Applied Free Probability Theory

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The goal of this work is to develop and apply modern free probability theory (FPT) to advance understanding of applied problems that traditionally are thought to be too difficult to tackle. The difficulty with quantum many-body systems (QMBS) research is two-fold: 1. The spectra of the sum of local interactions is not easily obtained from the spectra of the summands and 2. The exponential growth with the number of particles of the Hilbert space makes exact diagonalization feeble for any serious study of matter. The nature of the work will be both analytical and computational. The techniques are either entirely unknown to natural scientists or that their potential applicability is obscured by formal presentations. The purpose of this work is to apply an alternative methodology in probability to new physical problems.