  Imposing the Born rule as a fundamental principle of quantum mechanics would
require the existence of normalizable wave functions also for relativistic
particles. Indeed, the Fourier transforms of normalized k-space amplitudes
yield normalized x-space wave packets which reproduce the standard k-space
expectation values for energy and momentum from local momentum
pseudo-densities. However, in the case of bosonic fields, the wave packets are
nonlocally related to the corresponding relativistic quantum fields, and
therefore the canonical local energy-momentum densities differ from the
pseudo-densities and appear nonlocal in terms of the wave packets. We examine
the relation between the canonical energy density, the canonical charge
density, the energy pseudo-density, and the Born density for the massless free
Klein-Gordon field. We find that those four proxies for particle location are
tantalizingly close even in this extremely relativistic case: In spite of their
nonlocal mathematical relations, they are mutually local in the sense that
their maxima do not deviate beyond a common position uncertainty $\Delta x$.
Indeed, they are practically indistinguishable in cases where we would expect a
normalized quantum state to produce particle-like position signals, viz. if we
are observing quanta with momenta $p\gg\Delta p\ge\hbar/2\Delta x$. We also
translate our results to massless Dirac fields. Our results confirm and
illustrate that the normalized energy density provides a suitable measure for
positions of bosons, whereas normalized charge density provides a suitable
measure for fermions.
