Master's Defence

Much of the structure of quantum field theory (QFT) is predicated on the principle of locality. Adherence to locality is pursuant to convictions deduced from relativity, and is achieved in QFT by the association of regions of spacetime with algebras of observables. Although, by construction, the observables of QFT are local objects, one may also consider characterizing the spatial or spacetime features of a state. For example, if we have a single-particle state, how can we say that the particle is localized in a certain region of space? It turns out that such a characterization is obstructed by a collection of no-go theorems, which imply the absence of any suitable position operator or local number operator in the local algebra of observables. These difficulties seem to suggest that relativistic QFT cannot support an ontology in terms of localizable particles.

Looking towards low energies, one finds the widespread applicability of non-relativistic quantum mechanics (NRQM), a theory in which particle states are localizable by means of their wavefunction. This seems to imply that NRQM can support a particle ontology, so it is natural to ask whether one can make contact between the NRQM description of particles and some appropriate notion in the latent QFT. Admittedly QFT amd NRQM are very different theories, both at the dynamical and kinematical level, and recovering one from the other cannot come with no cost. The main undertaking of this thesis will be to illuminate this connection, by starting with a relativistic QFT and making suitable approximations to recover features of NRQM.