2022/12/12 by Herrera, David
#46L60 #47A55 (Primary) 15A27 #47B37 #81P15 #81R05 (Secondary) #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph) #Operator Algebras (math.OA) #Quantum Physics (quant-ph)
paper · doi:10.48550/arxiv.2212.06012
Resolving a conjecture of von Neumann, Ogata's theorem in arXiv:1111.5933 showed the highly nontrivial result that arbitrarily many matrices corresponding to macroscopic observables with N sites and a fixed site dimension d are asymptotically nearby commuting observables as N → ∞. In this paper, we develop a method to construct nearby commuting matrices for normalized highly reducible representations of su(2) whose multiplicities of irreducible subrepresentations exhibit a certain monotonically decreasing behavior. We then provide a constructive proof of Ogata's theorem for site dimension d=2 with explicit estimates for how close the nearby observables are. Moreover, motivated by the application to time-reversal symmetry explored in arXiv:1012.3494, our construction has the property that real macroscopic observables are asymptotically nearby real commuting observables.