The scientific role of Alfred Wehrl

The scientific role of Alfred Wehrl

Alfred Wehrl was born and grew up in Vienna. When he started his scientific career, he studied representations of the Bardeen Cooper Schrieffer model with Prof. Walter Thirring. Within his first two publications, he found the correct time evolution of the field operator when taking the thermodynamic limit in the strong sense. After that, he studied mixing properties of density matrices. In this way, he became interested in statistical mechanics models - a subject, to which he devoted his whole scientific life. After some contributions of more technical type, Wehrl studied the classical properties of entropy functionals. This included invariance, additivity, concavity, subadditivity, strong subadditivity and convergence properties. Next steps concerned the comparison of classical to quantum entropy.

His work culminated in the developement of a new notion of entropy at the end of the 1970s, which was later named after him: The Wehrl entropy. It is defined as the integral of -Q*logQ over the phase space, where Q is the Husimi-function (roughly speaking, Q is the expectation value of the density matrix in a ''Glauber coherent state''). Wehrl conjectured that the smallest possible value of his entropy is 1, which occurs if and only if the density matrix is a pure state projector onto any Glauber coherent state. This conjecture was soon proved by Lieb. One possible generalization is the so-called ''generalized Wehrl conjecture'' - one replaces  the function -Q*log Q in the entropy definition by any concave function and asks again for a minimal entropy value. Another possible generalization is the definition of the Wehrl entropy for Bloch coherent states (replacing ``Glauber'' coherent states in Q). The generalized Wehrl conjecture as well as its analogue for Bloch coherent states have been proven in 2012 by Lieb and Solovej. This shows, that there is still a lively development connected to the entropy notion Wehrl developed.