Modular theory and algebraic quantum physics

The description of a quantum physical system, in the algebraic approach, is given through a von Neumann algebra of observables A and a state w on it. In this context, the study of entanglement of quantum systems is relevant. That requires an appropriate assignment of an entropy to the algebraic states. This entropy can be obtained through the Gelfand-Naimark-Segal (GNS) construction, which leads to a density operator associated to the state. Recently, Balachandran et al. (2013) used the algebraic approach to deal with entanglement in systems of identical particles. As is well known, the standard approach fails in these systems due to the fact that partial trace loses its intended meaning, since the Hilbert space is not a simple tensor product. Instead of partial trace, they considered the restriction of a state to a subsystem, which in the algebraic formulation becomes particularly clear. By means of the GNS construction, they construct density operators such that their restriction to the algebra A coincide with w. Then, the von Neumann entropy of these density operators can be regarded as the entropy of the algebraic state. However, this approach is ambiguous and assigns multiple density operators to the same state. This occurs whenever the irreducible components of the representation appear in the GNS Hilbert space H with multiplicities different from one. In this dissertation, we used Tomita-Takesaki (modular) Theory (TTT) to develop an interpretation of this phenomenon as an emergent gauge symmetry, in the sense of Doplicher, Haag, and Roberts. The gauge group arises from the action of unitaries in the commutant of the representation via TTT. We characterize the ambiguity in the entropy through the modular objects, in particular the modular conjugation. Moreover, we provide a quantum operation which implements the gauge group and increases entropy. In this way, we relate the realm of quantum information theory to that of the theory of gauge fields. We apply the above for two fundamental cases. In the first case, we consider general finite dimensional algebras and we give a physical interpretation in terms of an equivalent description of the system as a bipartite system. In the second case, we consider quantum systems whose classical configuration spaces are homogeneous spaces of the form Q=G/H, where G is a compact Lie group. In this case, the von Neumann algebra is obtained by quantizing Q through an approach based on the use of the transformation group C*-algebras. We prove that the emergent gauge group contains the classical gauge group of Q. Additionally, we also study other applications of TTT to spin chains in the context of Araki's self-dual formalism. In this case, for a given thermal KMS state, the modular theory (through the GNS construction) provides a purification of the KMS state. This construction preserves the canonical anticommutation relations of the fermionic algebra, unlike the one obtained using thermofield dynamics. Possible applications of this approach to the study of quantum phases of matter are also discussed.

• Autores: Tabban Sabbagh, Souad María
• Tipo de recurso: Doctoral thesis
• Fecha de publicación: 2022
• Institución: Universidad de los Andes
• Repositorio: Séneca: repositorio Uniandes
• Idioma: eng
• OAI Identifier: oai:repositorio.uniandes.edu.co:1992/54545
• Acceso en línea: http://hdl.handle.net/1992/54545
• Palabra clave: AQFT; Operator algebras; Tomita-Takesaki modular theory; Entropy; Módulos (Algebra); Teoría cuántica; Teoría de campos (Física); Física
• Rights: openAccess
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/