In this paper, a conserved current for Klein–Gordon equation is derived. It is shown, for ($1+1$) dimensions, the first component of this current is non-negative and reduces to ${\left|\varphi \right|}^2$ in nonrelativistic limit. Therefore, it can be interpreted as the probability density of spinless particles. In addition, main issues pertaining to localization in relativistic quantum theory are discussed, with a demonstration on how this definition of probability density can overcome such obstacles. Our numerical study indicates that the probability density deviates significantly from ${\left|\varphi \right|}^2$ only when the uncertainty in momentum is greater than $m_{0}c$.

• Received 10 March 2018

DOI:https://doi.org/10.1103/PhysRevA.98.012125