Heisenberg uncertainty relations for relativistic bosons

TL;DR

This paper derives sharp Heisenberg uncertainty relations for relativistic bosons with spin 0 and 1, highlighting fundamental differences from fermions and emphasizing the role of energy density.

Contribution

It completes the derivation of uncertainty relations for relativistic particles, specifically for bosons, and clarifies the limitations of previous approaches based on four-current components.

Findings

Uncertainty relations for bosons differ significantly from those for fermions.

Energy density is a suitable basis for formulating uncertainty relations for relativistic bosons.

Using the time-component of the four-current leads to contradictions and is not viable.

Abstract

This work completes the program started in \cite{bb1,bb2,bb3} to derive the Heisenberg uncertainty relation for relativistic particles. Sharp uncertainty relations for massive relativistic particles with spin 0 and spin 1 are derived. The main conclusion is that the uncertainty relations for relativistic bosons are markedly different from those for relativistic fermions. The uncertainty relations for bosons are based on the energy density. It is shown that the uncertainty relations based on the time-component of the four-current, as was have done in \cite{bb3} for electrons, are untenable because they lead to contradictions.