One of the classical problems in quantum mechanics is the search for a phase space representation of the wave function of a quantum state. This is usually done using the Wigner transform, which can be defined for fermionic wave functions in position, and also for more general density operators. However, the resulting phase space representations do not satisfy the three probability axioms and are therefore treated as quasiprobabilities. The Wigner representation is therefore considered 'semiclassical'.
In this talk we will first see that the classical Gabor transform can be used instead of the Wigner transform to obtain a quadratic representation of a N-body wave function in phase space, which defines a fermionic random process in phase space for which all the relevant statistics can be computed classically from true probability distributions. We will provide examples with the variance and the entanglement entropy (which we show to be proportional, at least in the 1-body case).
In the second part of the talk we will discuss how the same problem can be tackled using the quantum Gabor transform with operator window. The result is a completely new type of fermionic random point process, which takes into account correlations among the building blocks of the density operators. Moreover, as in the classical Gabor transform construction of the previous paragraph, the statistics of the resulting process can be computed classical. We will provide some examples and a general estimate for the variance, showing that the resulting process is hyperuniform. This second part of the talk is joint work with Simon Halvdansson.