Brownian bridge in quantum probability

The Classical Brownian Bridge is constructed in Symmetric Fock space over an appropriate base Hilbert space. While the representation of the classical Ito-Wiener integral with respect to the increments of the Brownian bridge implements the unitary isomorphism between the Fock space and the (classical) L² space of the Brownian bridge (as is the case with the standard Brownian motion (SBM)), the quantum Ito-integrals with respect to the associated creation and annihilation bridge processes give different left-and right-integrals. This essentially displays the feature that the Brownian Bridge is not a process of independent increments.