Gauge inequivalence, energy inequalities and entanglement in algebraic quantum field theory

This PhD thesis in Mathematical Physics results from a three year long stay at the Universit`a degli studi di Roma Tor Vergata in Rome as Doctoal student of the Istituto Nazionale di Alta Matematica (INdAM) under supervision of Prof. Roberto Longo. During this time, the author was part of the group working on Operator algebras and Quantum Field Theory (QFT). The stay began in November 2018, ended in early 2022, and was briefly interrupted in spring 2020 due to the impact of the COVID pandemic. This thesis contains both a general overview and some original research results about mathematical aspects in QFT. In particular, the algebraic approach to QFT or Algebraic Quantum Field Theory (AQFT) is treated. It is based on a set of axioms - the Haag-Kastler axioms - that attempts to put QFT on a solid conceptual footing. In this thesis, the author concentrates on the role of gauge theory in the Quantisation procedure and the interplay of QFT and Quantum Information Theory (QIT). From the mathematical point of view, it involves mainly functional analysis, operator algebras, and representation theory. Although some of the results are purely mathematical, the author’s main interest is the logical structure of QFT.