This work completes the program started by I. Bialynicki-Birula and Z. Bialynicka-Birula [Uncertainty relation for photons, Phys. Rev. Lett. 108, 140401 (2012); Heisenberg uncertainty relation for photons, Phys. Rev. A 86, 022118 (2012); Heisenberg uncertainty relation for relativistic electrons, New J. Phys. 21, 073036 (2019)] 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 we have done previously for electrons, are untenable because they lead to contradictions.

• Received 3 March 2021
• Accepted 5 May 2021

DOI:https://doi.org/10.1103/PhysRevA.103.052211