Redefinition of the energy-momentum operator: Motivation and implications

Abstract

The paper is motivated by our idea to re-define the momentum operator in quantum physics through the sum of mechanical and electromagnetic momenta for the system “charged particle in an electromagnetic field” instead of its canonical momentum, which occurred successful in describing quantum phase effects for charges and dipoles (A.L. Kholmetskii, et al. Ann. Phys. 392, 49 (2018); Sci. Rep. 8, 11937 (2018)). Furthermore, we show how a recently obtained expression for the “point-by-point” quantum phase of a charged particle in the framework of a fully quantized model of the Aharonov-Bohm effect (C. Marletto, V. Vedral, Phys. Rev. Lett. 125, 040401 (2020)) supports the re-definition of the momentum operator in quantum mechanics from the theoretical side. These results motivated us to re-analyse the fundamental equations of relativistic quantum mechanics with a new energy-momentum operator. In this contribution, we solve the Dirac equation for an electrically bound electron with a new energy-momentum operator and extend the obtained solutions to the precise physics of simple atoms in the form of an effective theory, which does not touch the diagram technique of QED. We find that for majority of problems of precise physics of simple atoms, both definitions of the energy-momentum operator yield indistinguishable results with modern measurement precision. An important exception is the spectroscopy of ortho-positronium, which occurs crucial in choosing the correct expression for the energy-momentum operator, and show that the available measurement data rather support the new definition of this operator.