Trajectory-based conservation laws for massive spin-zero relativistic quantum particles in 1 + 1 spacetime

We present novel aspects of a trajectory-based theory of massive spin-zero relativistic quantum particles. In this approach, the quantum trajectory ensemble is the fundamental entity. It satisfies its own action principle, leading to a dynamical partial differential equation (via the Euler-Lagrange procedure), as well as to conservation laws (via Noether's theorem). In this paper, we focus on the derivation of the latter. In addition to the usual expected energy and momentum conservation laws, there is also a third law that emerges, associated with the conditions needed to maintain global simultaneity. We also show that the nonrelativistic limits of these conservation laws match those of the earlier, nonrelativistic quantum trajectory theory [J. Chem. Phys. 136, 031102 (2012)].