On the Impossibility of Obtaining Time-Independent, Three-Dimensional, Spherically-Symmetric Densities of Confined Systems of Relativistically Moving Constituents

TL;DR

This paper demonstrates how to define consistent, time-independent two-dimensional densities for confined relativistic systems using light-front formalism, highlighting limitations of traditional three-dimensional density methods.

Contribution

It introduces a method to obtain valid two-dimensional densities in relativistic systems and critiques existing three-dimensional density approaches for violating fundamental principles.

Findings

Two-dimensional densities are consistent with quantum and relativistic restrictions.

Traditional three-dimensional density methods violate fundamental restrictions.

Using wave packets with zero spatial extent leads to densities vanishing over time.

Abstract

The quantum mechanical definition of probability, the uncertainty principle and Poincare 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 are obtained. 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 manners. Furthermore, the use of the latter leads to densities that vanish almost everywhere in space as time increases from an initial value.