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Emergence of quantum dynamics from chaos: The case of prequantum cat maps

Javier Echevarría Cuesta

TL;DR

The paper extends Faure–Tsujii's prequantization framework to hyperbolic symplectic toral automorphisms (cat maps) on $ ext{T}^{2n}$, constructing a concrete prequantum bundle and explicit prequantum cat maps. It then links prequantum and quantum Hilbert spaces via the Schrödinger representation and a decomposition map, enabling a tensorial factorization of the prequantum transfer operator. The main contribution is a precise relation: for hyperbolic $M$ with $oldsymbol{ abla}$, the Pollicott–Ruelle resonances of the prequantum transfer operator $F_N$ factor as products of quantum cat map eigenvalues and resonances of a metaplectic/expanding component, thereby revealing how quantum dynamics emerges dynamically in correlation functions. This provides a rigorous dynamical mechanism for quantum behavior in chaotic systems on the torus and generalizes previous $n=1$ results, with explicit constructions and an illustrative example.

Abstract

Faure and Tsujii recently proposed a new quantization theory for symplectic Anosov diffeomorphisms. It combines prequantization with the study of the Pollicott--Ruelle resonances of an associated transfer operator. We apply this framework to the hyperbolic symplectic automorphisms of the $2n$-dimensional torus, the so-called cat maps. Our main result gives an explicit relation between the resonances of the prequantum transfer operator and the eigenvalues of the standard quantum cat maps, generalizing the case $n=1$ previously treated by Faure.

Emergence of quantum dynamics from chaos: The case of prequantum cat maps

TL;DR

The paper extends Faure–Tsujii's prequantization framework to hyperbolic symplectic toral automorphisms (cat maps) on , constructing a concrete prequantum bundle and explicit prequantum cat maps. It then links prequantum and quantum Hilbert spaces via the Schrödinger representation and a decomposition map, enabling a tensorial factorization of the prequantum transfer operator. The main contribution is a precise relation: for hyperbolic with , the Pollicott–Ruelle resonances of the prequantum transfer operator factor as products of quantum cat map eigenvalues and resonances of a metaplectic/expanding component, thereby revealing how quantum dynamics emerges dynamically in correlation functions. This provides a rigorous dynamical mechanism for quantum behavior in chaotic systems on the torus and generalizes previous results, with explicit constructions and an illustrative example.

Abstract

Faure and Tsujii recently proposed a new quantization theory for symplectic Anosov diffeomorphisms. It combines prequantization with the study of the Pollicott--Ruelle resonances of an associated transfer operator. We apply this framework to the hyperbolic symplectic automorphisms of the -dimensional torus, the so-called cat maps. Our main result gives an explicit relation between the resonances of the prequantum transfer operator and the eigenvalues of the standard quantum cat maps, generalizing the case previously treated by Faure.
Paper Structure (14 sections, 13 theorems, 122 equations, 3 figures)

This paper contains 14 sections, 13 theorems, 122 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Statement of the results
  3. Prequantization of classical cat maps
  4. Construction of the prequantum bundle
  5. Construction of the prequantum cat maps
  6. The relationship between prequantum and quantum Hilbert spaces
  7. The Schrödinger representation
  8. Appearance of the quantum Hilbert space
  9. Decomposition of the prequantum Hilbert spaces
  10. Study of the prequantum transfer operator
  11. Quantum cat maps
  12. Factorization of the prequantum transfer operator
  13. Pollicott--Ruelle spectrum of the prequantum transfer operator
  14. Example

Key Result

Theorem 2.6

Let $M\in \textup{Sp}(2n,\mathbb{Z})$ be hyperbolic. We can find $E\in \textup{GL}(n,\mathbb{R})$ satisfying $\Vert E^{-1}\Vert <1$ and $|\det E|>1$ such that $M$ is symplectically conjugate to Suppose that $\varphi_M=0$. Then, for any prequantum transfer operator $F$, the Pollicott--Ruelle resonances of $F_N$ with $N\in \mathbb{N}^*$ are, up to a global phase, given by where $\{\lambda_j\}_{j\i

Figures (3)

  • Figure 1: Spectra for a cat map $M\in \mathrm{Sp}(6,\mathbb{Z})$, with $N=2$.
  • Figure 2: Eigenvalues of the hyperbolic matrix $M\in \text{Sp}(6, \mathbb{Z})$.
  • Figure 3: Resonances of the operator $L_E$ and the annuli \ref{['eq:band-structure-2']}.

Theorems & Definitions (36)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Theorem 2.6
  • Remark 2.7
  • Theorem 2.8
  • Proposition 3.1
  • proof
  • ...and 26 more