Preprints

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.

The objective of this text is to present (and encourage the use of) the Williamson theorem and its consequences in several contexts in physics. The demonstration of the theorem is performed using only basic concepts of linear algebra and symplectic matrices. The immediate application is to place the study of small oscillations in the Hamiltonian scenario, where the theorem shows itself as a useful and practical tool for revealing the normal-mode coordinates and frequencies of the system. A modest introduction of the symplectic formalism in quantum mechanics is presented, which consequently opens up the use of the theorem to study quantum normal modes and quantum small oscillations, allowing the theorem to be applied to the canonical distribution of thermodynamically stable systems described by quadratic Hamiltonians. As a last example, a more advanced topic concerning uncertainty relations is developed to show once more its utility in a distinct and modern perspective.