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Pseudo-Differential Operators and Related Topics pp 161–172Cite as

Distributions and Pseudo-Differential Operators on Infinite-Dimensional Spaces with Applications in Quantum Physics

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Part of the book series: Operator Theory: Advances and Applications ((OT,volume 164))

Abstract

This paper is the review on theory of infinite-dimensional pseudo-differential operators(PDO) and their application to quantization of systems with the infinite number of degrees of freedom. There are considered various approaches to the theory of infinite-dimensional PDO (e.g., Berezin’s approach based on polynomial operators). There is presented in details the calculus of PDO based on the theory of distributions on infinite-dimensional spaces (general locally convex spaces). This calculus is based on the “Feynman measure” on the phase space (introduced by Smolyanov in 80th). Symbols of the most important PDO in the representation of second quantization are calculated.

This paper was partly supported by EU-Network “QP and Applications” and Nat. Sc. Found., grant N PHY99-07949 at KITP, Santa-Barbara, and visiting professor fellowship at Russian State Humanitarian University.

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References

  1. H. Weyl, The Theory of Groups and Quantum Mechanics, Dover, New-York, 1931.

    Google Scholar 

  2. M. A. Shubin, Pseudo-Differential Operators and Spectral Theory, English Transl., Springer Verlag, 1986; “Nauka”, Moscow, 1978.

    Google Scholar 

  3. P. M. Blekher and M. I. Vishik, On a class of pseudo-differential operators with infinitely many variables and applications, Math. USSR Sb. 15 (1971), 446–494.

    Google Scholar 

  4. O. G. Smolyanov, Linear differential operators in spaces of measures and functions on Hilbert space, Uspekhi Mat. Nauk. 28 (1973), 251–252.

    MATH  Google Scholar 

  5. Yu. M. Berezanskij, Self Adjoint Operators in Spaces of Functions of an Infinite Number of Variables, Amer. Math. Soc., Providence, R. I., 1986.

    Google Scholar 

  6. A. Yu. Khrennikov, Equations with infinite-dimensional pseudo-differential operators, Dokl. Akad. Nauk SSSR 267 (1982), 1313–1318.

    MATH  MathSciNet  Google Scholar 

  7. F. A. Berezin, The Method of Second Quantization, Academic Press, 1966.

    Google Scholar 

  8. F. A. Berezin, Representation of operators by means of functionals, Trudy Moscow. Mat. Obschch. 17 (1967), 117–196.

    MATH  MathSciNet  Google Scholar 

  9. G. S. Agarwal and E. Wolf, Calculus for functions of noncommuting operators and general phase-space methods in quantum mechanics I, II, III, Phys. Rev. D (3) 2 (1970), 2161–2186, 2187–2205, 2206–2225.

    MathSciNet  Google Scholar 

  10. O. G. Smolyanov, Infinite-dimensional pseudo-differential operators and Schrödinger quantization, Dokl. Akad. Nauk SSSR 263 (1982), 558–562.

    MATH  MathSciNet  Google Scholar 

  11. A. Yu. Khrennikov, The Feynman measure in phase space and symbols of infinite-dimensional pseudo-differential operators, Math. Notes 37 (1985), 734–742.

    MATH  MathSciNet  Google Scholar 

  12. A. Yu. Khrennikov, Second quantization and pseudo-differential operators, Theoret. Math. Phys. 66 (1986), 339–349.

    MATH  MathSciNet  Google Scholar 

  13. O. G. Smolyanov and A. Yu. Khrennikov, An algebra of infinite-dimensional pseudo-differential operators, Soviet Math. Dokl. 35 (1987), 1310–1314.

    MathSciNet  Google Scholar 

  14. A.Yu. Khrennikov, Superanalysis, Nauka, Fizmatlit, Moscow, 1997 (in Russian). English Translation: Kluwer, Dordreht, 1999.

    Google Scholar 

  15. V. S. Vladimirov, I. V. Volovich and E. I. Zelenov, p-adic Analysis and Mathematical Physics, World Scientific, Singapore, 1993.

    Google Scholar 

  16. A. Yu. Khrennikov, p-Adic Valued Distributions and Their applications to the Mathematical Physics, Kluwer, Dordreht, 1994.

    Google Scholar 

  17. A. Yu. Khrennikov, Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models, Kluwer, Dordreht, 1997.

    Google Scholar 

  18. C. De Witt-Morette, Feynman’s path integral. Definition without limiting procedure, Comm. Math. Phys. 28 (1972), 47–67.

    MathSciNet  Google Scholar 

  19. S. A. Albeverio and R. J. Höegh-Krohn, Mathematical theory of Feynman path integrals, Lecture Notes in Math., 523, Springer-Verlag, Berlin-Heidelberg, 1976.

    Google Scholar 

  20. A. A. Slavnov and L. D. Faddeev, Introduction to the Quantum Theory of Gauge Fields, Nauka, Moscow, 1978, English Transl., Gauge Fields, Benjamin/Cummings, Reading, Mass., 1980.

    Google Scholar 

  21. A. V. Uglanov, Feynman measures with correlation operators of indefinite sign, Soviet Math. Dokl. 25 (1982), 37–40.

    MATH  MathSciNet  Google Scholar 

  22. O. G. Smolyanov and A. Yu. Khrennikov, The central limit theorem for generalized measures on infinite-dimensional spaces, Soviet Math. Dokl. 31 (1985), 279–283.

    MathSciNet  Google Scholar 

  23. A. Yu. Khrennikov, Integration with respect to generalized measures on linear topological spaces, Trans. Moscow Math. Soc. 49 (1987), 113–129.

    MATH  MathSciNet  Google Scholar 

  24. A. Yu. Khrennikov, Infinite-dimensional pseudo-differential operators, Izv. Akad. Nauk USSR, ser. Math. 51 (1987), 46–68.

    Google Scholar 

  25. J. Glimm and A. Jaffe, Boson quantum field models, in Mathematics of Contemporary Physics, Proc. Instructional Conf., Editor: R. F. Streater, Academic Press, 1972, 77–143.

    Google Scholar 

  26. B. Simon, The P(Φ)2Euclidean (Quantum) Field Theory, Princeton University Press, Princeton, N. J., 1974.

    Google Scholar 

  27. V. S. Vladimirov and I. V. Volovich, Superanalysis I: Differential calculus, Theoret. Math. Phys. 59 (1984), 3–27.

    MathSciNet  Google Scholar 

  28. V. S. Vladimirov and I. V. Volovich, Superanalysis II: Integral calculus, Theoret. Math. Phys. 60 (1984), 169–198.

    MathSciNet  Google Scholar 

  29. A. Yu. Khrennikov, Superanalysis: theory of generalized functions and pseudo-differential operators, Theoret. Math. Phys. 72 (1987), 420–429.

    MathSciNet  Google Scholar 

  30. A. Yu. Khrennikov, Pseudo-differential equations in functional superanalysis, 1. Method of Fourier transform, Diff. Equations 24 (1988), 2144–2154.

    MATH  MathSciNet  Google Scholar 

  31. A. Yu. Khrennikov, Pseudo-differential equations in functional superanalysis, 2. Formula of Feynman-Kac, Diff. Equations 25 (1989) 314–324.

    MathSciNet  Google Scholar 

  32. A. Yu. Khrennikov, Equations on the superspace, Izvestia Academii Nauk USSR, ser. Math. 54 (1990), 556–606.

    Google Scholar 

  33. S. Albeverio and A. Yu. Khrennikov, p-adic Hilbert space representation of quantum systems with an infinite number of degrees of freedom, Int. J. of Modern Phys. B 10 (1996), 1665–1673.

    MathSciNet  Google Scholar 

  34. A. Yu. Khrennikov and M. Nilsson, p-Adic Deterministic and Random Dynamical Systems. Kluwer, Dordreht, 2004.

    Google Scholar 

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Author information

Authors and Affiliations

  1. International Center for Mathematical Modeling in Physics and Cognitive Sciences, University of Växjö, Sweden

    Andrei Khrennikov

Authors
  1. Andrei Khrennikov
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Editor information

Editors and Affiliations

  1. Dipartimento di Matematica, Università di Torino, Via Carlo Alberto, 10, 10123, Torino, Italy

    Paolo Boggiatto  & Luigi Rodino  & 

  2. School of Mathematics and Systems Engineering, Växjö University, SE-351 95, Växjö, Sweden

    Joachim Toft

  3. Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, Ontario, M3J 1P3, Canada

    M. W. Wong

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© 2006 Birkhäuser Verlag Basel/Switzerland

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Khrennikov, A. (2006). Distributions and Pseudo-Differential Operators on Infinite-Dimensional Spaces with Applications in Quantum Physics. In: Boggiatto, P., Rodino, L., Toft, J., Wong, M.W. (eds) Pseudo-Differential Operators and Related Topics. Operator Theory: Advances and Applications, vol 164. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7514-0_12

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