Kinematic mass of a composite in the many-particle Dirac model

TL;DR

This paper derives a relativistic energy-momentum relation for a composite particle in many-particle Dirac models, introducing a momentum-dependent kinematic mass and a modified deBroglie relation, despite the model's non-covariance.

Contribution

It presents a novel approach to define kinematic energy and derive an energy-momentum relation in non-covariant many-particle Dirac models, including a momentum-dependent mass.

Findings

Derived a modified deBroglie relation incorporating a kinematic mass.

Defined a momentum-dependent kinematic mass for the composite.

Separated relative and center of mass contributions in a non-covariant framework.

Abstract

We are interested in the energy-momentum relation for a moving composite in relativistic quantum mechanics in many-particle Dirac models. For a manifestly covariant model one can apply the Lorentz transform to go from the rest frame to a moving frame to establish an energy-momentum relation of the form $\sqrt{(M^*c^2)^2+c^2|{\bf P}|^2}$ where $M^*$ is the kinematic mass. However, the many-particle Dirac model is not manifestly covariant, and some other approach is required. We have found a simple approach that allows for a separation of relative and center of mass contributions to the energy. We are able to define the associated kinematic energy and determine the energy-momentum relation. Our result can be expressed as a modified deBroglie relation of the form $$ \hbar \omega ({\bf P}) = <\Phi' | \sum_j {m_j \over M} \beta_j | \Phi' >~ \sqrt{[M^*({\bf P}) c^2]^2 + c^2 |{\bf P}|^2} $$ where the kinematic mass $M^*$ will depend on the total momentum ${\bf P}$ for a general noncovariant potential. The prefactor that occurs we associate with a time dilation effect, the existence of which has been discussed previously in the literature.