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.