Localisation in Quantum Field Theory

Abstract

In nonrelatistic quantum mechanics, Born's principle of localistion is as follows: For a single particle, if a wave function $\psi_K$ vanishes outside a spatial region $K$, it is said to be localised in $K$. In particular if a spatial region $K'$ is disjoint from $K$, a wave function $\psi_{K'}$ localised in $K'$ is orthogonal to $\psi_K$. Such a principle of localisation does not exist compatibly with relativity and causality in quantum field theory (Newton and Wigner) or interacting point particles (Currie,Jordan and Sudarshan).It is replaced by symplectic localisation of observables as shown by Brunetti, Guido and Longo, Schroer and others. This localisation gives a simple derivation of the spin-statistics theorem and the Unruh effect, and shows how to construct quantum fields for anyons and for massless particles with `continuous' spin. This review outlines the basic principles underlying symplectic localisation and shows or mentions its deep implications. In particular, it has the potential to affect relativistic quantum information theory and black hole physics.