Unitary representations of the hyperfinite Heisenberg group and the logical extension methods in physics

Izumi Ojima1, Masanao Ozawa2
1Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan 606
2Research Institute for Mathematical Sciences, Kyoto University, 606, Kyoto, Japan

Tóm tắt

Từ khóa


Tài liệu tham khảo

Proc. 1st, 2nd, and 3rd Int. Symp. Foundations of Quantum Mechanics. Japan Society of Physics, Tokyo, (1983, 1986, 1989); P. Tombesi and E. R. Pike, editors,Squeezed and Nonclassical Light, Plenum, New York, 1989; O. Hirota C. Bendjaballah and S. Reynaud, editors,Proc. Int. Workshop on Quantum Aspects of Optical Communications, Lecture Notes in Physics No. 378, Springer-Verlag, Berlin, (1991);

L. Smarr, editor,Sources of Gravitational Radiation, Cambridge UP, Cambridge, 1979; P. Meystre and M. O. Scully, editors,Quantum Optics, Experimental Gravitation and Measurement Theory, Plenum, New York, 1983.

E. B. Davies,Quantum Theory of Open Systems, Academic Press, London, 1976; C. W. Helstrom,Quantum Detection and Estimation Theory, Academic Press, New York, 1976; A. S. Holevo,Probabilistic and Statistical Aspects of Quantum Theory, North-Holland, Amsterdam, 1982; M. Ozawa, J. Math. Phys.25, 79–87, (1984);26, 1948–1955, (1985);27, 759–763, (1986); Publ. RIMS Kyoto Univ.21, 279–295, (1985); P. J. Lahti P. Busch and P. Mittelstaedt,The Quantum Theory of Measurement, Lecture Notes in Physics, Springer-Verlag, Berlin, 1991.

O. Bratteli and D. W. Robinson,Operator Algebras and Quantum Statistical Mechanics I, Springer-Verlag, New York, 1979; II, 1979.

M. Ozawa,Cat Paradox for C*-dynamical Systems, Preprint Series, Dept. Math., Coll. Gen. Ed., Nagoya University, 1992.

M. Ozawa, Infinitesimal analysis and the measurement problem (in Japanese), inSoryuushi-ron Kenkyuu; Prog. Theor. Phys., in preparation.

R.C. Tolman,The Principles of Statistical Mechanics, Oxford University Press, Inc., 1938; (Dover edition) Dover Publications, Inc., 1979; I.M. Lifshitz and L.P. Pitaevskii.Landaru and Lifshitz: Course of Theoretical Physics Vol.5Statistical Physics, Part 1 (3rd edition), (English edition) Pergamon Press, 1976.

A. Robinson,Non-Standard Analysis, North-Holland, Amsterdam, 1966.

K. D. Stroyan and W. A. J. Luxemburg,Introduction to the Theory of Infinitesimals, Academic Press, New York, 1976; M. Davis,Applied Nonstandard Analysis, Wiley, New York, 1977; A. E. Hurd and P. A. Loeb,An Introduction to Nonstandard Real Analysis, Academic Press, Orland, 1985.

H. J. Keisler, Mem. Amer. Math. Soc.48, No. 297, 1984; K. D. Stroyan and J. M. Bayod,Foundations of Infinitesimal Stochastic Analysis, North-Holland, Amsterdam, 1986; S. Albeverioet. al., Nonstandard Methods in Stochastic Analysis and Mathematical Physics, Academic Press, Orland, 1986.

P. J. Cohen,Set Theory and the Continuum Hypothesis, Benjamin, New York, 1966.

J. L. Bell,Boolean-Valued Models and Independence Proofs in Set Theory, 2nd ed., Oxford University Press, Oxford, 1985.

D. Scott, InApplications of Model Theory to Algebra, Analysis, and Probability, Holt, Reinhart and Winston, 1969.

G. Takeuti,Two Applications of Logic to Mathematics, Princeton University Press, Princeton, 1978.

G. Takeuti, J. Symbolic Logic44, 417–440, (1979); Japan. J. Math.9, 207–245, (1983); M. Ozawa, J. Math. Soc. Japan36, 589–608, (1984); J. London Math. Soc. (2)32, 141–148, (1985);33, 347–354, (1986); K. Smith, J. Symbolic Logic49, 281–297, (1984); H. Nishimura, J. Symbolic Logic56, 731–741, (1991).

G. Takeuti, J. Math. Soc. Japan35, 1–21, (1983).

M. Ozawa, J. Math. Soc. Japan35, 609–627, (1983).

T. Hinokuma and M. Ozawa, Preprint Series 6, Dept. Math., Coll. Gen. Ed., Nagoya University, 1986; Ill. J. Math., to appear.

M. Ozawa, Journal of the Japan Association for Philosophy of Science18, 35–43, (1986).

I. Ojima, unpublished, 1991.

P. A. Loeb, Trans. Amer. Math. Soc.211, 113–122, (1975).

G. Takeuti, Proc. Japan Acad.38, 414–418, (1962); P. J. Kelemen and A. Robinson, J. Math. Phys.13, 1870–1874, 1875–1878, (1972); R. Kambe, Prog. of Theor. Phys.52, 688–706, (1974); M. O. Farrukh, J. Math. Phys.16, 177–200, (1975); Ph. Blanchard and J. Tarski, Acta Physica Austriaca49, (1978); R. Fittler, Helv. Phys. Acta57, 579–609, (1984);60, 881–902, (1987).

C. C. Chang and H. J. Keisler,Model Theory, Third edition, North-Holland, Amsterdam, 1990.

G. B. Folland,Harmonic Analysis in Phase Space, Princeton University Press, Princeton, 1989.

S. Machida and M. Namiki, Prog. Theor. Phys.63, 1457–1473, 1833–1847, (1980).

H. Araki, Prog. Theor. Phys.64, 719–730, (1980).

M. M. Yanase,Gendai Buturigaku to Atarashii Sekaizou [Modern Physics and New Outlook of the World] (in Japanese), Iwanami, Tokyo, 1984.

G. Takeuti,Chokkan-Shugi-teki Sugoron [Intuitionistic Set Theory] (in Japanese), Kinokuni-ya, Tokyo, 1980.

M. Ozawa, InAbstracts of the Special Session in Set Theory, American Mathematical Society, Honolulu, 1987.

M. Ozawa, In H. Araki, et al. ed.,Current Topics in Operator Algebras, pp. 52–58, World Scientific, Singapore, 1991.