Weyl-Wigner Representation of Canonical Equilibrium States

TL;DR

This paper develops a comprehensive Weyl-Wigner framework for representing canonical thermal equilibrium states of quadratic Hamiltonians, linking quantum states to classical symplectic structures and analyzing different dynamical regimes.

Contribution

It introduces a method to obtain Weyl-Wigner representations of thermal states for all quadratic Hamiltonians via Wick rotation, connecting quantum states to classical symplectic matrices.

Findings

Thermal states are fully characterized by complex symplectic matrices.

Different Hamiltonian dynamics categories are analyzed and classified.

Semiclassical and high-temperature approximations are derived and compared.

Abstract

The Weyl-Wigner representations for canonical thermal equilibrium quantum states are obtained for the whole class of quadratic Hamiltonians through a Wick rotation of the Weyl-Wigner symbols of Heisenberg and metaplectic operators. The behavior of classical structures inherently associated to these unitaries is described under the Wick mapping, unveiling that a thermal equilibrium state is fully determined by a complex symplectic matrix, which sets all of its thermodynamical properties. The four categories of Hamiltonian dynamics (Parabolic, Elliptic, Hyperbolic, and Loxodromic) are analyzed. Semiclassical and high temperature approximations are derived and compared to the classical and/or quadratic behavior.