  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.
