Contemporary Physics*, R. F. Streater, ed. (Academic Press, New York, 1972), pp. 77–143. Reprinted in Ref. 8.Google Scholar - 7.J. Glimm and A. Jaffe, “The resummation of one particle lines,”
*Commun. Math. Phys.***67**, 267–293 (1979).Google Scholar - 8.J. Glimm and A. Jaffe,
*Quantum Field Theory and Statistical Mechanics—Expositions*(Birkhäuser, Boston, 1985).Google Scholar - 9.R. F. Streater and A. S. Wightman,
*PCT, Spin and Statistics, and All That*(Benjamin, New York, 1964).Google Scholar - 10.O. Bratteli and D. W. Robinson,
*Operator Algebras and Quantum Statistical Mechanics*(Springer, Berlin), Vol. I, 1979; Vol. II, 1981.Google Scholar - 11.E. Seiler,
*Gauge Theories as a Problem of Constructive Quantum Field Theory and Statistical Mechanics*(Springer, Berlin, 1982).Google Scholar - 12.R. Haag and B. Schroer, “Postulates of quantum field theory,”
*J. Math. Phys.***3**, 248–256 (1962).Google Scholar - 13.R. Haag and D. Kastler, “An algebraic approach to quantum field theory,”
*J. Math. Phys.***5**, 848–861 (1964).Google Scholar - 14.W. Driessler, S. J. Summers, and E. H. Wichmann, “On the connection between quantum fields and von Neumann algebras of local operators,”
*Commun. Math. Phys.***105**, 49–84 (1986).Google Scholar - 15.D. Buchholz and E. H. Wichmann, “Causal independence and the energy-level density of states in local quantum field theory,”
*Commun. Math. Phys.***106**, 321–344 (1986).Google Scholar - 16.H. Roos, “Independence of local algebras in quantum field theory,”
*Commun. Math. Phys.***16**, 238–246 (1970).Google Scholar - 17.H. Ekstein, “Presymmetry II,”
*Phys. Rev.***184**, 1315–1337 (1969); correction in*Phys. Rev. D***1**, 185(E) (1970).Google Scholar - 18.B. De Facio and D. C. Taylor, “Commutativity and causal independence,”
*Phys. Rev. D***8**, 2729–2731 (1973).Google Scholar - 19.D. Buchholz, C. D'Antoni, and K. Fredenhagen, “The universal structure of local algebras,”
*Commun. Math. Phys.***111**, 123–135 (1987).Google Scholar - 20.W.-D. Garber, “The connexion of duality and causal properties for generalized free fields,”
*Commun. Math. Phys.***42**, 195–208 (1975).Google Scholar - 21.H. J. Borchers, “Über die Vollständigkeit lorentzinvarianter Felder in einer zeitartigen Röhre,”
*Nuovo Cimento***19**, 787–793 (1961).Google Scholar - 22.H. Araki, “A generalization of Borchers' theorem,”
*Helv. Phys. Acta***36**, 132–139 (1963).Google Scholar - 23.S. Stratila and L. Zsido,
*Lectures on von Neumann algebras*(Abacus, Tunbridge Wells, 1979).Google Scholar - 24.L. J. Cohen,
*An Introduction to the Philosophy of Induction and Probability*(Oxford University Press, Oxford, 1989).Google Scholar - 25.A. Daneri, A. Loinger, and G. M. Prosperi, “Quantum theory of measurement and ergodicity conditions,
*Nucl. Phys.***33**, 297–319 (1962). Reprinted in Ref. 33.Google Scholar - 26.K. Hepp, “Quantum theory of measurement and macroscopic observables,”
*Helv. Phys. Acta***45**, 237–248 (1972).Google Scholar - 27.B. Whitten-Wolfe and G. G. Emch, “A mechanical quantum measuring process,”
*Helv. Phys. Acta***49**, 45–55 (1976).Google Scholar - 28.S. Machida and M. Namiki, “Theory of measurement in quantum mechanics,” I.
*Prog. Theor. Phys.***63**, 1457–1473 (1980); II.*Prog. Theor. Phys.***63**, 1833–1847 (1980).Google Scholar - 29.H. Araki, “A remark on the Machida-Namiki theory of measurement,”
*Prog. Theor. Phys.***64**, 719–730 (1980).Google Scholar - 30.H. Araki, “A continuous superselection rule as a model of classical measuring apparatus in quantum mechanics,” in
*Fundamental Aspects of Quantum Theory*, V. Gorini and A. Frigerio, eds. (Plenum, New York, 1986), pp. 23–33.Google Scholar - 31.W. H. Zurek, “Pointer basis of quantum apparatus: Into what mixture does the wave packet collapse?”
*Phys. Rev. D***24**, 1516–1525 (1981).Google Scholar - 32.W. H. Zurek, “Environment-induced superselection rules,”
*Phys. Rev. D***26**, 1862–1880 (1982).Google Scholar - 33.J. A. Wheeler and W. H. Zurek,
*Quantum Theory and Measurement*(Princeton University Press, Princeton, 1983).Google Scholar - 34.I. N. Sanov, “On the probability of large deviations of random variables,”
*Mat. Sb.***42**, 11–44 (1957); translation for the Institute of Mathematical Statistics, published by the American Mathematical Society in*Selected Translations in Mathematical Statistics and Probability***1**, 213–244 (1961).Google Scholar - 35.
- 36.M. Takesaki,
*Theory of Operator Alegbras, Vol. 1*(Springer, Berlin, 1979).Google Scholar - 37.M. Reed and B. Simon,
*Methods of Modern Mathematical Physics, Vol. 1: Functional Analysis*(Academic Press, New York, 1972).Google Scholar - 38.H. Araki, “Relative entropy of states of von Neumann algebras,”
*Publ. Res. Inst. Math. Sci. (Kyoto)***11**, 809–833 (1976).Google Scholar - 39.H. Umegaki, “Conditional expectation in an operator algebra. IV: Entropy and information,”
*Kodai Math. Sem. Rep.***14**, 59–85 (1962).Google Scholar - 40.G. Lindblad, “Expectations and entropy inequalities for finite quantum systems,”
*Commun. Math. Phys.***39**, 111–119 (1974).Google Scholar - 41.D. Petz, “Characterization of the relative entropy of states of matrix algebras,” preprint.Google Scholar
- 42.F. Hiai and D. Petz, “The proper formula for relative entropy and its asymptotics in quantum probability,”
*Commun. Math. Phys.***143**, 99–114 (1991).Google Scholar - 43.J. E. Shore and R. W. Johnson, “Axiomatic derivation of the principle of maximum entropy and the principle of minimum cross-entropy,”
*IEEE Trans. Inform. Theory***26**, 26–37 (1980),Google Scholar - 44.J. E. Shore and R. W. Johnson, “Principles of cross-entropy minimization,”
*IEEE Trans. Inform. Theory***27**, 472–482 (1981).Google Scholar - 45.H. Araki, “Relative entropy for states of von Neumann algebras, II,”
*Publ. Res. Inst. Math. Sci. (Kyoto)***13**, 173–192 (1977).Google Scholar - 46.P. Exner,
*Open Quantum Systems and Feynman Integrals*(Reidel, Dordrecht, 1985).Google Scholar