Non-positivity of Groenewold operators

Abstract

A central feature in the Hilbert space formulation of classical mechanics is the quantisation of classical Liouville densities, leading to what may be termed term 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.

9 pages, 1 figures, submitted to Europhysics Letters

References in corpus (4)

• Admissible states in quantum phase space
• The Quantum-Classical Transition in Nonlinear Dynamical Systems
• Bounds on Integrals of the Wigner Function
• Quantum symmetries and the Weyl-Wigner product of group representations

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