2015/07/16 by Alfredo M. Ozorio de Almeida, Olivier Brodier
Computer Science · Mathematics · Physics and Astronomy · #Correlation function (quantum field theory) #Hilbert space #Kernel (algebra) #Mathematical physics #Mathematics #Nonlinear Dynamics and Pattern Formation #Observable #Operator (biology) #Phase space #Physics #Pure mathematics #Quantum #Quantum chaos and dynamical systems #Quantum mechanics #Semiclassical physics #Spectroscopy and Quantum Chemical Studies #Unitary operator #Unitary transformation #Wigner distribution function #quant-ph
paper · pdf · doi:10.1088/1751-8113/49/18/185302
18 pages, 6 figures
arxiv created 2015/07/16 · openalex publication_date 2016/03/22 · arxiv updated 2016/04/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
The trace of an arbitrary product of quantum operators with the density operator is rendered as a multiple phase space integral of the product of their Weyl symbols with the Wigner function. Interspersing the factors with various evolution operators, one obtains an evolving correlation. The kernel for the matching multiple integral that evolves within the Weyl representation is identified with the trace of a single compound unitary operator. Its evaluation within a semiclassical approximation then becomes a sum over the periodic trajectories of the corresponding classical compound canonical transformation. The search for periodic trajectories can be bypassed by an exactly equivalent initial value scheme, which involves a change of integration variable and a reduced compound unitary operator. Restriction of all the operators to observables with smooth non-oscillatory Weyl symbols leads naturally to classical evolution within a single phase space integral, if each observable undergoes independent Heisenberg evolution. The linear response function for infrared spectroscopy provides an example depending on a double phase space integral. This is compared to an alternative scheme that employs the widely used Herman-Kluk propagator.