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Negativity witness for the quantum ergodic conjecture

For an ergodic system, the time average of a classical observable coincides with that obtained via the Liouville probability density, a delta-function on the energy shell. Reinterpreting this distribution as a Wigner function, that is, the Weyl representation of a density operator, the quantum ergodic conjecture identifies this classical construction as an approximate representation of the quantum eigenstate of the same energy. It is found that this reasonable hypothesis, as far as expectations for observables are concerned, does not satisfy the requirement of positivity, so that the delta-function on the shell cannot be a true Wigner function. This result was first presented by N. Balazs for the special case of a single degree of freedom, such that the system is simultaneously integrable and ergodic. Here, it is shown, for a delta-function on an arbitrary curved energy shell in a $2N$-dimensional phase space, that there exists a positive operator for which the conjectured Wigner function predicts a negative expectation.