Relativistic probability densities for location

Abstract

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.