Index and dynamics of quantized contact transformations

Steven Zelditch

Annales de l'institut Fourier (1997)

  • Volume: 47, Issue: 1, page 305-363
  • ISSN: 0373-0956

Abstract

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Quantized contact transformations are Toeplitz operators over a contact manifold ( X , α ) of the form U χ = Π A χ Π , where Π : H 2 ( X ) → L 2 ( X ) is a Szegö projector, where χ is a contact transformation and where A is a pseudodifferential operator over X . They provide a flexible alternative to the Kähler quantization of symplectic maps, and encompass many of the examples in the physics literature, e.g. quantized cat maps and kicked rotors. The index problem is to determine ind ( U χ ) when the principal symbol is unitary, or equivalently to determine whether A can be chosen so that U χ is unitary. We show that the answer is yes in the case of quantized symplectic torus automorphisms g —by showing that U g duplicates the classical transformation laws on theta functions. Using the Cauchy-Szegö kernel on the Heisenberg group, we calculate the traces on theta functions of each degree N . We also study the quantum dynamics generated by a general q.c.t. U χ , i.e. the quasi-classical asymptotics of the eigenvalues and eigenfunctions under various ergodicity and mixing hypotheses on χ . Our principal results are proofs of equidistribution of eigenfunctions ϕ N j and weak mixing properties of matrix elements ( B ϕ N i , ϕ N j ) for quantizations of mixing symplectic maps.

How to cite

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Zelditch, Steven. "Index and dynamics of quantized contact transformations." Annales de l'institut Fourier 47.1 (1997): 305-363. <http://eudml.org/doc/75230>.

@article{Zelditch1997,
abstract = {Quantized contact transformations are Toeplitz operators over a contact manifold $(X,\alpha )$ of the form $U_\{\chi \} = \Pi A \chi \Pi $, where $\Pi : H^2(X) \rightarrow L^2(X)$ is a Szegö projector, where $\chi $ is a contact transformation and where $A$ is a pseudodifferential operator over $X$. They provide a flexible alternative to the Kähler quantization of symplectic maps, and encompass many of the examples in the physics literature, e.g. quantized cat maps and kicked rotors. The index problem is to determine $\{\rm ind\}(U_\{\chi \})$ when the principal symbol is unitary, or equivalently to determine whether $A$ can be chosen so that $U_\{\chi \}$ is unitary. We show that the answer is yes in the case of quantized symplectic torus automorphisms $g$—by showing that $U_g$ duplicates the classical transformation laws on theta functions. Using the Cauchy-Szegö kernel on the Heisenberg group, we calculate the traces on theta functions of each degree $N$. We also study the quantum dynamics generated by a general q.c.t. $U_\{\chi \}$, i.e. the quasi-classical asymptotics of the eigenvalues and eigenfunctions under various ergodicity and mixing hypotheses on $\chi .$ Our principal results are proofs of equidistribution of eigenfunctions $\phi _\{Nj\}$ and weak mixing properties of matrix elements $(B\phi _\{Ni\}, \phi _\{Nj\})$ for quantizations of mixing symplectic maps.},
author = {Zelditch, Steven},
journal = {Annales de l'institut Fourier},
keywords = {theta function; quantized symplectic torus; quantized contact transformations; Toeplitz operators; Szegö projector; pseudodifferential operator; Kähler quantization; index problem; Cauchy-Szegö kernel; Heisenberg group; quantum dynamics; mixing symplectic maps},
language = {eng},
number = {1},
pages = {305-363},
publisher = {Association des Annales de l'Institut Fourier},
title = {Index and dynamics of quantized contact transformations},
url = {http://eudml.org/doc/75230},
volume = {47},
year = {1997},
}

TY - JOUR
AU - Zelditch, Steven
TI - Index and dynamics of quantized contact transformations
JO - Annales de l'institut Fourier
PY - 1997
PB - Association des Annales de l'Institut Fourier
VL - 47
IS - 1
SP - 305
EP - 363
AB - Quantized contact transformations are Toeplitz operators over a contact manifold $(X,\alpha )$ of the form $U_{\chi } = \Pi A \chi \Pi $, where $\Pi : H^2(X) \rightarrow L^2(X)$ is a Szegö projector, where $\chi $ is a contact transformation and where $A$ is a pseudodifferential operator over $X$. They provide a flexible alternative to the Kähler quantization of symplectic maps, and encompass many of the examples in the physics literature, e.g. quantized cat maps and kicked rotors. The index problem is to determine ${\rm ind}(U_{\chi })$ when the principal symbol is unitary, or equivalently to determine whether $A$ can be chosen so that $U_{\chi }$ is unitary. We show that the answer is yes in the case of quantized symplectic torus automorphisms $g$—by showing that $U_g$ duplicates the classical transformation laws on theta functions. Using the Cauchy-Szegö kernel on the Heisenberg group, we calculate the traces on theta functions of each degree $N$. We also study the quantum dynamics generated by a general q.c.t. $U_{\chi }$, i.e. the quasi-classical asymptotics of the eigenvalues and eigenfunctions under various ergodicity and mixing hypotheses on $\chi .$ Our principal results are proofs of equidistribution of eigenfunctions $\phi _{Nj}$ and weak mixing properties of matrix elements $(B\phi _{Ni}, \phi _{Nj})$ for quantizations of mixing symplectic maps.
LA - eng
KW - theta function; quantized symplectic torus; quantized contact transformations; Toeplitz operators; Szegö projector; pseudodifferential operator; Kähler quantization; index problem; Cauchy-Szegö kernel; Heisenberg group; quantum dynamics; mixing symplectic maps
UR - http://eudml.org/doc/75230
ER -

References

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  1. [At] M. ATIYAH, The Geometry and Physics of Knots, Lezioni Lincee, Cambridge Univ. Press, 1990. Zbl0729.57002MR92b:57008
  2. [AT] L. AUSLANDER and R. TOLIMIERI, Is computing with the finite Fourier transform pure or applied mathematics, Bull. AMS, 1 (1979), 847-897. Zbl0475.42014MR81e:42020
  3. [A] L. AUSLANDER, Lecture Notes on Nil-theta Functions, CBMS series no. 34, AMS Publications (1977). Zbl0421.22001MR57 #6289
  4. [AdPW] S. AXELROD, S. DELLA PIETRA, and E. WITTEN, Geometric quantization of the Chern-Simons gauge theory, J.D.G., 33 (1991), 787-902. Zbl0697.53061MR92i:58064
  5. [Bai] W. BAILY, Classical theory of θ-functions, in AMS Proc.Symp.Pure. Math. IX, AMS (1966), 306-311. Zbl0178.55002MR34 #7827
  6. [B] F. BENATTI, Deterministic Chaos in Infinite Quantum Systems, Trieste Notes in Physics, Springer-Verlag (1993). 
  7. [BNS] F. BENATTI, H. NARNHOFER, and G.L. SEWELL, A non-commutative version of the Arnold cat map, Lett. Math. Phys., 21 (1991), 157-172. Zbl0722.46033MR92m:46103a
  8. [BPU] D. BORTHWICK, T. PAUL, A. URIBE, Legendrian distributions with applications to relative Poincaré series, Invent. Math., 122 (1995), 359-402. Zbl0859.58015MR97a:58188
  9. [B] L. BOUTET DE MONVEL, Toeplitz operators—an asymptotic quantization of symplectic cones, in : Stochastic Processes and Their Applications, S. Albeverio (Ed.), Kluwer Acad. Pub. Netherlands, 1990. Zbl0735.47014MR92h:58190
  10. [BG] L. BOUTET DE MONVEL and V. GUILLEMIN, The Spectral Theory of Toeplitz Operators, Ann. Math. Studies 99, Princeton U. Press (1981). Zbl0469.47021MR85j:58141
  11. [BS] L. BOUTET DE MONVEL and J. SJÖSTRAND, Sur la singularité des noyaux de Bergmann et de Szegö, Astérisque 34-35, (1976), 123-164. Zbl0344.32010
  12. [BdB] A. BOUZOUINA and S. DE BIÈVRE, Equipartition of the eigenfunctions of quantized ergodic maps on the torus, to appear in Comm.Math.Phys. Zbl0876.58041
  13. [BR] O. BRATTELI and D.W. ROBINSON, Operator Algebras and Quantum Statistical Mechanics I, Springer-Verlay, 1979. Zbl0421.46048MR81a:46070
  14. [C] P. CARTIER, Quantum mechanical commutation relations and theta functions, in Algebraic Groups and Discontinuous Subgroups, Proc. Symp. Pure Math. AMS (1966). Zbl0178.28401MR35 #7654
  15. [CV] Y. COLIN DE VERDIÈRE, Ergodicité et functions propres du Laplacian, Comm. Math. Phys., 102 (1985), 497-502. Zbl0592.58050MR87d:58145
  16. [D] I. DAUBECHIES, Coherent states and projective representations of the linear canonical transformations, J. Math. Phys., 21 (1980), 1377-1389. Zbl0453.22012MR81f:81023
  17. [dEGI] M. D'EGLI ESPOSTI, S. GRAFFI, and S. ISOLA, Stochastic properties of the quantum Arnold cat in the classical limit, Comm. Math. Phys., 167 (1995), 471-509. 
  18. [Do] R.G. DOUGLAS, C*-Algebra Extensions and K-Homology, Ann. Math. Studies no. 95, Princeton Univ. Press, Princeton, 1980. Zbl0458.46049MR82c:46082
  19. [F] G. FOLLAND, Harmonic Analysis in Phase Space, Ann. Math. Studies, no. 122, Princeton Univ. Press, 1989. Zbl0682.43001MR92k:22017
  20. [FS] G. FOLLAND and E. STEIN, Estimates for the ∂b complex and analysis on the Heisenberg group, Comm. P.A.M., 27 (1974), 429-522. Zbl0293.35012MR51 #3719
  21. [G1] V. GUILLEMIN, Residue traces for certain algebras of Fourier Integral operators, J. Fun. Anal., 115 (1993), 381-417. Zbl0791.35162MR95a:58123
  22. [G2] V. GUILLEMIN, A non-elementary proof of quadratic reciprocity (unpublished manuscript). 
  23. [HB] J.H. HANNAY and M.V. BERRY, Quantization of linear maps on a torus, Physica D1 (1980), 267. Zbl1194.81107
  24. [H] E.J. HELLER, In : Chaos and Quantum Physics, Les Houches 1989 (ed. by M.J. Giannoni, A. Voros and J. Zinn-Justin), Amsterdam, North Holland, 1991. 
  25. [Herm] C. HERMITE, Sur quelques formules relatives a la transformation des fonctions elliptiques, Journal de Liouville, III (1858), 26. 
  26. [JP] V. JAKSIC and C.A. PILLET, On a model of quantum friction III : Ergodic properties of the spin-boson system, Comm. Math. Phys. 178 (1996), 627-651. Zbl0864.47049MR98e:82035
  27. [K] V. KAC, Infinite Dimensional Lie Algebras, 3rd ed. Cambridge : Cambridge Univ. Press, 1990. Zbl0716.17022
  28. [KP] V. KAC and D.H. PETERSON, Infinite dimensional Lie algebras, theta functions and modular forms, Adv in Math., 53 (1984), 125-264. Zbl0584.17007MR86a:17007
  29. [Ke] J. KEATING, The cat maps : quantum mechanics and classical motion, Nonlinearity, 4 (1991), 309-341. Zbl0726.58037MR92m:81055
  30. [Kloo] H.D. KLOOSTERMAN, The behaviour of general theta functions under the modular group and the characters of binary modular congruence groups. I, Ann. Math., 47 (1946), 317. Zbl0063.03262
  31. [M] D. MUMFORD, Tata Lectures on Theta III, Progress in Math. 97, Birkhauser, Boston (1991). Zbl0744.14033MR93d:14065
  32. [NT1] H. NARNHOFER and W. THIRRING, Transitivity and ergodicity of quantum systems, J.Stat.Phys., 52 (1988), 1097-1112. Zbl1084.46054MR89i:46073
  33. [NT2] H. NARNHOFER and W. THIRRING, Mixing properties of quantum systems, J.Stat.Phys., 57 (1989), 811-825. Zbl0716.60133MR90k:82020
  34. [R] D. RUELLE, Statistical Mechanics, Benjamin, 1969. 
  35. [Sn] A.I. SNIRELMAN, Ergodic properties of eigenfunctions, Usp. Math. Nauk., 29 (1974), 181-182. 
  36. [S] E. STEIN, Harmonic Analysis, Princeton: Princeton Univ. Press, 1993. 
  37. [Su] T. SUNADA, Quantum ergodicity, preprint 1994. Zbl0891.58015
  38. [Th] W. THIRRING, A Course in Mathematical Physics, vol. 4 : Quantum Mechanics of Large Systems, Springer-Verlag, New York, 1983. Zbl0491.46057
  39. [UZ] A. URIBE and S. ZELDITCH, Spectral statistics on Zoll surfaces, Comm. Math. Phys. 154 (1993), 313-346. Zbl0791.58102MR95b:58161
  40. [W] P. WALTERS, An Introduction to Ergodic Theory, Graduate Texts in Math. 79, Springer-Verlag, NY (1982). Zbl0475.28009MR84e:28017
  41. [Wei] A. WEINSTEIN, Fourier Integral Operators, quantization, and the spectrum of a Riemannian manifold, Colloques Internationaux C.N.R.S. 237, Géométrie Symplectique et Physique (1976). 
  42. [We] J. WEITSMAN, Quantization via real polarization of the moduli space of flat connections and Chern-Simons gauge theory in genus one, Comm.Math.Phys., 137 (1991), 175-190. Zbl0717.53065MR92f:58071
  43. [Z1] S. ZELDITCH, Quantum ergodicity of C*-dynamical systems, (Comm.Math.Phys., 177 (1996), 507-528. Zbl0856.58019MR98c:46145
  44. [Z2] S. ZELDITCH, Quantum Mixing, J. Fun. Anal., 140 (1996), 68-86. Zbl0858.58049MR98f:58197
  45. [Z3] S. ZELDITCH, Quantum transition amplitudes for ergodic and for completely integrable systems, J. Fun. Anal., 94 (1990), 415-436. Zbl0721.58051MR92b:58181

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