Constructing quantum localization operators using conformal symmetry

We review the difficulties in merging quantum theory with relativity. In particular, we discuss the issue of localization in quantum mechanics. We introduce the conformal group, a supergroup of the Poincar\'{e} group and give its generators and the corresponding algebra. We then illustrate how this allows us to create a space-time localization operator that is consistent with special relativity and quantum theory. We give an explicit expression of the localization operators in 1+1 dimensions and discuss the challenges of generalizing it to higher dimensions.