Three-dimensional, boost-invariant formalism for systems of relativistically moving constituents

Light front quantum mechanics in three dimensions can be used to construct boost-invariant wave functions for the internal structure of relativistic systems. The Miller-Brodsky variable $\tilde{z}$ -- which is canonically conjugate to the momentum fraction $x$ -- allows a spatial description of the longitudinal degree of freedom. We show how $\tilde{z}$ can be constructed as an operator and prove its boost invariance. A relativistic harmonic oscillator potential from Li, Maris, Zhao and Vary [Phys Lett B 758 (2016) 118] is used as an example of a two-body interaction that can be constructed using $\tilde{z}$ and for which closed-form analytic solutions can be found. We systematically explore the conditions in which the non-relativistic harmonic oscillator solutions are reproduced and the conditions in which relativistic corrections are significant. Harmonic oscillator states are commonly used as a basis for nuclear many-body calculations. The present effort may provide a basis for providing light-front wave functions of nuclei.