Applications of modular inclusions in quantum field theory and quantum gravity
Ceyhan, Fikret Ali
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https://hdl.handle.net/2142/130016
Description
Title
Applications of modular inclusions in quantum field theory and quantum gravity
Author(s)
Ceyhan, Fikret Ali
Issue Date
2025-07-07
Director of Research (if dissertation) or Advisor (if thesis)
Faulkner, Thomas
Doctoral Committee Chair(s)
Leigh, Robert G
Committee Member(s)
Draper, Patrick I
Pfaff, Wolfgang
Department of Study
Physics
Discipline
Physics
Degree Granting Institution
University of Illinois Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
entanglement
relative entropy
Abstract
In the last few decades, quantum information theory has expanded our understanding of the entanglement structure of space-time. The information theory quantities, namely the quantum relative entropy, a distinguishability measure between two quantum states, have played a key role in providing new insights into both gravitational systems and ordinary quantum field theory in flat space. What makes relative entropy a unique information measure is its well-defined nature for both discrete quantum systems and continuum field theories alike. One important result that the study of relative entropy has led to is a rigorous proof of Quantum Null Energy Condition (QNEC), a lower bound on the energy density of a local region in space (null components of the stress-energy tensor) by the second derivative of entanglement entropy with respect to shape variations in the light-cone direction. Although previous proofs of QNEC relied on the assumption of separable Hilbert spaces and the use of density matrices \cite{Balakrishnan_2019, Bousso_2016}, which are ill defined in QFT, a theory-independent proof requires recasting of QNEC in terms of relative entropy. To that end, in this thesis, we use the abstract machinery of operator algebras and modular theory, which properly account for infinite dimensions, to prove QNEC. We demonstrate the role played by the modular theory of von Neumann algebras and their inclusion properties in explaining this energy inequality and also draw a connection to the field of quantum error correction. Moreover, we present ongoing work on the role of modular theory in understanding correlation measures in Conformal Field Theories such as mutual information, which is a specific instance of relative entropy, and its connection to the thermal partition function and its breakdown. While our previous work proving QNEC focuses on applications of spacetime inclusions that are light-like, new work on mutual information focuses on inclusions of algebras (correspondingly regions) that are space-like.
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