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Impossibility of obtaining time-independent, three-dimensional, spherically symmetric densities of confined systems of relativistically moving constituents

Gerald A. Miller

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Phys. Rev. C 112, 045204 – Published 27 October, 2025

DOI: https://doi.org/10.1103/jkn7-4fzj

Abstract

The quantum-mechanical definition of probability, the uncertainty principle, and Poincaré invariance provide strong basic restrictions on the ability to define spatial densities associated with form factors describing the properties of confined systems of relativistically moving constituents. Despite this, many papers ignore one or more of these restrictions. Here I show how to obtain time-independent, two-dimensional densities that are consistent with the stated restrictions. This is done using the light-front, infinite momentum frame formalism. Two-dimensional density interpretations of the axial-vector form factor and all three gravitational form factors is obtained. The resulting mass radius is smaller than the charge radius. Additionally, an expression of a two-dimensional mass density related to the trace of the energy momentum tensor is obtained. I also show that all known methods for finding three-dimensional densities—using the Breit frame, Abel transformations, Wigner distributions, and spherically symmetric wave packets with vanishing spatial extent—violate the basic restrictions in different ways. Furthermore, the use of the latter leads to densities that vanish almost everywhere in space as time increases from an initial value.

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Phys. Rev. C 112, 045204– Published 27 October, 2025
  • Received 1 August 2025
  • Accepted 15 September 2025
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