20 jun 2016 -- 12:00
Aula Tricerri, DiMaI, Firenze
Ph.D. seminar
INdAM Intensive Period "Hypercomplex Function Theory and Applications"
Abstract.
The talk starts with the introduction - and slight modification - of the concept of a 'slice' in a general Clifford algebra. In this way a differential operator is obtained which establishes a representation of the Lie superalgebra $\mathfrak{osp}(1
2)$. This so-called 'slice Dirac operator' is used to construct Clifford-Hermite functions which are shown to be orthogonal with respect to a well-defined inner product. Moreover they obey a scalar differential equation, by which they qualify extremely well as the eigenfunctions of a slice Fourier transform. Given their orthogonality, it is a natural question to ask whether transforms can be constructed which map them to other orthogonal bases. This talk addresses the particular case of the Fock space. The goal of the talk is to construct a slice Clifford counterpart of the Segal-Bargmann transform, which classically maps Clifford-Hermite functions onto the monomial basis of the Fock space.