The classical Segal--Bargmann transform (SBT) is an isomorphism between a real Gaussian Hilbert space and a reproducing kernel Hilbert space of holomorphic functions.  It arises in quantum field theory, as a concrete witness of wave-particle duality.  Introduced originally in the 1960s, it has been generalized and extended to many contexts: Lie Groups (Hall, Driver, late 1980s and early 1990s), free probability (Biane, early 2000s), and more recently $q$-Gaussian factors (Cébron, Ho, 2018).

In this talk, I will discuss current work with Charlesworth and Ho on a version of the SBT in bifree probability, a "two faced" version of free probability introduced by Voiculescu in 2014.  Our work leads to some interesting new combinatorial structures ("stargazing partitions"), as well as a detailed analysis of the resultant family of reproducing kernels.  In the end, the bifree SBT has a surprising connection with the $q$-Gaussian version for some $q\ne 0$.