Summary

At a time when a stream of research was striving to reformulate quantum mechanics by abolishing operators and substituting functions, Wigner and Szilard proposed in 1932 a quasi-probability distribution defined on phase space thanks to wave functions. They did not explain its genesis.
The first part of our thesis proposes a genesis of this quasi-distribution, based on the natural conditions it must fulfill. It briefly examines a pathology it suffers from: exhibiting negative values in certain subdomains of the phase space (hence the "quasi"), a pathology that does no harm to the calculation of mean values. She then shows how, if we take spin into account, with wave functions giving way to spinners, we are led, thanks to the calculation of mean values of observables, to a generalization of this quasi-distribution in the form of a Hermitian matrix. This approach is extended to the Wigner cross transform, i.e. to weak values.
An important theorem, which has been the subject of a publication, is proved in the second part of our thesis. Using harmonic analysis, this result expresses weak values in terms of an integral over a Lie group acting on the Hilbert space under consideration. We give two particular examples: SU(2) and SO(3). The case of a quotient group is briefly discussed.
In a third section, we recall the well-known link between Clifford algebras and two important equations of quantum physics: the Klein-Gordon and Dirac equations, and its generalization to Riemannian spacetimes.
Finally, in a fourth section we introduce spin groups, and use the spin group Spin(3,2) in the context of the Wigner cross transform discussed in the first section.

Jury

  • Prof. André FÜZFA (UNamur), President
  • Prof. Yves CAUDANO (UNamur), Secretary
  • Dr. Thomas DURT (Institut Fresnel and Ecole Centrale Marseille, Marseille, France)
  • Prof. Romain MURENZI (Worcester Polytecnic Institute)
  • Prof. Dominique LAMBERT (UNamur)
  • Prof. Bertrand HESPEL (UNamur)
  • Prof. André HARDY (UNamur)