The qq-bit (III): Symmetric q-Jordan-Wigner embeddings

We prove that, replacing the left Jordan-Wigner q-embedding by the symmetric q-embedding described in Sec. 2, the result of the corresponding central limit theorem changes drastically with respect to those obtained in Ref. 5. In fact, in the former case, for any q ∈ ℂ, the limit space is precisely the 1-mode Interacting Fock Space (IFS) that realizes the canonical quantum decomposition of the limit classical random variable. In the latter case, this happens if and only if q = ±1. Furthermore, as shown in Sec. 4, the limit classical random variable turns out to coincide with the 1-mode version of the q Λ -deformed quantum Brownian introduced by Parthasarathy 8,9 and extended to the general context of bi-algebras by Schürman 10,11 . The last section of the paper (Appendix) describes this continuous version in white noise language, leading to a simplification of the original proofs, based on quantum stochastic calculus.