2014/03/10 by Luigi Accardi, Accardi, Luigi, Abdessatar Barhoumi +3
Mathematics · Physics and Astronomy · #FOS: Mathematics #Functional Analysis (math.FA) #Probability (math.PR) #Quantum Mechanics and Applications #Quantum optics and atomic interactions #Random Matrices and Applications
paper · pdf · doi:10.48550/arxiv.1403.2662
openalex publication_date 2014/03/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The identification mentioned in the title allows a formulation of the multidi mensional Favard Lemma different from the ones currently used in the literature and which exactly parallels the original one dimensional formulation in the sense that the positive Jacobi sequence is replaced by a sequence of positive Hermitean (square) matrices and the real Jacobi sequence by a sequence of Hermitean matri ces of the same dimension. Moreover, in this identification, the multidimensional extension of the compatibility condition for the positive Jacobi sequence becomes the condition which guarantees the existence of the creator in an interacting Fock space. The above result opens the way to the program of a purely algebraic clas sification of probability measures on ℝd with finite moments of any order. In this classification the usual Boson Fock space over ℂd is characterized by the fact that the positive Jacobi sequence is made up of identity matrices and the real Jacobi sequences are identically zero. The quantum decomposition of classical real valued random variables with all moments is one of the main ingredients in the proof.