Approximate Synthetic Spectra of Quantum Systems Associated with Compact Lie Groups

Abstract

How can noncommuting quantum observables retain the geometry of a classical phase space? Motivated by Mumford's question on approximately macroscopically unique (AMU) states and the work of Lin and Wang, we study synthetic spectra of position--momentum and angular-momentum systems. Under the hypotheses of their AMU theorems, nonempty synthetic spectra yield AMU states. For sufficiently small $\hbar$, the position--momentum spectrum fills the plane, while the angular-momentum spectrum contains any prescribed ball. The latter calculation leads to compact Lie groups: we prove that highest-weight coadjoint orbits lie in regional synthetic spectra under an explicit variance bound. Function-space representations of $SU(n)$, $SO(n)$, and $Sp(n)$ provide further examples.