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

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 $\sqrt{(M^{\ast }{c}^2{)}^2+c^2|\mathbf{P}{|}^2}$ where $M^{\ast }$ 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 $$\hslash \omega (\mathbf{P})=<{\Phi }^{\prime }|\sum _j\frac{m_j}{M}{\beta }_j|{\Phi }^{\prime }>\sqrt{[M^{\ast }(\mathbf{P})c^2{]}^2+c^2|\mathbf{P}{|}^2}$$ where the kinematic mass $M^{\ast }$ will depend on the total momentum $\mathbf{P}$ 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.

arXiv comment14 pages