Index and dynamics of quantized contact transformations

Quantized contact transformations are Toeplitz operators over a contact manifold (X,�) of the form U� = �A��, where � : H 2 (X)! L 2 (X) is a Szego projector, whereis a contact transformation and where A is a pseudodifferential operator over X. They provide a flexible alternative to the Kahler quantiza- tion 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�) when the principal symbol is unitary, or equivalently to determine whether A can be chosen so that Uis unitary. We show that the answer is yes in the case of quantized symplectic torus automorphisms g—by showing that Ug duplicates the classical transformation laws on theta functions. Using the Cauchy-Szego kernel on the Heisenberg group, we cal- culate the traces on theta functions of each degree N. We also study the quantum dynamics generated by a general q.c.t. U�, i.e. the quasi-classical asymptotics of the eigenvalues and eigenfunctions under various ergodicity and mixing hypotheses on �. Our principal results are proofs of equidistribution of eigenfunctions �Nj and weak mixing properties of matrix elements (BNi,�Nj) for quantizations of mixing symplectic maps. The problem of quantizing symplectic maps and of analyzing the dynamics of the quantum system is a very basic one in mathematical physics, and has been studied extensively, by both mathematicians and physicists, from many different points of view. The present article is concerned with one such quantization method, that of Toeplitz quantization, and with one particular viewpoint towards the ergodicity and mixing properties of the quantized maps, that of quantized Gelfand-Naimark-Segal systems in the sense of (Z.1). We will describe a method of quantizing contact transformations of a contact manifold (X, α) with periodic contact flow as unitary operators on an associated Hardy space H 2 (X) , and prove a number of results on the index and dynamics of the quantized contact transformations. The method, essentially a unitarized version of Boutet de Monvel's Toeplitz quantization (B)(B.G), is closely related to the geometric quantization of symplectic maps on Kahler manifolds and produces new examples of quantized GNS systems. The quantum ergodicity theorems follow in part from the general results of (Z.1), but also include some sharper ergodicity and mixing theorems analogous to those of (Z.2,3) in the case of wave groups. To illustrate the method and ergodicity results we will also study in detail the Toeplitz quantization of symplectic torus automorphisms ( 'cat maps') (§5), undoubtedly the most popular of maps to undergo quantization- see (A.d'P.W)(B.N.S)(H.B)(dB.B)(d'E.G.I)(K.P)(K)(Ke)(We) for just a few among the many treatments. As the reader is surely aware, quantization is not a uniquely defined process and it is not apriori clear how the plethora of quantizations defined in these articles are related to each other or to the quantization presented here. In fact, although it is not quite obvious, all but that of (B.N.S) are equivalent to the Toeplitz quantization presented here. We will describe the relations between them more precisely at the end of the introduction. What is more, it will be proved in §5 that the Toeplitz quantization of SL(2, Z) reproduces what must be the quantization of most ancient vintage—-namely, the Hermite-Jacobi action of SL(2, Z) (or more precisely its theta-subgroup SLθ(2, Z)) on spacesN of theta functions of any degree N (cf. (Herm), (Kloo) or, for a modern treatment, (K)(K.P)). We construct this action by lifting g ∈ SLθ(2, Z) to a contact transformation χg of NR/NZ, the quotient of the Heisenberg group by its integral subgroup, and compressing the latter to the spacesN. This connection develops the long chain of links between theta functions and harmonic