Entanglement Entropy and Algebra in Quantum Field Theory

Abstract

Quantum Field Theory (QFT) represents a vast generalization of Quantum Mechanics (QM), as it deals with systems that have an infinite number of degrees of freedom. The Stone-von Neumann theorem, which establishes the equivalence of irreducible represen- tations of the canonical commutation relations (CCR) in QM, does not extend to QFT. Consequently, QFT admits multiple inequivalent irreducible representations, leading to a much richer algebraic structure. This essay aims to explore the physics of QFT from the operator algebra perspective, particularly focusing on entanglement entropy. We discuss the role of von Neumann algebras of different types in QFT, describe the local operator algebra approach to QFT, and explain how entanglement entropy can be defined in terms of the algebra of observables. Additionally, we explore the benefits of this approach in concrete applications, specifically in quantum field theory on curved spacetime.