Operators central limit theorem

The extension to a noncommutative algebra of the concept of cumulants, developed in a recent work [1] (hereafter BB20), opens up the possibility of applying to the field of operators (Liouvillians, quantum density matrices, Schrodinger’s equation, etc...) the many important results on the statistics of commuting quantities obtained in the 19th and 20th centuries. Among these, the Central Limit Theorem (CLT) occupies a prominent place. Here, starting with BB20, we show that the extension of CLT to operators (for this reason renamed M-CLT) can be effectively achieved and its application to concrete cases leads to a robust derivation, and generalization, of classical results from Statistical Mechanics and Quantum Mechanics. An example is the Fokker-Planck equation, which, being so derived as the lowest order of a series of cumulative-operators, does not involve the Pawula’s theorem [2] concerning the classic Kramers-Moyall expansion. However, once CLT is introduced into the field of operators, it will be particularly significant if extended to cases where the strict “independent-identically distributed” assumption is released, at least in part, as has already been done for c-numbers (see, e.g., [3]). This is a promising field of research worthy of exploration, which this work leaves open. We will mention only a few research directions and applications, particularly for the study of closed or thermalized multiparticle systems, that might be useful to explore.