The Wavefunction and the Wigner Function of the Universe

Frank J. Tipler

J H Hannay

In contrast to a stationary Gaussian random function of a real variable which is free to have any correlation function, the closest analogous analytic random function in the complex plane has no true freedom - it is (statistically) unique. Since it has arisen only recently, as an apparently universal feature in the physical context of quantum chaos, I refer to it here as `the chaotic analytic function'. I note that it is implied by the assumption that a quantum chaotic wavefunction has Gaussian randomness and has a constant value for the average of its Wigner function in phase space. Interpreted literally this shows that the chaotic analytic function is the Bargmann function of a pure `white noise' wavefunction. More physically, if `constant' is replaced by `smooth on the scale of a Planck area', these assumptions are the semiclassical ones made by Berry for chaotic eigenstates. The analysis shows that the chaotic analytic function is still obtained semiclassically.

Jan Mostowski, Joanna Pietraszewicz

The Wigner function is a quantum analogue of the classical joined distribution of position and momentum. As such is should be a good tool to study quantum-classical correspondence. In this paper, the classical limit of the Wigner function is shown using the quantum harmonic oscillator as an example. The Wigner function is found exactly for all states. The semi-classical wavefunctions for highly excited states are used as the approach to the classical limit. Therefore, one can found the classical limit of the Wigner function for highly excited states and shown that it gives the classical microcanonical ensemble.

Raymond Laflamme

Valerian I. Tatarskii

J. Žák

The Wigner function has a doubling feature which is expressed in the fact that an x, p-point in the Wigner function leads to a (2x, 2p)-point in the wavefunctions. This feature is of no special consequence in the continuous phase plane. But in the finite phase plane, it is of crucial importance. This doubling feature enables one to define the finite phase plane Wigner function directly from the Wigner function in the continuous phase plane. Bearing in mind the doubling feature of the Wigner function, we define the inversion operator not only around the points of the discrete phase plane lattice but also around their mid-points. Our approach makes it clear why there is a difference in constructing the Wigner function in odd- and even-dimensional phase spaces.