The q-Canonical Commutation Relation (q-CCR) is an interpolation between the CCR and the CAR with a parameter q. In the 1990s, M. Bożejko and R. Speicher found that the q-CCR is represented on the q-Fock space. The q-Gaussians are realized as the field operators with the vacuum state, which forms a non-commutative distribution. 

On the other hand, a conjugate system is a notion of free probability introduced by D. Voiculescu. This carries important information about a non-commutative distribution of given operators and has many implications for the generated von Neumann algebra.  

In this talk, I will present a concrete formula for the conjugate system for the q-Gaussians. This talk is based on the joint work with R. Speicher.