Algebra of Observables and States for Quantum Abelian Duality

TL;DR

This paper develops an algebraic framework for Abelian duality in quantum field theory on curved spacetimes, constructing the algebra of observables, analyzing its sectors, and demonstrating duality implementation via unitary operators.

Contribution

It introduces a novel algebraic construction of observables for Abelian duality in quantum field theory, including explicit examples and the implementation of duality at the Hilbert space level.

Findings

Constructed the algebra of observables as a direct sum of three pre-symplectic groups.

Proved the existence of a ground Hadamard state in 2D and 4D cases.

Showed that Abelian duality is implemented by unitary operators in the GNS representation.

Abstract

The study of dualities is a central issue in several modern approaches to quantum field theory, as they have broad consequences on the structure and on the properties of the theory itself. We call Abelian duality the generalisation to arbitrary spacetime dimension of the duality between electric and magnetic field in Maxwell theory. In the present thesis, in the framework of algebraic quantum field theory, the Abelian duality for quantum field theory on globally hyperbolic spacetime with compact Cauchy surface is tackled. Fistly, the algebra of observables is constructed. It is shown that it can be presented as the direct sum of three pre-symplectic Abelian groups, each corresponding to a different sector of the theory. As a consequence, it is possible to provide quantum states for the theory by building separate states on each direct summand. In particular, explicit examples in two and four dimensions are discussed thoroughly; a ground Hadamard state in a suitable sense is proved to exist for both of them. Lastly, it is shown that the Abelian duality is implemented by unitary operators at the level of the GNS Hilbert space.