Non-positivity of Groenewold operators

Details Non-positivity of Groenewold operators Non-positivity of Groenewold operators

2004-07-07

arxiv.org/abs/quant-ph/0407052v1

Europhys. Lett., 68 (1) pp. 1-7 (2004)

Physics

Quantum Physics

9 pages, 1 figures, submitted to Europhysics Letters

Scientific paper

10.1209/epl/i2004-10175-8

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.

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