Explore the foundations of modular theory and its applications in quantum field theory (QFT) in this comprehensive lecture by Nima Lashkari from Purdue University. Delve into the Tomita-Takesaki modular theory and its impact on constraining correlation functions of quantum fields. Discover how modular theory tools prove the absence of obstructions in sewing local states into a global pure state in QFT, unlike finite quantum systems. Cover key topics including qudit examples, algebra of observables, projection operators, canonical purification, Rech-Schlieder property, superselection sectors, and various algebraic structures in quantum mechanics. Gain insights into the intricate relationship between modular theory and QFT through this in-depth, 1 hour 41 minute lecture from the Infosys-ICTS String Theory Lectures series.