The q q-bit (I): Central limits with left q-Jordan-Wigner embeddings, monotone interacting Fock space, Azema random variable, probabilistic meaning of q

The qq-bit is the q-deformation of the q-bit. It arises canonically from the quantum decomposition of Bernoulli random variables and the q-parameter has a natural probabilistic and physical interpretation as asymmetry index of the given random variable. The connection between a new type of q-deformation (generalizing the Hudson-Parthasarathy bosonization technique and different from the usual one) and the Azema martingale was established by Parthasarathy. Inspired by this result, Schürmann first introduced left and right q-JW-embeddings of M2 (2 × 2 complex matrices) into the infinite tensor product ⊗n in; NM2, proved central limit theorems (CLT) based on these embeddings in the context of∗-bi-algebras and constructed a general theory of q-Levy processes on∗-bi-algebras. For q =-1, left q-JW-embeddings define the Jordan-Wigner transformation, used to construct a tensor representation of the Fermi anti-commutation relations (bosonization). For q = 1, they reduce to the usual tensor embeddings that were at the basis of the first quantum CLT due to von Waldenfels. The present paper is the first of a series of four in which we study these theorems in the tensor product context. We prove convergence of the CLT for all q in; C. The moments of the limit random variable coincide with those found by Parthasarathy in the case q [-1, +1). We prove that the space where the limit random variable is represented is not the Boson Fock space, as in Parthasarathy, but the monotone Fock space in the case q = 0 and a non-trivial deformation of it for q ≠-1, 0, +1. The main analytical tool in the proof is a non-trivial extension of a recently proved multi-dimensional, higher order Cesaro-type theorem. The present paper deals with the standard CLT, i.e. the limit is a single random variable. Paperdeals with the functional extension of this CLT, leading to a process. In paperthe left q-JW-embeddings are replaced by symmetric q-embeddings. The radical differences between the results of the present paper and those ofraise the problem to characterize those CLT for which the limit space provides the canonical decomposition of all the underlying classical random variables (see the Introduction, Lemma 4.5 and Sec. 5 of the present paper for the origin of this problem). This problem is solved in the paperfor CLT associated to states satisfying a generalized Fock property. The states considered in this series have this property.