Public Thesis Defense of Louis DE MAN - IRMP

Smooth Wave Front Set and Deformations of Tempered Distributions Algebras by Louis DE MAN

Lundi 10 février 2025 à 16h00 - Auditoire CYCL01, chemin du Cyclotron, 2 à 1348 Louvain-la-Neuve.

Experiments have shown that the microscopic world exhibits strange quantum phenomena that do not occur in classical mechanics. While the measurable quantities of a classical system—such as energy and position—are represented by functions that form a commutative algebra under pointwise multiplication, quantum quantities are more appropriately described using noncommutative algebras of operators. With this in mind, a natural question arises: How do quantum effects fade as we transition from the quantum to the classical world, that is, as the system's scale increases? From a mathematical perspective, this question has led to the development of deformation quantization, which addresses the issue by constructing a family of noncommutative associative products on an algebra of functions defined on the classical space. These products deform the usual commutative pointwise multiplication. This continuous deformation captures the scaling process of moving from the quantum realm (the noncommutative algebra) to the classical one (the commutative algebra). The deformed product that describes the quantum system is called a star-product.

Star-products are formal objects in the sense that they are defined as infinite power series. Since this form is impractical for most physical and mathematical applications, there has been an ongoing effort to define so-called non-formal star-products. Non-formal star-products are those that do not involve infinite power series but instead map two elements of the algebra directly to another element of the algebra. The construction of such objects often presents tedious analytical difficulties. In this thesis, we build on the work of Bahns and Schulz, employing the smooth wave front set to define non-formal star-products on algebras of tempered distributions.

The smooth wave front set is a microlocal object introduced by Hörmander in 1989, which simultaneously encodes the singular behavior at finite points of a tempered distribution and its asymptotic growth at infinity. In 2019, Bahns and Schulz observed that the smooth wave front set provides a sufficient condition to determine whether the Weyl star-product of two tempered distributions is well-defined in a non-formal sense. Using this criterion, they constructed non-formal algebras of tempered distributions under the Weyl star-product. We show that their work relies on the interaction between the smooth wave front set and the twist associated with the Weyl star-product (in the sense of Drinfeld's construction). This approach enables the extension of their result to other star-products on symplectic vector spaces, including the Kohn-Nirenberg star-product. The irreducible unitary representations of the Heisenberg group play a crucial role in this framework.

Membres du jury :

Prof. Pierre Bieliavsky (UCLouvain) (Promoteur)
Prof. Philippe Ruelle (UCLouvain) (Président)
Prof. Pedro Vaz (UCLouvain) (Secrétaire)
Prof. Heiner Olbermann (UCLouvain)
Prof. Dorothea Bahns (Georg-August-Universität Göttingen, Allemagne)
Prof. Victor Gayral (Université de Reims Champagne-Ardenne, France)

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