Index and dynamics of quantized contact transformations

Zelditch, Steven

Annales de l'Institut Fourier, Tome 47 (1997) no. 1, pp. 305-363.

Les transformations de contact quantifiées sont des opérateurs unitaires de Toeplitz de la forme $U_{\chi }=\Pi A\chi \Pi$ sur une variété $(X,\alpha )$ de contact. Ici, $\Pi :H^2\left(X\right)\to L^2\left(X\right)$ est un projecteur de Szegö, $\chi$ est une transformation de contact, et $A$ est un opérateur pseudodifférentiel sur $X$. On peut quantifier une transformation symplectique ${\chi }_o$ sur une variété symplectique $(M,\omega )$ de cette façon lorsque ${\chi }_o$ se relève en une transformation de contact $\chi$ sur le fibré “pré-quantique” en cercles $X\to M$. On montre que les automorphismes symplectiques ${\chi }_o$ d’un tore $(M,\sum d{x}_i\wedge d{\xi }_i)$ sont de ce type : le fibré $X$ est alors le quotient du groupe de Heisenberg par son réseau entier $\delta$, le projecteur $\Pi$ est le noyau de Szegö, et, à une constante près, $\Pi \chi \Pi$ définit une des lois de transformation de Hermite–Jacobi sur les fonctions thêta. Il en résulte que les applications quantiques du chat (telles qu’elles sont connues dans la littérature physique) ne sont autres que l’action métaplectique du groupe de thêta sur les fonctions thêta. Il résulte aussi que les indices de ces applications symplectiques sont nuls. On donne par ailleurs des résultats généraux sur l’ergodicité quantique des transformations de contact quantifiées, c’est-à-dire, sur les propriétés asymptotiques des valeurs et fonctions propres de $\Pi A\chi \Pi$.

Quantized contact transformations are Toeplitz operators over a contact manifold $(X,\alpha )$ of the form $U_{\chi }=\Pi A\chi \Pi$, where $\Pi :H^2\left(X\right)\to L^2\left(X\right)$ 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 $\mathrm{ind}\left(U_{\chi }\right)$ 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.

[At] M. Atiyah, The Geometry and Physics of Knots, Lezioni Lincee, Cambridge Univ. Press, 1990. | MR | Zbl

[AT] L. Auslander and R. Tolimieri, Is computing with the finite Fourier transform pure or applied mathematics, Bull. AMS, 1 (1979), 847-897. | MR | Zbl

[A] L. Auslander, Lecture Notes on Nil-theta Functions, CBMS series no. 34, AMS Publications (1977). | MR | Zbl

[AdPW] S. Axelrod, S. Della Pietra, and E. Witten, Geometric quantization of the Chern-Simons gauge theory, J.D.G., 33 (1991), 787-902. | MR | Zbl

[Bai] W. Baily, Classical theory of θ-functions, in AMS Proc.Symp.Pure. Math. IX, AMS (1966), 306-311. | MR | Zbl

[B] F. Benatti, Deterministic Chaos in Infinite Quantum Systems, Trieste Notes in Physics, Springer-Verlag (1993).

[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. | MR | Zbl

[BPU] D. Borthwick, T. Paul, A. Uribe, Legendrian distributions with applications to relative Poincaré series, Invent. Math., 122 (1995), 359-402. | EuDML | MR | Zbl

[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. | MR | Zbl

[BG] L. Boutet De Monvel and V. Guillemin, The Spectral Theory of Toeplitz Operators, Ann. Math. Studies 99, Princeton U. Press (1981). | MR | Zbl

[BS] L. Boutet De Monvel and J. Sjöstrand, Sur la singularité des noyaux de Bergmann et de Szegö, Astérisque 34-35, (1976), 123-164. | Numdam | Zbl

[BdB] A. Bouzouina and S. De Bièvre, Equipartition of the eigenfunctions of quantized ergodic maps on the torus, to appear in Comm.Math.Phys. | Zbl

[BR] O. Bratteli and D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics I, Springer-Verlay, 1979. | MR | Zbl

[C] P. Cartier, Quantum mechanical commutation relations and theta functions, in Algebraic Groups and Discontinuous Subgroups, Proc. Symp. Pure Math. AMS (1966). | MR | Zbl

[CV] Y. Colin De Verdière, Ergodicité et functions propres du Laplacian, Comm. Math. Phys., 102 (1985), 497-502. | MR | Zbl

[D] I. Daubechies, Coherent states and projective representations of the linear canonical transformations, J. Math. Phys., 21 (1980), 1377-1389. | MR | Zbl

[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.

[Do] R.G. Douglas, C*-Algebra Extensions and K-Homology, Ann. Math. Studies no. 95, Princeton Univ. Press, Princeton, 1980. | MR | Zbl

[F] G. Folland, Harmonic Analysis in Phase Space, Ann. Math. Studies, no. 122, Princeton Univ. Press, 1989. | MR | Zbl

[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. | MR | Zbl

[G1] V. Guillemin, Residue traces for certain algebras of Fourier Integral operators, J. Fun. Anal., 115 (1993), 381-417. | MR | Zbl

[G2] V. Guillemin, A non-elementary proof of quadratic reciprocity (unpublished manuscript).

[HB] J.H. Hannay and M.V. Berry, Quantization of linear maps on a torus, Physica D1 (1980), 267.

[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.

[Herm] C. Hermite, Sur quelques formules relatives a la transformation des fonctions elliptiques, Journal de Liouville, III (1858), 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. | MR | Zbl

[K] V. Kac, Infinite Dimensional Lie Algebras, 3rd ed. Cambridge : Cambridge Univ. Press, 1990. | Zbl

[KP] V. Kac and D.H. Peterson, Infinite dimensional Lie algebras, theta functions and modular forms, Adv in Math., 53 (1984), 125-264. | MR | Zbl

[Ke] J. Keating, The cat maps : quantum mechanics and classical motion, Nonlinearity, 4 (1991), 309-341. | MR | Zbl

[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. | Zbl

[M] D. Mumford, Tata Lectures on Theta III, Progress in Math. 97, Birkhauser, Boston (1991). | MR | Zbl

[NT1] H. Narnhofer and W. Thirring, Transitivity and ergodicity of quantum systems, J.Stat.Phys., 52 (1988), 1097-1112. | MR | Zbl

[NT2] H. Narnhofer and W. Thirring, Mixing properties of quantum systems, J.Stat.Phys., 57 (1989), 811-825. | MR | Zbl

[R] D. Ruelle, Statistical Mechanics, Benjamin, 1969.

[Sn] A.I. Snirelman, Ergodic properties of eigenfunctions, Usp. Math. Nauk., 29 (1974), 181-182.

[S] E. Stein, Harmonic Analysis, Princeton: Princeton Univ. Press, 1993.

[Su] T. Sunada, Quantum ergodicity, preprint 1994. | Zbl

[Th] W. Thirring, A Course in Mathematical Physics, vol. 4 : Quantum Mechanics of Large Systems, Springer-Verlag, New York, 1983. | Zbl

[UZ] A. Uribe and S. Zelditch, Spectral statistics on Zoll surfaces, Comm. Math. Phys. 154 (1993), 313-346. | MR | Zbl

[W] P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Math. 79, Springer-Verlag, NY (1982). | MR | Zbl

[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).

[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. | MR | Zbl

[Z1] S. Zelditch, Quantum ergodicity of C*-dynamical systems, (Comm.Math.Phys., 177 (1996), 507-528. | MR | Zbl

[Z2] S. Zelditch, Quantum Mixing, J. Fun. Anal., 140 (1996), 68-86. | MR | Zbl

[Z3] S. Zelditch, Quantum transition amplitudes for ergodic and for completely integrable systems, J. Fun. Anal., 94 (1990), 415-436. | MR | Zbl