Quantum correlations and an intertwining operator on two dimensional hyperbolic space.

Abstract

In this thesis, we study an intertwining operator on two dimensional hyperbolic space introduced by Anantharaman and Zelditch in \cite{ana.zel.2}. This operator intertwines two natural flows: one is the geodesic flow on the cotangent bundle of hyperbolic space, which can be thought of as the evolution of a classical observable under classical dynamics. The other flow is the flow on the space of quantum observables (operators) given by the conjugation of a quantum observable with the free Schr\"{o}dinger propagator on hyperbolic space. This can be thought of as the evolution of a quantum observable under quantum dynamics.

In the first part of the thesis, we modify the construction of the intertwining operator from \cite{ana.zel.2}. The modified intertwining operator is an isometry on the space of square integrable symbols in a particularly natural way without destroying the intertwining property. This allows us to exactly relate two dynamical quantities on the noncompact hyperbolic space, the quantum correlation and the classical correlation, with no error term. Then for the purposes of trying to extend this exact relation to a compact quotient, we explore the structure of the intertwining operator as a Fourier Integral Operator and we describe some features of the canonical relation associated to the intertwining operator.

We will also study the behaviour of a matrix element $\langle \Op(a)\phi_j, \Op(b)\phi_k\rangle$ when we pull the symbol of $\Op(a)$ back via geodesic flow, i.e. we study the quantity $\langle \Op(g_t^*a)\phi_j, \Op(b)\phi_k\rangle$. The trace over the diagonal matrix elements of $\langle \Op(g_t^*a)\phi_j, \Op(b)\phi_k\rangle$ can be considered the ``principal part'' of the time $t$ quantum correlation between operators $\Op(a)$ and $\Op(b)$. Here $\phi_j$ are a basis of Laplace eigenfunctions on a compact hyperbolic surface and $\Op(a)$ is given by the Zelditch quantisation. We show these matrix elements decay exponentially as $|t|\rightarrow\infty$ under certain assumptions on the symbols. Consequently, the trace also decays in a similar manner under similar assumptions on the symbols. To prove this, we use ideas developed recently on the microlocal analysis of hyperbolic dynamical systems.

Lastly, we provide a chapter exploring similar ideas in the contrasting geometric context of the Euclidean torus. The significance here is that we have the well known Weyl quantisation in this context which satisfies the desirable ``exact Egorov" property we hope to develop an analogy of in the hyperbolic case. We apply our analysis in the Euclidean case towards understanding toral quantum ergodicity at small length scales. Show more