Local quanta, unitary inequivalence, and vacuum entanglement

In this work we develop a formalism for describing localised quanta for a real-valued Klein-Gordon field in a one-dimensional box [0,R]. We quantise the field using non-stationary local modes which, at some arbitrarily chosen initial time, are completely localised within the left or the right side of the box. In this concrete set-up we directly face the problems inherent to a notion of local field excitations, usually thought of as elementary particles. Specifically, by computing the Bogoliubov coefficients relating local and standard (global) quantisations, we show that the local quantisation yields a Fock representation of the Canonical Commutation Relations (CCR) which is unitarily inequivalent to the standard one. In spite of this, we find that the local creators and annihilators remain well defined in the global Fock space FG, and so do the local number operators associated to the left and right partitions of the box. We end up with a useful mathematical toolbox to analyse and characterise local features of quantum states in FG. Specifically, an analysis of the global vacuum state |0G〉∈FG in terms of local number operators shows, as expected, the existence of entanglement between the left and right regions of the box. The local vacuum |0L〉∈FL, on the contrary, has a very different character. It is neither cyclic (with respect to any local algebra of operators) nor separating and displays no entanglement between left and right partitions. Further analysis shows that the global vacuum also exhibits a distribution of local excitations reminiscent, in some respects, of a thermal bath. We discuss how the mathematical tools developed herein may open new ways for the analysis of fundamental problems in local quantum field theory.