Semiclassical trace formulae using coherent states

Abstract

We derive semiclassical trace formulae including Gutzwiller's trace formula using coherent states. This formulation has several advantages over the usual coordinate-space formulation. Using a coherent-state basis makes it immediately obvious that classical periodic orbits make separate contributions to the trace of the quantum-mechanical time evolution operator. In addition, our approach is manifestly canonically invariant at all stages, and leads to the simplest possible derivation of Gutzwiller's formula.

Spectral statistics of chaotic many-body systems

“…If we want the hopping and interaction terms in a Bose-Hubbard like system to remain comparable we then have to adjust the corresponding coefficients in a way similar to the case If the Hamiltonian is written in Weyl instead of normal ordering one obtains an analogous result however without a SK phase. This is in line with [24] as well as results for the propagator in [27,33]. A derivation along the lines followed here will be given in appendix B; it uses the analogue of the discretised action stated in [33] and the corresponding Hessian is again evaluated with the help of [29].…”

“…is related to but somewhat simpler than the path integral for matrix elements of coherent state time evolution operator [27]; our use of tr e Ht i is motivated by [24,25]. In case of normal ordering the path integral can be derived by splitting the time interval t into  steps of width t  t = and then using the result for the short-time propagator…”

Symplectic Geometry and Quantum Mechanics

“…We will show that provided we choose the integer ν correctly this is the Weyl representation of ± S ∈ Mp(n) such that π Mp ( S) = S (the definition of the operators (7.62) is due to Mehlig and Wilkinson [121], who however do not make precise the integer ν). Anyhow, formula (7.62) defines a Weyl operator with complex Gaussian twisted symbol…”

A Phase-Space Study of the Quantum Loschmidt Echo in the Semiclassical Limit

“…In this caseR 1 (F ) has a smooth Weyl symbol given by the following formula: (see [10] where we have named this formula the "Mehlig-Wilkinson formula", according to the physics literature [22]) R(F, X) = e iπν |det(1l + F )| −1/2 exp −iJ(1l − F )(1l + F ) −1 X · X (4.17)…”

“…These estimates are non-perturbative, and are carefully calculated in terms of parameters (z, δ, t, ). The main tools we have used and developed in this respect are 1) semiclassical coherent states propagation estimates ( [9]) 2) a beautiful formula inspired by B. Mehlig and M. Wilkinson ( [22]) about the Weyl symbol of a metaplectic operator, and thus of its expectation value in coherent states as a simple Gaussian phase-space integral ( see [10] where we have completed the proof of Mehlig-Wilkinson, and treated in particular the case where the monodromy operator has eigenvalue 1). Note that very recently, J. Bolte and T. Schwaibold have independently obtained a similar result about semiclassical estimates of the Quantum Fidelity ( [2]).…”