Index and Dynamics of Quantized Contact Transformations

TL;DR

This paper studies quantized contact transformations as Toeplitz operators on contact manifolds, analyzing their index, trace formulas, and quantum dynamics, with applications to symplectic maps and quantum ergodicity.

Contribution

It introduces a framework for quantized contact transformations, computes their index and traces, and analyzes eigenfunction distribution and mixing properties in quantum dynamics.

Findings

Index is computable for quantized symplectic torus automorphisms.

Trace formulas on theta functions are derived using the Heisenberg group kernel.

Eigenfunctions are shown to be equidistributed under certain ergodic conditions.

Abstract

Quantized contact transformations are Toeplitz operators over a contact manifold $(X,\alpha)$ of the form $U_{\chi} = \Pi A \chi \Pi$, where $\Pi : H^2(X) \to L^2(X)$ is a Szego projector, where $\chi$ is a contact transformation and where $A$ is a pseudodifferential operator over $X$. They provide a flexible alternative to the Kahler 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 $ind(U_{\chi})$ 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-Szego 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.