  We investigate the construction of spin-1/2 fermionic and spin-0 bosonic
wave-packets having compact spatial support in the framework of a computational
quantum field theory (QFT) scheme offering space-time solutions of the
relativistic wave equations in background fields. In order to construct
perfectly localized wave-packets, we introduce a spatial density operator
accounting for particles of both positive and negative charge. We examine
properties of the vacuum and single-particle expectation values of this
operator and compare them to the standard QFT particle and anti-particle
spatial densities. The formalism is illustrated by computing numerically the
Klein tunneling dynamics of strictly localized wave-packets impinging on a
supercritical electrostatic step. The density operator introduced here could be
useful to model situations in which it is desirable to avoid dealing with the
infinite spatial tails intrinsic to pure particle or anti-particle
wave-packets.
