| Summary: | Imposing the Born rule as a fundamental principle of quantum mechanics would require the existence of normalizable wave functions ψ ( x , t ) also for relativistic particles. Indeed, the Fourier transforms of normalized k -space amplitudes $\psi ({\boldsymbol{k}},t)=\psi ({\boldsymbol{k}})\exp (-{\rm{i}}{\omega }_{{\boldsymbol{k}}}t)$ yield normalized functions ψ ( x , t ) which reproduce the standard k -space expectation values for energy and momentum from local momentum (pseudo-)densities ℘ _μ ( x , t ) = ( ℏ /2i)[ ψ ^+ ( x , t )∂ _μ ψ ( x , t ) − ∂ _μ ψ ^+ ( x , t ) · ψ ( x , t )]. However, in the case of bosonic fields, the wave packets ψ ( x , t ) are nonlocally related to the corresponding relativistic quantum fields ϕ ( x , t ), and therefore the canonical local energy-momentum densities ${ \mathcal H }({\boldsymbol{x}},t)=c{{ \mathcal P }}^{0}({\boldsymbol{x}},t)$ and ${ \mathcal P }({\boldsymbol{x}},t)$ differ from ℘ _μ ( x , t ) and appear nonlocal in terms of the wave packets ψ ( x , t ). We examine the relation between the canonical energy density ${ \mathcal H }({\boldsymbol{x}},t)$ , the canonical charge density ϱ ( x , t ), the energy pseudo-density $\tilde{{ \mathcal H }}({\boldsymbol{x}},t)=c{\wp }^{0}({\boldsymbol{x}},t)$ , and the Born density ∣ ψ ( x , t )∣ ^2 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 Δ 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 ≫ Δ p ≥ ℏ /2Δ x . We also translate our results to massless Dirac fields. Our results confirm and illustrate that the normalized energy density ${ \mathcal H }({\boldsymbol{x}},t)/E$ provides a suitable measure for positions of bosons, whereas normalized charge density ϱ ( x , t )/ q provides a suitable measure for fermions.
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