Relativistic probability densities for location

• URL: http://arxiv.org/abs/2304.08540v1
• Date: Mon, 17 Apr 2023 18:14:42 GMT
• Title: Relativistic probability densities for location
• Authors: Joshua G. Fenwick, Rainer Dick
• Abstract summary: We study the Born rule as a fundamental principle of quantum mechanics for relativistic particles. We find that those four proxies for particle location are tantalizingly close even in this extremely relativistic case. 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.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• 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.

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