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In this paper we investigate the multivariate orthogonal polynomials based onthe theory of interacting Fock spaces. Our framework is on the same stream lineof the recent paper by Accardi, Barhoumi, and Dhahri \cite{ABD}. The(classical) coordinate variables are decomposed into non-commuting (quantum)operators called creation, annihilation, and preservation operators, in theinteracting Fock spaces. Getting the commutation relations, which follow fromthe commuting property of the coordinate variables between themselves, we candevelop the reconstruction theory of the measure, namely the Favard's theorem.We then further develop some related problems including the marginaldistributions and the rank theory of the Jacobi operators. We will see that thedeficiency rank of the Jacobi operator implies that the underlying measure issupported on some algebraic surface and vice versa. We will provide with someexamples.

Multivariate orthogonal polynomials: quantum decomposition, deficiency rank and support of measure