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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) quantizations, we show that the local quantisation yields a Fock space $\mathfrak F^L$ which is unitarily inequivalent to the standard one $\mathfrak F^G$. In spite of this, we find that the local creators and annihilators remain well defined in the global Fock space $\mathfrak F^G$, 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 $\mathfrak F^G$. Specifically, an analysis of the global vacuum state $|0_G\rangle\in\mathfrak F^G$ 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 $|0_L\rangle\in\mathfrak F^L$, on the contrary, has a very different character. It is neither cyclic nor separating and displays no entanglement. 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.