Segal CFT: Hilbert Space Applications

This lecture discusses the applications of Segal's Conformal Field Theory (CFT) in the context of Hilbert spaces. The instructor begins by introducing the concept of Hilbert spaces and their significance in quantum mechanics. The lecture covers various mathematical formulations, including the Verma modules and their role in the representation theory of the Virasoro algebra. The instructor explains how these concepts relate to the structure of CFT and the importance of understanding the Hamiltonian in this framework. The discussion includes examples of Gaussian functions and their properties within Hilbert spaces, emphasizing the mathematical rigor required for these applications. The lecture also touches on the implications of these theories in physics, particularly in the context of quantum field theories and statistical mechanics. Throughout the lecture, the instructor provides insights into the mathematical tools necessary for working with these advanced concepts, ensuring a comprehensive understanding of the subject matter.