Abstract: 

The transition from quantum mechanics to the classical macroscopic world remains a profound puzzle in physics. To mathematically characterize this emergence of classicality, David Mumford proposed the concept of Approximately Macroscopically Unique (AMU) states, which behave classically for a set of almost-commuting macroscopic observables. In this talk, I will present recent joint work with Huaxin Lin to rigorously establish the existence of AMU states for quantum systems consisting of unbounded self-adjoint operators with small commutators. To tackle the inherent lack of exact joint eigenvectors for non-commuting observables, we employ the framework of the synthetic spectrum. To bring these abstract concepts down to earth, we will walk through explicit calculations of fundamental physical paradigms, showing exactly how this mathematical machinery works in practice.