# The Time Machine Factory [unspeakable, speakable] on Time Travel -TM2019

22-25 September 2019
Turin
Europe/Rome timezone

## Locally Covariant Quantum Field Theory on Causally Compatible Sets

Sep 25, 2019, 5:32 PM
39m
Turin

#### Turin

Monday, 23 Sept. * Aula Magna, Dip. di Scienze della Vita e Biologia dei Sistemi, Via Accademia Albertina, 13 * Tuesday, 24 Sept.- Wednesday, 25 Sept. * Aula Magna, Palazzo del Rettorato, Via Verdi, 8 * @Università degli Studi di Torino

### Speaker

Prof. James Vickers (Mathematical Sciences, University of Southampton, UK)

### Description

In this talk I describe the mathematics required in order to provide a description of the observables for quantum fields on low-regularity spacetimes. The first step involves constructing low-regularity advanced and retarded Green operators as maps between suitable function spaces. In specifying these we need to use graph norms on Sobolev spaces to ensure that the Green operators are well-defined inverses. The causal propagator is then used to define a symplectic form on a topological vector space $V(M)$. A key point is the way in which the causal propagator on a (non-smooth) globally hyperbolic spacetime restricts to the causal propagator on a smaller causally compatible submanifold and therefore induces a symplectic map between the vector spaces. This property enables one to provide a locally covariant description of the quantum fields in terms of the elements of quasi-local $C^*$-algebras on which one may define canonical commutation relations. I end with a brief discussion on the choice of Sobolev micro-local spectrum condition used to single out the physical states in the low-regularity setting.