Non-positivity of Groenewold operatorsA. J. Bracken and J. G. Wood
Centre for Mathematical Physics - Department of Mathematics University of Queensland - Brisbane, Australia 4072
received 8 July 2004; accepted 4 August 2004
A central feature in the Hilbert space formulation of classical mechanics is the quantisation of classical Liouville densities, leading to what may be termed Groenewold operators. We investigate the spectra of the Groenewold operators that correspond to Gaussian and to certain uniform Liouville densities. We show that when the classical coordinate-momentum uncertainty product falls below Heisenberg's limit, the Groenewold operators in the Gaussian case develop negative eigenvalues and eigenvalues larger than 1. However, in the uniform case, negative eigenvalues are shown to persist for arbitrarily large values of the classical uncertainty product.
03.65.Ta - Foundations of quantum mechanics; measurement theory.
03.65.Sq - Semiclassical theories and applications.
45.20.-d - Formalisms in classical mechanics.
© EDP Sciences 2004