Liberation of Projections

We study the liberation process for projections: $(p,q)\mapsto (p_t,q)= (u_tpu_t^\ast,q)$where$u_t$ is a free unitary Brownian motion freely independent from $\{p,q\}$. Its action on the operator-valued angle $q{p}_{t}q$ between the projections induces a flow on the corresponding spectral measures ${\mu }_t$; we prove that the Cauchy transform of the measure satisfies a holomorphic PDE. We develop a theory of subordination for the boundary values of this PDE, and use it to show that the spectral measure ${\mu }_t$ possesses a piecewise analytic density for any $t>0$ and any initial projections of trace $\frac{1}{2}$. We us this to prove the Unification Conjecture for free entropy and information in this trace $\frac{1}{2}$ setting.