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Weyl calculus in Wiener spaces and in QED

  • L. Amour
  • R. Lascar
  • J. NourrigatEmail author
Article
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Abstract

The concern of this article is a semiclassical Weyl calculus on an infinite dimensional Hilbert space H. If (iHB) is a Wiener triplet associated to H, the quantum state space will be the space of \(L^2\) functions on B with respect to a Gaussian measure with h / 2 variance, where h is the semiclassical parameter. We prove the boundedness of our pseudodifferential operators (PDO) in the spirit of Calderón–Vaillancourt with an explicit bound, a Beals type characterization, and metaplectic covariance. An application to a model of quantum electrodynamics is added in the last section (Sect. 7), for fixed spin 1 / 2 particles interacting with the quantized electromagnetic field (photons). We prove that some observable time evolutions, the spin evolutions, the magnetic and electric evolutions when subtracting their free evolutions, are PDO in our class.

Keywords

Pseudodifferential operators Semiclassical analysis Infinite dimensional analysis Calderón–Vaillancourt bounds Beals characterization Covariance Metaplectic group Spins interaction Quantum electrodynamics Photon number Bogoliubov transforms 

Notes

References

  1. 1.
    Amour, L., Jager, L., Nourrigat, J.: On bounded Weyl pseudodifferential operators in Wiener spaces. J. Funct. Anal. 269, 2747–2812 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Amour, L., Lascar, R., Nourrigat, J.: Beals characterization of pseudodifferential operators in Wiener spaces. Appl. Math. Res. Express 1, 242–270 (2017)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Amour, L., Lascar, R., Nourrigat, J.: Weyl calculus in QED I. The unitary group. J. Math. Phys. 58(1), 013501 (2017).  https://doi.org/10.1063/1.4973742 MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bach, V., Fröhlich, J., Sigal, I.M.: Quantum electrodynamics of confined nonrelativistic particles. Adv. Math. 137(2), 299–395 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Beals, R.: Characterization of pseudodifferential operators and applications. Duke Math. J. 44(1), 45–57 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bony, J.M.: Caractérisation des opd. Séminaire EDP, X. Exposé 23, p. 17 (1996-1997)Google Scholar
  7. 7.
    Bony, J.M.: Characterization of pseudo-differential operators. In: Progress in Non linear Differential Equations and Their Applications. Vol. 84, pp. 21–34. Birkhaüser Springer, New York (2013)Google Scholar
  8. 8.
    Bony, J.M., Chemin, J.Y.: Espaces fonctionnels associés au calcul de Weyl-Hörmander. Bull. Soc. Math. Fr. 122(1), 77–118 (1994)CrossRefzbMATHGoogle Scholar
  9. 9.
    Boutet de Monvel, L., Kree, P.: Pseudodifferential operators and Gevrey classes. Ann. Inst. Fourier 17, 295–323 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bruneau, L., Dereziński, J.: Bogoliubov Hamiltonians and one-parameter groups of Bogoliubov transformations. J. Math. Phys. 48(2), 022101 (2007). 24 ppMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Calderón, A.P., Vaillancourt, R.: A class of bounded pseudo-differential operators. Proc. Natl. Acad. Sci. U.S.A. 69, 1185–1187 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Combescure, M., Robert, D.: Coherent states and applications in mathematical physics. In: Theoretical and Mathematical Physics. pp. xiv+415. Springer, Dordrecht (2012). ISBN: 978-94-007-0195-3Google Scholar
  13. 13.
    Dereziński, J., Gérard, C.: Asymptotic completeness in quantum field theory. Massive Pauli–Fierz Hamiltonians. Rev. Math. Phys. 11(4), 383–450 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gohberg, I.C., Krein, M.G.: Introduction to the Theory of Linear Nonselfadjoint Operators. Translations of Mathematical Monographs, vol. 18. American Mathematical Society, Providence (1969)zbMATHGoogle Scholar
  15. 15.
    Gross, L.: Abstract Wiener spaces. In: Proceedings of the 5th Berkeley Sym. Math. Stat. Prob, vol. 2, pp. 31–42 (1965)Google Scholar
  16. 16.
    Hiroshima, F., Sasaki, I., Spohn, H., Suzuki, A.: Enhanced Binding in Quantum Field Theory. Kyushu University COE Lecture Note 38 (2012). arXiv:1203.1136
  17. 17.
    Hörmander, L.: The Analysis of Linear Partial Differential Operators, III edn. Springer, Berlin (1985)zbMATHGoogle Scholar
  18. 18.
    Jager, L.: Stochastic extensions of symbols in Wiener spaces and heat operator. arXiv:1607.02253 (2016)
  19. 19.
    Janson, S.: Gaussian Hilbert Spaces. Cambridge Tracts in Maths, vol. 129. Cambridge Univ. Press, Cambridge (1997)CrossRefzbMATHGoogle Scholar
  20. 20.
    Kuo, H.H.: Gaussian Measures in Banach Spaces. Lecture Notes in Mathematics, vol. 463. Springer, Berlin-New York (1975)CrossRefGoogle Scholar
  21. 21.
    Lascar, B.: Une classe d’opérateurs elliptiques du second ordre sur un espace de Hilbert. J. Funct. Anal. 35(3), 316–343 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Lerner, N.: Metrics on the Phase Space and Non-selfadjoint Pseudo-differential Operators, Pseudo-Differential Operators. Theory and Applications, vol. 3. Birkhuser Verlag, Basel (2010)CrossRefzbMATHGoogle Scholar
  23. 23.
    Ramer, R.: On nonLinear transformations of Gaussian measures. J. Funct. Anal. 15, 166–187 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics. Academic Press, New York, London (1978)zbMATHGoogle Scholar
  25. 25.
    Shale, D.: Linear symmetries of free boson fields. Trans. Am. Math. Soc. 103, 149–167 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Simon, B.: The \(P(\varphi )_2\) Euclidean (Quantum) Field Theory. Princeton Series in Physics. Princeton University Press, Princeton (1974)Google Scholar
  27. 27.
    Taylor, M.E.: Pseudodifferential Operators. Princeton Mathematical Series, vol. 34. Princeton University Press, Princeton (1981)CrossRefzbMATHGoogle Scholar
  28. 28.
    Unterberger, A.: Oscillateur harmonique et opérateurs pseudo-différentiels. Ann. Inst. Fourier (Grenoble) 29(3), 201–221 (1979). xiMathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Zworski, M.: Semiclassical Analysis. Graduate Studies in Mathematics, vol. 138. American Mathematical Society, Providence (2012)zbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.LMR FRE 2011Université de Reims Champagne-ArdenneReims Cedex 2France
  2. 2.Laboratoire J.A. Dieudonné, UMR CNRS 7351Université de Nice Sophia-AntipolisNice Cedex 02France

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